Geometric design and modeling system using control geometry

• US 7,196,702 B1
• Filed: 07/23/1999
• Issued: 03/27/2007
• Est. Priority Date: 07/23/1998
• Status: Expired due to Term

• Current Assignee: Cad-Sense LLC
• Original Assignee: Freedesign, Inc.
• Inventors: Hagen, Lance, Lee, John N., Hagen, Scott, Rockwood, Alyn
• Primary Examiner(s): Nguyen; Kimbinh T.
• Application Number:
  US09/360,029
• Time in Patent Office: 2,804 Days
• Field of Search: 345/419, 345/964, 345/420
• US Class Current: 345/419
• CPC Class Codes: G06T 17/30 Polynomial surface description
• CPC Class Codes: Y10S 715/964 CAD or CAM, e.g. interactive design tools

Abstract

A method and system for computer aided design (CAD) is disclosed for designing geometric objects. The present invention interpolates and/or blends between such geometric objects sufficiently fast so that real time deformation of such objects occurs while deformation data is being input. Thus, a user designing with the present invention obtains immediate feedback to input modifications without separately entering a command for performing such deformations. The present invention utilizes novel computational techniques for blending between geometric objects, wherein weighted sums of points on the geometric objects are used in deriving a new blended geometric object. The present invention is particularly useful for designing the shape of surfaces. Thus, the present invention is applicable to various design domains such as the design of, e.g., bottles, vehicles, and watercraft. Additionally, the present invention provides for efficient animation via repeatedly modifying surfaces of an animated object such as a representation of a face.

Claims

1. A method for generating a geometric object representation for display on a computational system, comprising:

providing a representation of a curve;

obtaining, for each of a plurality of points on the curve, data indicative of a shape of a first surface at the point, wherein said data is invariant between different graphical views of at least one of said curve and said first surface;

determining a representation of said first surface using said data for each of said points, wherein said first surface contains the curve;

generating a representation of a second surface having a contour that changes with a change in a collection of points of said first surface, said points of said collection not being on said curve, and not being on said second surface;

wherein said contour is invariant between different graphical views;

displaying, on a graphical display device, a graphical display of said second surface representation having a modified contour determined using said change in the collection of points.

2. The method as claimed in claim 1, wherein said first surface includes an isocline ribbon for said second surface, and said curve is a profile for said isocline ribbon.

3. The method of claim 1, wherein said first and second surfaces have said curve as a boundary, and said first and second surfaces have identical tangent planes at each point of said curve.

4. The method of claim 1, wherein said step of determining includes providing said second surface as one of:

a developable surface, and a label surface.

5. The method of claim 1, wherein said step of generating includes determining said second surface as a blended surface between said first surface and at least one additional surface, wherein substantially every point on said second surface is a function of a point on said first surface and a point on said additional surface.

6. A method for generating a representation of a geometric object, comprising:

obtaining a first surface wherein a portion thereof is a developable surface;

constructing a closed curve boundary for said developable surface, wherein said boundary identifies an interior of said closed curve on said first surface,

wherein said step of constructing includes generating said boundary as a profile curve having a corresponding isocline ribbon for use in deriving a blended surface having said boundary as a boundary for said blended surface;

trimming said first surface to approximately said boundary so that substantially only said interior of said first surface is graphically displayed;

graphically attaching a label to said interior so that said label substantially covers said interior.

7. A method for generating a representation of a geometric object, comprising:

obtaining a first surface wherein a portion thereof is a developable surface;

constructing a closed curve boundary for said developable surface, wherein said boundary identifies an interior of said closed curve on said first surface,

wherein said step of constructing includes generating said boundary as a profile curve having a corresponding isocline ribbon for use in deriving a blended surface having said boundary as a boundary for said blended surface;

wherein points of said blended surface are derived from a weighted sum of points from said isocline ribbon and at least one other surface;

trimming said first surface to approximately said boundary so that substantially only said interior of said first surface is graphically displayed;

graphically attaching a label to said interior so that said label substantially covers said interior.

8. A method for modifying a representation of a geometric object, comprising:

first displaying a first geometric object representation having a dimension greater than or equal to two;

second displaying one or more additional geometric object representations, wherein one or more of a shape and a position of said additional geometric object representations are indicative of a shape of said first geometric object representation, and wherein one or more of said additional geometric object representations have a dimension greater than or equal to 1;

simultaneously performing the following steps (A1) and (A2);

(A1) changing one of a shape and a location of one or more of said additional geometric object representations in response to input, by a user, of a substantially continuous time series of change requests for changing said additional geometric object representations;

(A2) redisplaying said first geometric object representation with modifications indicative of said changes to said additional geometric object representations;

wherein during an inputting of one of said change requests, said redisplaying step simultaneously performs a substep of graphically displaying, for one or more previously input change requests of said time series, corresponding modifications to said first geometric object representation;

wherein at least one of said first displaying and redisplaying steps includes a step of determining said first geometric object representation as a function of at least some of said additional geometric object representations; and

wherein said determining step includes computing a weighted sum of points, P, for obtaining a point, q, of said first geometric object representation, wherein said points P are obtained using said additional geometric object representations.

