Moving an object in a three-dimensional workspace

• US 5,513,303 A
• Filed: 08/05/1994
• Issued: 04/30/1996
• Est. Priority Date: 08/02/1990
• Status: Expired due to Term

• Current Assignee: Xerox Corporation (Xerox Holdings Corp.)
• Original Assignee: Xerox Corporation (Xerox Holdings Corp.)
• Inventors: Card, Stuart K., Robertson, George G., Mackinlay, Jock
• Primary Examiner(s): Bayerl, Raymond J.
• Application Number:
  US08/286,764
• Time in Patent Office: 634 Days
• Field of Search: 395/119, 395/127, 395/161, 395/138, 395/152, 395/155, 395/136, 345/139, 345/122, 345/121
• US Class Current: 345/419
• CPC Class Codes: G06F 3/04815 Interaction with a metaphor-based environment or interaction object displayed as three-dimensional, e.g. changing the user viewpoint with respect to the environment or object
• CPC Class Codes: G06F 3/04845 for image manipulation, e.g. dragging, rotation, expansion or change of colour
• CPC Class Codes: G06T 13/00 Animation
• CPC Class Codes: Y10S 345/953 Geometric processing
• CPC Class Codes: Y10S 345/959 Object path adherence
• CPC Class Codes: Y10S 345/96 Iterative display of preconfigured images

Abstract

In response to user signals requesting motion of a displayed object, a system presents a sequence of images, each including an object perceptible as a moved continuation of the previously displayed object. The user can independently request radial motion and lateral motion, and the system presents a sequence of images in which the object is displaced radially by the requested radial motion and laterally by the requested lateral motion. The user can request lateral motion by operating a mouse and can request radial motion by operating keys on a keyboard, with one key requesting motion toward a radial source and another key requesting motion away from the radial source. The radial source can be the viewpoint. The object'"'"'s motion toward the viewpoint includes two phases. In the first phase, the object follows an acceleration path, enabling the user to control motion near its starting point and providing increasingly rapid motion; in the second phase, it follows an asymptotic path, enabling the user to control its motion as it approaches the viewpoint and preventing it from passing the viewpoint. The displacements between positions on the asymptotic path can follow a logarithmic function, with each displacement a proportion of the distance from the previous position to the viewpoint. The phases can be produced by using the logarithmic function to clip an acceleration function. The same rate of acceleration can be applied when the user requests motion away from the viewpoint. The processor can perform an animation loop, each step of which receives user signals and presents another image.

Claims

1. A method of operating a system that includes a display, a user input device set for providing signals, and a processor connected for receiving signals from the user input device set and for presenting images on the display;

the user input device set including a first device through which a user can provide radial motion signals and a second device through which a user can provide lateral motion signals;

the radial motion signals requesting radial motion along rays and the lateral motion signals requesting lateral motion in directions lateral to the rays;

the first and second devices being structured so that a user viewing an image presented on the display that includes an object in a three-dimensional workspace can request presentation of a sequence of images in which the object moves along a path within the three-dimensional workspace by operating the first and second devices to request concurrent radial motion and lateral motion independently;

the radial motion being along rays extending from radial sources, with the user being able to request radial motion along any of a set of rays that extend in different directions from each radial source;

the requested radial motion and lateral motion in combination being equivalent to motion along the path;

the method comprising;

presenting a first image on the display;

the first image including a first object that is perceptible as having a first position within a three-dimensional workspace;

receiving a first radial motion signal from the first device and a first lateral motion signal from the second device;

the first radial motion signal and the first lateral motion signal requesting a first radial motion of the first object and a first lateral motion of the first object;

the first radial motion signal and the first lateral motion signal requesting that the first radial motion and the first lateral motion be concurrent;

the first lateral motion signal indicating one of a set of rays that extends from a radial source in a direction that is not perpendicular to the display;

the first radial motion and the first lateral motion in combination being equivalent to motion along a first path within the three-dimensional workspace; and

in response to the first radial motion signal and the first lateral motion signal, presenting a second image on the display;

the second image including a second object that is perceptible as a moved continuation of the first object at a second position within the three-dimensional workspace;

the second position being perceptible as concurrently displaced along the first path from the first position both radially toward or away from the radial source along the indicated one of the set of rays and laterally in a direction lateral to the indicated one of the set of rays.

2. The method of claim 1 in which the first and second objects are perceptible as being viewed from a viewpoint, the viewpoint being the radial source;

the second position being perceptible as displaced from the first position radially along the indicated one of the set of rays toward or away from the viewpoint.

3. The method of claim 2 in which the first radial motion is toward the viewpoint.

4. The method of claim 3 in which the first radial motion follows an acceleration path.

5. The method of claim 2 in which the first radial motion is away from the viewpoint.

6. The method of claim 5 in which the first radial motion follows an acceleration path.

7. The method of claim 1 in which the first lateral motion is in a plane.

8. A system comprising:

a display;

a user input device set for providing signals; and

a processor connected for receiving signals from the user input device set and for presenting images on the display;

the user input device set including a first device through which a user can provide radial motion signals and a second device through which a user can provide lateral motion signals;

the radial motion signals requesting radial motion along rays and the lateral motion signals requesting lateral motion in directions lateral to the rays;

the first and second devices being structured so that a user viewing an image presented on the display that includes an object in a three-dimensional workspace can request presentation of a sequence of images in which the object moves along a path within the three-dimensional workspace by operating the first and second devices to request concurrent radial motion and lateral motion independently;

the radial motion being along rays extending from radial sources, with the user being able to request radial motion along any of a set of rays that extend in different directions from each radial source;

the requested radial motion and lateral motion in combination being equivalent to motion along the path;

the processor being operable to present a first image on the display;

the first image including a first object that is perceptible as having a first position within a three-dimensional workspace;

the processor further being operable to receive a first radial motion signal from the first device and a first lateral motion signal from the second device;

the first radial motion signal requesting a first radial motion of the first object and the first lateral motion signal requesting a first lateral motion of the first object;

the first radial motion signal and the first lateral motion signal requesting that the first radial motion and the first lateral motion be concurrent;

the first lateral motion signal indicating one of a set of rays that extends from a radial source in a direction that is not perpendicular to the display;

the first radial motion and the first lateral motion in combination being equivalent to motion along a first path within the three-dimensional workspace;

the processor responding to the first radial motion signal and the first lateral motion signal by presenting a second image on the display;

the second image including a second object that is perceptible as a moved continuation of the first object at a second position within the three-dimensional workspace;

the second position being perceptible as concurrently displaced along the first path from the first position both radially toward or away from the radial source along the indicated one of the set of rays and laterally in a direction lateral to the indicated one of the set of rays.

9. The system of claim 8 in which the the user input device set comprises a mouse, the user requesting lateral motion by operating the mouse, the first lateral motion signal including data indicating mouse operation.

10. The system of claim 8 in which the user input device set comprises a keyboard, the user requesting radial motion by operating the keyboard, the first radial motion signal including data indicating keyboard operation.

11. The system of claim 10 in which the user requests radial motion toward or away from a radial source, the keyboard having a first key and a second key, the user operating the first key to request motion toward the radial source, the user operating the second key to request motion away from the radial source.

12. The system of claim 8 in which the first lateral motion signal indicates two-dimensional coordinates, the processor producing coordinates of the second position within the three-dimensional workspace by operating on the two-dimensional coordinates.