9. The method as claimed in claim 8, wherein said weighted sum includes weights wherein each weight is obtained using a corresponding blending function, each said blending function having a range of 0 to 1, and each said blending function being dependent upon a parametric pre-image of q for determining the weight to which the blending function corresponds.

10. A method for modifying a representation of a geometric object, comprising:

providing two or more particular geometric object representations, wherein one or more of a shape and a position of said particular geometric object representations are used in determining a shape of a first geometric object representation relative to a coordinate system, wherein said first geometric object representation has a dimension greater than or equal to two in the coordinate system, and wherein one or more of said particular geometric object representations have a dimension greater than or equal to 1 in the coordinate system;

first displaying, to a user, a first display of said first geometric object;

iteratively performing said steps of changing and second displaying following;

changing one of a shape and a orientation of at least one of said one or more particular geometric object representations thereby obtaining a changed one or more of said particular geometric object representations;

second displaying, to the user, a second display of a geometric object representation, wherein the geometric object representation for said second display is determined using said changed one or more particular geometric object representations, and wherein a shape difference between said first geometric object representation and the geometric object representation for said second display is determined using a value for said one of the orientation and shape changed in said step of changing, the geometric object representation for the second display includes a representation whose display includes a point, P₁, derived using said changed particular geometric object representations, wherein said point P₁ is not representative of a point of said changed particular geometric objects; and

wherein each iteration of said changing step includes inputting, by the user, a next portion of a substantially continuous time series of change requests for changing said particular geometric object representations;

wherein said iteratively performed steps of changing and second displaying are interleaved so that the user perceives a substantially real time deformation of said first geometric object representation during said continuous time series of change requests.

11. The method as claimed in claim 10, wherein said particular geometric object representations have graphical representations that are presented as connected to said first display of said first geometric object representation when said particular geometric object representations are concurrently displayed with said first display on a computer graphics output device.

12. The method as claimed in claim 10, wherein said particular geometric object representations include one or more of user selectable points, vectors, curves and surfaces.

13. A method as claimed in claim 11, wherein at least one of said first displaying and second displaying steps includes a step of determining said first geometric object representation as a blending of at least some of said particular geometric object representations.

14. A method as claimed in claim 11, wherein at least one of said steps of first displaying and second displaying includes computing a weighted sum of points, P, for obtaining a point, q, wherein said points P are obtained using said particular geometric object representations.

15. A method as claimed in claim 14, wherein said weighted sum includes weights wherein each weight is obtained using a corresponding blending function, each said blending function having a predetermined range, and each said blending function being dependent upon a parametric pre-image of q for determining the weight to which the blending function corresponds.

16. A method as claimed in claim 11, wherein said coordinate system is for a three-dimensional coordinate space.

17. The method of claim 10, wherein there is at least one point, P, wherein

(i) said first display of said first geometric object, and

(ii) said second display of the geometric object are different.

18. The method of claim 10, wherein said particular geometric object representations include one of:

a parametric surface and a plurality of points obtained from a parametric surface.

19. The method of claim 10, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

20. The method of claim 10, wherein the coordinate system is for an object space having said first geometric object representation, and said geometric object representation coordinatized therein.

21. The method of claim 10, further including a step of determining three dimensional coordinates of points of said geometric object representation for said second display.

30. The method of claim 20, wherein said one dimensional geometric object includes a vector.

31. The method of claim 30, wherein said vector proceeds away from a point on said reference geometric object.

22. A method for generating a geometric object comprising:

selecting a rendering of a surface on a display;

subsequently, identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a, corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.

23. The method of claim 22, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

24. The method of claim 22, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

25. The method of claim 24, wherein said identified point resides at a location on said reference geometric object.

26. The method of claim 25, wherein said identified point is slidable within an extent for said reference geometric object.

27. The method of claim 24, wherein said step of identifying includes a step of the user selecting said point from the display.

28. The method of claim 22, wherein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects.

29. The method of claim 28, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

32. The method of claim 28, wherein said one dimensional geometric object has a non-zero curvature.

33. The method of claim 22, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects.

34. The method of claim 33, wherein said two dimensional geometric object is parameterized according to two independent parameters.

35. The method of claim 33, wherein said two dimensional geometric object includes a parametric surface.

36. The method of claim 33, wherein said two dimensional geometric object includes an isocline ribbon.

37. The method of claim 22, further including:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

38. The method of claim 37, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

39. The method of claim 37, wherein, the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

41. The method of claim 37, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

42. The method of claim 22, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

43. The method of claim 42, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

44. The method of claim 22, wherein said reference geometric object is one of a curve, and a profile.

45. The method of claim 22, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

46. The method of claim 22, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

47. The method of claim 22, wherein said predetermined function performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

48. The method of claim 22, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

49. The method of claim 22, further including a step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

50. The method of claim 22, further including a step of determining said weightings by evaluating a fiction having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

51. The method of claim 22, further including identifying, by a user, said second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of;

locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set;

third generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape.

52. The method of claim 51, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

53. The method of claim 52, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object;

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

54. The method of claim 53, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

40. The method of 37, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one of more additional geometric objects.

55. A method for generating a geometric object for display, comprising:

means for identifying, by a user, at least one reference geometric object from a display of a graphical display device;

means for obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

means for first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

means for second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein for each point (p) of at least some points of said derived geometric object, p is determined by;

(a) obtaining a first result by applying a first weighting to first data obtained using a point on said one or more additional geometric objects, (b) obtaining a second result by applying a second weighting to second data obtained using at least one point on said second reference geometric object, and (c) combining said first results and said second results in a predetermined way to obtain a representation of p;

wherein said first weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.

56. A method for generating a geometric object for display, comprising:

providing a representation of a curve having an extent along said curve;

obtaining, for each of a plurality of points on the curve extent, corresponding data indicative of a shape of a first surface in at least a neighborhood of the point;

wherein said first surface follows a shape of the curve throughout said extent;

generating a representation of a second surface having a contour that is a function of a collection of points provided on a said first surface, said points of said collection are not on said extent;

determining data representative of a change in location of at least one point of said collection of points, said at least one point not being a point on said second surface;

modifying a shape of said second surface by using said data;

displaying said second surface with said modified shape on a graphical display.

57. An apparatus for generating data for a geometric object, comprising:

a user input capability for entering information related to selecting two or more reference geometric objects displayed on a graphical display device;

at least one processor for performing the following steps;

(a) obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

(b) first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

(c) second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein for each point (p) of at least some points of said derived geometric object, p is (a) obtaining a first result by applying a first weighting to first data obtained using a point on said one or more additional geometric objects, (b) obtaining a second result by applying a second weighting to second data obtained using at least one point on said second reference geometric object, and (c) combining said first results and said second results in a predetermined way to obtain a representation of p;

wherein said first weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

a graphical display device for displaying said derived geometric object on said display of said graphical display device.

58. A medium for storing a program for performing computer aided design, wherein said program for performing the following steps:

storing two or more particular geometric object representations, wherein one or more of a graphical shape and a position of said particular geometric object representations are used in determining a shape of a first geometric object representation relative to a coordinate system, wherein said first geometric object representation has a dimension greater than or equal to two in the coordinate system, and wherein one or more of said particular geometric object representations have a dimension greater than or equal to 1 in the coordinate system;

first displaying, to a user, a first display of said first geometric object;

changing one of a shape and a location of at least one of said one or more particular geometric object representations thereby obtaining a changed one or more of said particular geometric object representations;

second displaying, to the user, a second display of a geometric object representation, wherein the geometric object representation for said second display is determined using said changed one or more particular geometric object representations, and wherein a shape difference between said first geometric object representation and the geometric object representation for said second display is determined using a value for said one of the location and shape changed in said step of changing, the geometric object representation for the second display includes a representation whose display includes a point, P₁, derived using said changed particular geometric object representations, wherein said point P₁ is not representative of a point of said changed particular geometric objects.

59. A medium for storing a program for performing computer aided design, wherein said program for performing the following steps:

identifying, at least one reference geometric object selected by a user from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for use in determining a corresponding orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein for each point (p) of at least some points of said derived geometric object, p is determined by;

(a) obtaining a first result by applying a first weighting to first data obtained using a point on said one or more additional geometric objects, (b) obtaining a second result by applying a second weighting to second data obtained using at least one point on said second reference geometric object, and (c) combining said first results and said second results in a predetermined way to obtain a representation of p;

wherein said first weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.

60. The medium for storing of claim 59, wherein the program thereon further performs the following steps:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

61. A method for generating a geometric object on a computational system, comprising:

providing a representation of a curve;

obtaining, for each of a plurality of points on the curve, data indicative of a shape of a first surface at the point;

determining a representation of said first surface wherein said first surface contains the curve, and wherein said first surface includes an isocline ribbon for said second surface, and said curve is a profile for said isocline ribbon; and

displaying, on a graphical display device, a representation of a second surface having a contour that is a function of points of said first surface.

62. A method for modifying a representation of a geometric object, comprising:

providing two or more particular geometric object representations, wherein one or more of a shape and a position of said particular geometric object representations are used in determining a shape of a first geometric object representation relative to a coordinate system, wherein said first geometric object representation has a dimension greater than or equal to two in the coordinate system, and wherein one or more of said particular geometric object representations have a dimension greater than or equal to 1 in the coordinate system;

first displaying, to a user, a first display of said first geometric object;

changing one of a shape and a orientation of at least one of said one or more particular geometric object representations thereby obtaining a changed one or more of said particular geometric object representations;

second displaying, to the user, a second display of a geometric object representation, wherein the geometric object representation for said second display is determined using said changed one or more particular geometric object representations, and wherein a shape difference between said first geometric object representation and the geometric object representation for said second display is determined using a value for said one of the orientation and shape changed in said step of changing, the geometric object representation for the second display includes a representation whose display includes a point, P₁, derived using said changed particular geometric object representations, wherein said point P₁ is not representative of a point of said changed particular geometric objects

wherein at least one of said steps of first displaying and second displaying includes computing a weighted sum of points, P, for obtaining a point, q, wherein said points P are obtained using said particular geometric object representations.