13. The system of claim 8 in which the first device comprises a keyboard.

14. The system of claim 8 in which the second device comprises a mouse.

15. A method of operating a system that includes a display, a user input device, and a processor connected for receiving signals from the user input device and for presenting images on the display;

the user input device providing motion requesting signals from a user viewing images presented on the display that include objects in a three-dimensional workspace, the motion requesting signals requesting presentation of a sequence of images in which objects move along paths within the three-dimensional workspace;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, each following iteration having a next preceding iteration;

the first iteration comprising;

presenting a respective image on the display;

the respective image of the first iteration including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal requesting motion of the respective object of the next preceding iteration toward a destination position within the three-dimensional workspace;

the motion requesting signal requesting concurrent radial and lateral motion from the next preceding iteration'"'"'s respective position toward the destination position;

the requested radial motion and the requested lateral motion in combination being equivalent to motion along a path toward the destination position within the three-dimensional workspace; and

presenting a respective image on the display;

each respective image including a respective object that is perceptible as having a respective position within the three-dimensional workspace;

the respective object of each following iteration being perceptible as a continuation of the respective object of the next preceding iteration, concurrently displaced both radially and laterally from the next preceding iteration'"'"'s respective position;

the respective position of each following iteration being displaced both radially and laterally along the path by a respective displacement from the respective position of the next preceding iteration;

the act of presenting a respective image comprising;

operating the processor to automatically perform an asymptotic function to obtain the respective displacement;

the following iterations together obtaining a sequence of respective displacements so that the respective positions of the following iterations approach the destination position asymptotically along the path.

16. The method of claim 15 in which the respective objects are perceptible as being viewed from a viewpoint, the destination position being the viewpoint, the respective positions approaching the viewpoint asymptotically along the path.

17. The method of claim 15 in which each following iteration further comprises:

performing an animation loop, the acts of receiving the respective motion requesting signal and of presenting the respective image being included in the animation loop.

18. A method of operating a system that includes a display, a user input device, and a processor connected for receiving signals from the user input device and for presenting images on the display;

the user input device providing motion requesting signals from a user viewing images presented on the display that include objects in a three-dimensional workspace, the motion requesting signals requesting presentation of a sequence of images in which objects move along paths within the three-dimensional workspace;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, each following iteration having a next preceding iteration;

the first iteration comprising;

presenting a respective image on the display, the respective image of the first iteration including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal requesting motion of the respective object of the next preceding iteration;

the motion requesting signal requesting concurrent radial and lateral motion;

the requested radial motion and the requested lateral motion in combination being equivalent to motion along a path within the three-dimensional workspace; and

presenting a respective image on the display;

each respective image including a respective object that is perceptible as having a respective position within the three-dimensional workspace;

the respective object of each of the following iterations being perceptible as a continuation of the respective object of the next preceding iteration, concurrently displaced along the path both radially and laterally from the next preceding iteration'"'"'s respective position;

the respective motion requesting signal of each of a first set of the following iterations requesting motion of the respective object of the next preceding iteration toward a destination position;

the first set of following iterations including first phase iterations and second phase iterations;

the first phase iterations preceding the second phase iterations;

the respective position of each of the first set of following iterations being displaced both radially and laterally by a respective displacement from the respective position of the next preceding iteration;

in each of the second phase iterations, the act of presenting a respective image comprising;

operating the processor to automatically perform an asymptotic function to obtain the respective displacement;

the following iterations together obtaining a sequence of respective displacements so that the respective positions of the first phase iterations accelerate toward the destination position along the path and so that the respective positions of the second phase iterations approach the destination position asymptotically along the path.

19. The method of claim 18 in which the respective objects are perceptible as being viewed from a viewpoint, the destination position being the viewpoint, the respective positions approaching the viewpoint asymptotically along the path.

20. The method of claim 18 in which each following iteration comprises:

performing an animation loop, the acts of receiving the respective motion requesting signal and of presenting the respective image being included in the animation loop.

21. The method of claim 18 in which the respective motion requesting signal of each of a second set of the following iterations requests motion of the respective object of the next preceding iteration away from the destination position;

the respective positions of the second set of following iterations accelerating away from the destination position along the path.

22. The method of claim 21 in which the respective positions of the first phase iterations have a first rate of acceleration and the respective positions of the second set of following iterations have a second rate of acceleration;

the first and second rates of acceleration having the same magnitude.