63. The method as claimed in claim 62, wherein said particular geometric object representations have graphical representations that are presented as connected to said first display of said first geometric object representation when said particular geometric object representations are concurrently displayed with said first display on a computer graphics output device.

64. The method as claimed in claim 62, wherein said particular geometric object representations include one or more of user selectable points, vectors, curves and surfaces.

65. A method as claimed in claim 62, further including iteratively performing said steps of changing and second displaying, wherein each iteration of said changing step includes inputting, by the user, a next portion of a substantially continuous time series of change requests for changing said particular geometric object representations;

wherein said iteratively performed steps of changing and second displaying are interleaved so that the user perceives a substantially real time deformation of said first geometric object representation during said continuous time series of change requests.

66. A method as claimed in claim 62, wherein at least one of said first displaying and second displaying steps includes a step of determining said first geometric object representation as a blending of at least some of said particular geometric object representations.

67. A method as claimed in claim 62, wherein said weighted sum includes weights wherein each weight is obtained using a corresponding blending function, each said blending function having a predetermined range, and each said blending function being dependent upon a parametric pre-image of q for determining the weight to which the blending function corresponds.

68. A method as claimed in claim 62, wherein said coordinate system is for a three-dimensional coordinate space.

69. The method of claim 62, wherein there is at least one point, P, wherein

(i) said first display of said first geometric object, and

(ii) said second display of the geometric object are different.

70. The method of claim 62, wherein said particular geometric object representations include one of:

a parametric surface and a plurality of points obtained from a parametric surface.

71. The method of claim 62, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

72. The method of claim 62, wherein the coordinate system is for an object space having said first geometric object representation, and said geometric object representation coordinatized therein.

73. The method of claim 62, further including a step of determining three dimensional coordinates of points of said geometric object representation for said second display.

74. A method for modifying a representation of a geometric object, comprising:

providing two or more particular geometric object representations, wherein one or more of a shape and a position of said particular geometric object representations are used in determining a shape of a first geometric object representation relative to a coordinate system, wherein said first geometric object representation has a dimension greater than or equal to two in the coordinate system, and wherein one or more of said particular geometric object representations have a dimension greater than or equal to 1 in the coordinate system;

first displaying, to a user, a first display of said first geometric object;

changing one of a shape and a orientation of at least one of said one or more particular geometric object representations thereby obtaining a changed one or more of said particular geometric object representations;

second displaying, to the user, a second display of a geometric object representation, wherein the geometric object representation for said second display is determined using said changed one or more particular geometric object representations, and wherein a shape difference between said first geometric object representation and the geometric object representation for said second display is determined using a value for said one of the orientation and shape changed in said step of changing, the geometric object representation for the second display includes a representation whose display includes a point, P₁, derived using said changed particular geometric object representations, wherein said point P₁ is not representative of a point of said changed particular geometric objects

wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

75. The method as claimed in claim 74, wherein said particular geometric object representations have graphical representations that are presented as connected to said first display of said first geometric object representation when said particular geometric object representations are concurrently displayed with said first display on a computer graphics output device.

76. The method as claimed in claim 74, wherein said particular geometric object representations include one or more of user selectable points, vectors, curves and surfaces.

77. A method as claimed in claim 74, further including iteratively performing said steps of changing and second displaying, wherein each iteration of said changing step includes inputting, by the user, a next portion of a substantially continuous time series of change requests for changing said particular geometric object representations;

wherein said iteratively performed steps of changing and second displaying are interleaved so that the user perceives a substantially real time deformation of said first geometric object representation during said continuous time series of change requests.

78. A method as claimed in claim 74, wherein at least one of said first displaying and second displaying steps includes a step of determining said first geometric object representation as a blending of at least some of said particular geometric object representations.

79. A method as claimed in claim 74, wherein at least one of said steps of first displaying and second displaying includes computing a weighted sum of points, P, for obtaining a point, q, wherein said points P are obtained using said particular geometric object representations.

80. A method as claimed in claim 79, wherein said weighted sum includes weights wherein each weight is obtained using a corresponding blending function, each said blending function having a predetermined range, and each said blending function being dependent upon a parametric pre-image of q for determining the weight to which the blending function corresponds.

81. A method as claimed in claim 74, wherein said coordinate system is for a three-dimensional coordinate space.

82. The method of claim 74, wherein there is at least one point, P, wherein

(i) said first display of said first geometric object, and

(ii) said second display of the geometric object are different.

83. The method of claim 74, wherein said particular geometric object representations include one of:

a parametric surface and a plurality of points obtained from a parametric surface.

84. The method of claim 74, wherein the coordinate system is for an object space having said first geometric object representation, and said geometric object representation coordinatized therein.

85. The method of claim 74, further including a step of determining three dimensional coordinates of points of said geometric object representation for said second display.

86. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device;

wherein one or more of (i) through (iii) following hold;

(i) said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects, said two dimensional geometric object includes an isocline ribbon;

(ii) said initial collection of geometric objects includes one of;

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

(iii) said additional geometric objects includes one of;

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

87. The method of claim 86, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

88. The method of claim 86, further including selecting a rendering of a surface on the display prior to said step of identifying.

89. The method of claim 86, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

90. The method of claim 89, wherein said identified point resides at a location on said reference geometric object.

91. The method of claim 90, wherein said identified point is slidable within an extent for said reference geometric object.

92. The method of claim 89, wherein said step of identifying includes a step of the user selecting said point from the display.

93. The method of claim 86, herein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects.

94. The method of claim 93, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

95. The method of claim 93, wherein said one dimensional geometric object includes a vector.

96. The method of claim 95, wherein said vector proceeds away from a point on said reference geometric object.

97. The method of claim 93, wherein said one dimensional geometric object has a non-zero curvature.

98. The method of claim 86, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects, said two dimensional geometric object includes an isocline ribbon.

99. The method of claim 86, wherein said two dimensional geometric object is parameterized according to two independent parameters.

100. The method of claim 86, wherein said two dimensional geometric object includes a parametric surface.

101. The method of claim 86, her including:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

102. The method of claim 101, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

103. The method of claim 101, wherein, the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

104. The method of claim 101, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one, of more additional geometric objects.

105. The method of claim 101, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

106. The method of claim 86, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

107. The method of claim 106, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

108. The method of claim 86, wherein said reference geometric object is one of a curve, and a profile.

109. The method of claim 86, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

110. The method of claim 86, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

111. The method of claim 86, wherein said predetermined fiction performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

112. The method of claim 86, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

113. The method of claim 86, further including a step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

114. The method of claim 86, further including a step of determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

115. The method of claim 86, further including

identifying, by a user, said second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of;

locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set;

third generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape.

116. The method of claim 115, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

117. The method of claim 116, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object; and

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

118. The method of claim 117, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

119. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric-object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing;

wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

120. The method of claim 119, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

121. The method of claim 119, further including selecting a rendering of a surface on the display prior to said step of identifying.

122. The method of claim 119, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

123. The method of claim 122, wherein said identified point resides at a location on said reference geometric object.

124. The method of claim 123, wherein said identified point is slidable within an extent for said reference geometric object.

125. The method of claim 122, wherein said step of identifying includes a step of the user selecting said point from the display.

126. The method of claim 119, wherein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects.

127. The method of claim 126, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

128. The method of claim 126, wherein said one dimensional geometric object includes a vector.

129. The method of claim 128, wherein said vector proceeds away from a point on said reference geometric object.

130. The method of claim 126, wherein said one dimensional geometric object has a non-zero curvature.

131. The method of claim 119, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects.

132. The method of claim 131, wherein said two dimensional geometric object is parameterized according to two independent parameters.

133. The method of claim 131, wherein said two dimensional geometric object includes a parametric surface.

134. The method of claim 131, wherein said two dimensional geometric object includes an isocline ribbon.

135. The method of claim 119, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

136. The method of claim 119, wherein, the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

138. The method of claim 119, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

139. The method of claim 138, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

140. The method of claim 119, wherein said reference geometric object is one of a curve, and a profile.

141. The method of claim 119, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

142. The method of claim 119, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

143. The method of claim 119, wherein said predetermined function performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

144. The method of claim 119, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

145. The method of claim 119, further including a step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

146. The method of claim 119, further including a step of determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

147. The method of claim 119, further including

identifying, by a user, said second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of;

locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set;

third generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape.

148. The method of claim 147, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

149. The method of claim 148, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object; and

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

150. The method of claim 149, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

137. The method of 119, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one of more additional geometric objects.

151. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object;

wherein one or more of (i) and (ii) following hold;

(i) at least one of said steps of first and second generating includes a step of performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting; and

(ii) said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object; and

displaying said derived geometric object on said display of said graphical display device.

152. The method of claim 151, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

153. The method of claim 151, further including selecting a rendering of a surface on the display prior to said step of identifying.

154. The method of claim 151, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

155. The method of claim 154, wherein said identified point resides at a location on said reference geometric object.

156. The method of claim 155, wherein said identified point is slidable within an extent for said reference geometric object.

157. The method of claim 154, wherein said step of identifying includes a step of the user selecting said point from the display.