23. A method of operating a system that includes a display, a user input device, and a processor connected for receiving signals from the user input device and for presenting images on the display;

the user input device providing motion requesting signals from a user viewing images presented on the display that include objects in a three-dimensional workspace, the motion requesting signals requesting presentation of a sequence of images in which objects move along paths within the three-dimensional workspace;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, each following iteration having a next preceding iteration;

the first iteration comprising;

presenting a respective image on the display, the respective image of the first iteration including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal requesting motion of the respective object of the next preceding iteration;

the motion requesting signal requesting concurrent radial and lateral motion;

the requested radial motion and the requested lateral motion in combination being equivalent to motion along a path toward the destination position within the three-dimensional workspace; and

presenting a respective image on the display;

each respective image including a respective object that is, perceptible as having a respective position within the three-dimensional workspace;

the respective object of each of the following iterations being perceptible as a continuation of the respective object of the next preceding iteration, concurrently displaced along the path both radially and laterally from the next preceding iteration'"'"'s respective position;

the respective motion requesting signal of each of a first set of the following iterations requesting motion of the respective object of the next preceding iteration toward a destination position;

the respective position of each of the first set of following iterations being displaced both radially and laterally from the respective position of the next preceding iteration by a respective displacement;

each of the following iterations in the first set further comprising;

determining the respective displacement by operating the processor to automatically perform a clipping operation on first and second functions, the first function being an acceleration function defining a first displacement to a first position that is on the path toward the destination position in the workspace;

the second function being an asymptotic function defining a second displacement to a second position that is on the path toward the destination position in the three-dimensional workspace;

the clipping operation choosing one of the first displacement and the second displacement;

the respective object of each following iteration being perceptible as accelerating along the path when the clipping operation chooses the first displacement and as approaching the destination position asymptotically along the path when the clipping operation chooses the second displacement.

24. The method of claim 23 in which the respective objects are perceptible as being viewed from a viewpoint, the destination position being the viewpoint.

25. The method of claim 23 in which each following iteration comprises:

performing an animation loop, the acts of receiving the respective motion requesting signal and of presenting the respective image being included in the animation loop.

26. The method of claim 23 in which the respective motion requesting signal of each of a second set of the following iterations requests motion of the respective object of the next preceding iteration away from the destination position;

the respective position of each of the second set of following iterations being displaced from the respective position of the next preceding iteration by a respective displacement;

each of the following iterations in the second set further comprising;

determining the respective displacement using a third function, the third function being an acceleration function defining the respective displacement to the respective position so that the respective position is perceptible as accelerating away from the destination position.

27. The method of claim 26 in which the first function has a first rate of acceleration and the third function has a second rate of acceleration;

the first and second rates of acceleration having the same magnitude.

28. A method of operating a system that includes a display, a user input device set for providing signals, and a processor connected for receiving signals from the user input device set and for presenting images on the display;

the use input device set providing motion requesting signals requesting motion of objects;

the motion requesting signals requesting radial motion along rays that each extend from a radial source and lateral motion in directions lateral to the rays;

the user input device set being structured so that a user viewing an image presented on the display that includes an object in a three-dimensional workspace can request presentation of a sequence of images in which the object moves along a path within the three-dimensional workspace by operating the user input device set to request concurrent radial motion and lateral motion independently;

the radial motion being along rays extending from radial sources, with the user being able to request radial motion along any of a set of rays that extend in different directions from each radial source;

the requested radial motion and lateral motion in combination being equivalent to motion along the path;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, each following iteration having a next preceding iteration;

the number of following iterations being two or more;

the first iteration comprising;

presenting a respective image on the display;

the respective image including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal from the user input device set, the motion requesting signal requesting radial motion and lateral motion of the next preceding iteration'"'"'s respective object;

the motion requesting signal requesting that the radial motion and the lateral motion be concurrent;

the motion requesting signal indicating one of a set of rays that extends from a radial source in a direction that is not perpendicular to the display; and