158. The method of claim 151, wherein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects.

159. The method of claim 158, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

160. The method of claim 158, wherein said one dimensional geometric object includes a vector.

161. The method of claim 160, wherein said vector proceeds away from a point on said reference geometric object.

162. The method of claim 158, wherein said one dimensional geometric object has a non-zero curvature.

163. The method of claim 151, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects.

164. The method of claim 163, wherein said two dimensional geometric object is parameterized according to two independent parameters.

165. The method of claim 163, wherein said two dimensional geometric object includes a parametric surface.

166. The method of claim 163, wherein said two dimensional geometric object includes an isocline ribbon.

167. The method of claim 151, further including:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

168. The method of claim 167, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

169. The method of claim 167, wherein the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

171. The method of claim 167, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

172. The method of claim 151, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

173. The method of claim 151, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

174. The method of claim 151, wherein said reference geometric object is one of a curve, and a profile.

175. The method of claim 151, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

176. The method of claim 151, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

177. The method of claim 151, wherein said predetermined function performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

178. The method of claim 151, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

179. The method of claim 151, further including a step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

180. The method of claim 151, further including a step of determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

181. The method of claim 151, further including

identifying, by a user, said second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of;

locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set;

third generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape.

182. The method of claim 181, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

183. The method of claim 182, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object; and

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

184. The method of claim 183, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

170. The method of 167, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one of more additional geometric objects.

185. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined fiction are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object;

wherein one or more of steps (i) and (ii) following are performed;

(i) determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space;

(ii) determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.

186. The method of claim 185, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

187. The method of claim 185, further including selecting a rendering of a surface on the display prior to said step of identifying.

188. The method of claim 185, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

189. The method of claim 188, wherein said identified point resides at a location on said reference geometric object.

190. The method of claim 189, herein said identified point is slidable within an extent for said reference geometric object.

191. The method of claim 188, wherein said step of identifying includes a step of the user selecting said point from the display.

192. The method of claim 185, wherein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects.

193. The method of claim 192, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

194. The method of claim 192, wherein said one dimensional geometric object includes a vector.

195. The method of claim 194, wherein said vector proceeds away from a point on said reference geometric object.

196. The method of claim 192, wherein said one dimensional geometric object has a non-zero curvature.

197. The method of claim 185, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects.

198. The method of claim 197, wherein said two dimensional geometric object is parametenized according to two independent parameters.

199. The method of claim 197, wherein said two dimensional geometric object includes a parametric surface.

200. The method of claim 197, wherein said two dimensional geometric object includes an isocline ribbon.

201. The method of claim 185, further including:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

202. The method of claim 201, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

203. The method of claim 201, wherein, the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

205. The method of claim 201, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

206. The method of claim 185, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

207. The method of claim 206, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

208. The method of claim 185, wherein said reference geometric object is one of a curve, and a profile.

209. The method of claim 185, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

210. The method of claim 185, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

211. The method of claim 185, wherein said predetermined function performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

212. The method of claim 185, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

213. The method of claim 185, further including said step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

214. The method of claim 185, further including said step of determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

215. The method of claim 185, further including

identifying, by a user, said second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of;

locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set; and

third generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape.

216. The method of claim 215, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

217. The method of claim 216, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object; and

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

218. The method of claim 217, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

204. The method of 201, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one of more additional geometric objects.

219. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

identifying, by a user, a second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set;

second generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape;

third generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least said second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.

220. The method of claim 219, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

221. The method of claim 219, further including selecting a rendering of a surface on the display prior to said step of identifying.

222. The method of claim 219, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

223. The method of claim 222, wherein said identified point resides at a location on said reference geometric object.

224. The method of claim 223, wherein said identified point is slidable within an extent for said reference geometric object.

225. The method of claim 222, wherein said step of identifying includes a step of the user selecting said point from the display.

226. The method of claim 219, wherein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects.

227. The method of claim 226, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

228. The method of claim 226, wherein said one dimensional geometric object includes a vector.

229. The method of claim 228, wherein said vector proceeds away from a point on said reference geometric object.

230. The method of claim 226, wherein said one dimensional geometric object has a non-zero curvature.

231. The method of claim 219, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects.

232. The method of claim 231, wherein said two dimensional geometric object is parameterized according to two independent parameters.

233. The method of claim 231, wherein said two dimensional geometric object includes a parametric surface.

234. The method of claim 231, wherein said two dimensional geometric object includes an isocline ribbon.

235. The method of claim 219, further including:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

236. The method of claim 235, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

237. The method of claim 235, wherein the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

239. The method of claim 235, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

240. The method of claim 219, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

241. The method of claim 240, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

242. The method of claim 219, wherein said reference geometric object is one of a curve, and a profile.

243. The method of claim 219, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

244. The method of claim 219, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

245. The method of claim 219, wherein said predetermined function performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

246. The method of claim 219, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

247. The method of claim 219, further including a step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

248. The method of claim 219, further including a step of determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

249. The method of claim 219, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

250. The method of claim 249, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object;

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

251. The method of claim 250, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

238. The method of 235, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one of more additional geometric objects.

252. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

wherein said step of obtaining includes identifying a one dimensional geometric object as a geometric object of said initial collection, wherein a parameterization of said one dimensional geometric object is used in determining said additional geometric objects;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by applying weightings to results from a predetermined function of points on said one or more additional geometric objects;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.

253. The method of claim 252, wherein said identifying step includes graphically selecting a curve as said reference geometric object.

254. The method of claim 252, further including selecting a rendering of a surface on the display prior to said step of identifying.

255. The method of claim 252, wherein said step of obtaining includes identifying a point as a geometric object of said initial collection, said point including coordinate values for a coordinate system which can be displayed from different perspectives on the display.

256. The method of claim 255, wherein said identified point resides at a location on said reference geometric object.

257. The method of claim 256, wherein said identified point is slidable within an extent for said reference geometric object.

258. The method of claim 255, wherein said step of identifying includes a step of the user selecting said point from the display.

259. The method of claim 252, wherein said one dimensional geometric object is parameterized according to a single independent parameter.

260. The method of claim 252, wherein said one dimensional geometric object includes a vector.

261. The method of claim 260, wherein said vector proceeds away from a point on said reference geometric object.

262. The method of claim 252, wherein said one dimensional geometric object has a non-zero curvature.

263. The method of claim 252, wherein said step of first generating includes determining a two dimensional geometric object as one of said one or more additional geometric objects.

264. The method of claim 263, wherein said two dimensional geometric object is parameterized according to two independent parameters.

265. The method of claim 263, wherein said two dimensional geometric object includes a parametric surface.

266. The method of claim 263, wherein said two dimensional geometric object includes an isocline ribbon.

267. The method of claim 252, further including:

changing one of a shape and an orientation of at least one of said one or more additional geometric objects thereby obtaining a changed one or more of said additional geometric objects;

second displaying, to the user, a second display of a geometric object, wherein the geometric object for said second display is determined using said changed one or more additional geometric objects, and wherein a shape difference between said derived geometric object and the geometric object for said second display is determined using a value for said one of the location and shape changed in said step of changing.

268. The method of claim 267, wherein said geometric object for said second display includes a modified version of said derived geometric object, wherein said shape difference uses each change in the shape and location of the at least one of said one of more additional geometric objects.

269. The method of claim 267, wherein, the geometric object for the second display includes a point, P₁, derived using said changed additional geometric objects, wherein said point P₁ is not representative of a point of said changed additional geometric objects.

271. The method of claim 267, wherein there is no manual entry after said step of changing resulting in a performance of said step of second displaying.

272. The method of claim 252, wherein said geometric information is used to determine the position, orientation, and shape of each of said one or more additional geometric objects.

273. The method of claim 272, wherein at least one of said steps of first and second generating includes:

performing an interpolation for determining a first plurality of points from a plurality of said additional geometric objects, wherein for each of point of said first plurality of points there is a corresponding one of said weightings, wherein said step of second generating includes multiplying each point of said first plurality of points by its corresponding weighting.

274. The method of claim 252, wherein said reference geometric object is one of a curve, and a profile.

275. The method of claim 252, wherein said initial collection of geometric objects includes one of:

(a) a marker;

(b) a profile handle;

(c) an isocline handle; and

(d) an isocline rib.

276. The method of claim 252, wherein said additional geometric objects includes one of:

(a) a marker;

(b) an isocline ribbon;

(c) a profile;

(d) an isocline handle;

(e) a ribbon segment;

(f) an isocline boundary; and

(g) a boundary segment.

277. The method of claim 252, wherein said predetermined function performs one or more of:

(a) a step of determining said points on said additional geometric objects according to a predetermined collection of points in a parameter space from which said additional geometric objects are parameterized;

(b) a step of providing, as said results, object space coordinates of said points on said additional geometric objects; and

(c) a step of interpolating between said additional geometric objects for obtaining a plurality of points for results.

278. The method of claim 252, wherein said predetermined function performs a step of obtaining coordinates for at least one point, P, derived using said additional geometric objects, and wherein said point P is not representative of a point of said additional geometric objects, said point P is not representative of a point of said derived geometric object.

279. The method of claim 252, further including a step of determining said weightings by evaluating at least one blending function at a plurality of points in a parametric space wherein at least one of said additional geometric objects are parameterized according to points in the parametric space.

280. The method of claim 252, further including a step of determining said weightings by evaluating a function having parametric inputs corresponding to a parameterization of said derived geometric object, wherein said weightings are substantially a value of one when said parametric inputs correspond to points of said reference geometric object, and said weightings are substantially a value of zero when said parametric inputs correspond to points of said second reference geometric object.

281. The method of claim 280, wherein said step of second generating includes:

second determining of said derived geometric object by applying weightings to results from a second predetermined function of points on said one or more additional geometric objects of said second set;

wherein said weightings are such that they are applied so that when said results of the second predetermined function are from inputs indicative of said points of said one or more additional geometric objects of said second set wherein said inputs become progressively closer to said second reference geometric object, then points of said derived geometric object becomes progressively closer to said second reference geometric object.

282. The method of claim 281, wherein one of B₁ and B₂ includes one of a quintic polynomial a Bezier curve, and a trigonometric function.

283. The method of claim 252, further including

identifying, by a user, said second reference geometric object;

obtaining a second initial collection of geometric objects using one or more of;

locations represented on said second reference geometric object, and directional information indicative of one or more directions from said second reference geometric object;

wherein said geometric objects of said second initial collection provide geometric information for use in determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects of a second set;

third generating said one or more additional geometric objects of said second set using said geometric objects from said second initial collection, each said additional geometric object of said second set having its corresponding at least one of position, orientation and shape.

284. The method of claim 283, wherein denoting a function for obtaining said weightings for applying to results from the predetermined function as, B₁, and denoting a function for obtaining said weightings for applying to results from the second predetermined function as, B₂, then one or more of the following conditions hold:

(a) B₁ and B₂ are a partition of unity;

(b) a rate of change of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object;

(c) a second derivative of one or more of B₁ and B₂ reduces to substantially zero when said weightings for said one or more of B₁ and B₂ correspond to points that become progressively closer to one of;

said reference geometric object and said second reference geometric object.

270. The method of 267, wherein said step of changing includes receiving input from the user, said input initiating said change in the shape and location of the at least one of said one of more additional geometric objects.

285. A method for modifying a representation of a geometric object, comprising:

providing two or more particular geometric object representations, wherein one or more of a shape and a position of said particular geometric object representations are used in determining a shape of a first geometric object representation relative to a coordinate system, wherein said first geometric object representation has a dimension greater than or equal to two in the coordinate system such that said first geometric object representation has at least two parameters u and v whose values can be varied independently of one another for determining points of said first geometric object, wherein there is a plurality of curves defined by said first geometric object representation wherein;

(i) each of said curves includes a point, P₀, of said first geometric object, and (ii) for each of said curves, C, there is no segment thereof having P₀, with the segment being identical with a segment of another of said curves; and

(iii) at least a first of said curves is determined by varying only the u parameter as input to said first geometric object representation, and a second of said curves is determined by varying only the v parameter as input to said first geometric object representation;

wherein one or more of said particular geometric object representations have a dimension greater than or equal to 1 in the coordinate system;

first displaying, to a user, a first display of said first geometric object;

changing one of a shape and a orientation of at least one of said one or more particular geometric object representations thereby obtaining a changed one or more of said particular geometric object representations;

second displaying, to the user, a second display of a geometric object representation, wherein the geometric object representation for said second display is determined using said changed one or more particular geometric object representations, and wherein a shape difference between said first geometric object representation and the geometric object representation for said second display is determined using a value for said one of the orientation and shape changed in said step of changing, the geometric object representation for the second display includes a representation whose display includes a point, P₁, derived using said changed particular geometric object representations, wherein said point P₁ is not representative of a point of said changed particular geometric objects.

286. The method of claim 285, wherein for at least one geometric object (G) of said additional geometric objects, there is a parameterization for identifying at least some points of G, wherein said parameterization is obtained using a predetermined correspondence between (i) and (ii) following:

(i) a first geometric space, and (ii) a second geometric space having said derived geometric object;

wherein said parameterization is invariant between different graphical views of G, and wherein for each point (P) of said some points of G, there is at least one point (Q) of are Ax said first geometric space such that;

(a) P is determined by said parameterization applied to Q, and (b) there is at least one of said weightings applied to P that is determined using Q.

287. A method for generating a geometric object comprising:

identifying, by a user, at least one reference geometric object from a display of a graphical display device;

obtaining an initial collection of geometric objects using one or more of;

locations represented on said reference geometric object, and directional information indicative of one or more directions from said reference geometric object;

wherein said geometric objects of said initial collection provide geometric information for determining a corresponding at least one of a position, orientation, and shape for each of one or more additional geometric objects;

first generating said one or more additional geometric objects using said geometric objects from said initial collection, each said additional geometric object having its corresponding at least one of position, orientation and shape;

second generating a derived geometric object from said one or more additional geometric objects, wherein said derived geometric object spans a space between said reference geometric object and at least a second reference geometric object, wherein a first portion of said derived geometric object follows a shape of a predetermined extent of said reference geometric object and a second portion of said derived geometric object follows a shape of a predetermined extent of said second reference geometric object, and wherein points of said derived geometric object are determined by a step of applying weightings to values dependent upon points on said one or more additional geometric objects, wherein for applying at least two of said weightings, there is no user input therebetween;

wherein said weightings are such that they are applied so that when said results of the predetermined function are from inputs of said points of said one or more additional geometric objects wherein said inputs become progressively closer to said reference geometric object, then points of said derived geometric object becomes progressively closer to said reference geometric object; and

displaying said derived geometric object on said display of said graphical display device.