in response to the motion requesting signal, presenting a respective image on the display;

the image including a respective object that is perceptible as a moved continuation of the next preceding iteration'"'"'s respective object at a respective position within the three-dimensional workspace;

the respective position being perceptible as concurrently displaced from the next preceding iteration'"'"'s respective position both radially toward or away from the radial source along the indicated one of the set of rays and laterally in a direction lateral to the indicated one of the set of rays;

the motion requesting signal of each of the number of following iterations requesting radial motion in a first direction that is either toward or away from a radial source so that a request for radial motion in the first direction continues during the number of following iterations and the first iteration'"'"'s respective object is perceptible as continuing to move, during the number of following iterations, along a path in the three-dimensional workspace that is equivalent to the combination of the requested lateral motion with the requested radial motion in the first direction.

29. A system comprising:

a display;

a user input device set for providing signals; and

a processor connected for receiving signals from the user input device set and for presenting images on the display;

the user input device set providing motion requesting signals requesting motion of objects;

the motion requesting signals requesting radial motion along rays that each extend from a radial source and lateral motion in directions lateral to the rays;

the user input device set being structured so that a user viewing an image presented on the display that includes an object in a three-dimensional workspace can request presentation of a sequence of images in which the object moves along a path within the three-dimensional workspace by operating the user input device set to request concurrent radial motion and lateral motion independently;

the radial motion being along rays extending from radial sources, with the user being able to request radial motion along any of a set of rays that extend in different directions from each radial source;

the requested radial motion and lateral motion in combination being equivalent to motion along the path;

the processor being operable to present a first image on the display;

the first image including a first object that is perceptible as having a first position within a three-dimensional workspace;

the processor further being operable to receive a first motion requesting signal set from the user input device set, the first motion requesting signal set including two or more radial motion requesting signals;

each radial motion requesting signal in the first motion requesting signal set requesting radial motion in a first direction that is either toward or away from a radial source so that a request for radial mot, on in the first direction continues during the first motion requesting signal set;

each radial motion requesting signal indicating one of a set of rays that extends from a radial source in a direction that is not perpendicular to the display;

the first motion requesting signal set requesting a first radial motion of the first object and a first lateral motion of the first object;

the first motion requesting signal set requesting that the first radial motion and the first lateral motion be concurrent;

the processor responding to the first motion requesting signal set by presenting a sequence of images on the display;

each image in the sequence including an object that is perceptible as a moved continuation of the preceding image'"'"'s object at a position within the three-dimensional workspace;

each image'"'"'s position being perceptible as concurrently displaced both radially in the first direction from the radial source along the indicated one of the set of rays and laterally in a direction lateral to the indicated one of the set of rays from the preceding image'"'"'s position so that the first object is perceptible as continuing to move, during the sequence of images, along a first path in the three-dimensional workspace that is equivalent to the combination of the first lateral motion with the first radial motion;

the sequence of images including a last image with a last object in a last position that is perceptible as displaced along the first path from the first position to the last position.

30. A method of operating a system that includes a display, a user input device set, and a processor connected for receiving signals from the user input device set and for presenting images on the display;

the user input device set providing motion requesting signals from a user viewing images presented on the display that include objects in a three-dimensional workspace, the motion requesting signals requesting presentation of a sequence of images in which objects move along paths within the three-dimensional workspace;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, the number of following iterations being two or more;

each following iteration having a next preceding iteration;

the first iteration comprising;

presenting a respective image on the display;

the respective image of the first iteration including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal from the user input device set requesting motion of the respective object of the next preceding iteration toward a destination position within the three-dimensional workspace;

the motion requesting signal indicating the destination position;

the motion requesting signal requesting concurrent radial and lateral motion from the next preceding iteration'"'"'s respective position toward the destination position;

the requested radial motion and the requested lateral motion in combination being equivalent to motion along a path toward the destination position within the three-dimensional workspace;

using the destination position indicated by the motion requesting signal to automatically obtain a respective position within the three-dimensional workspace;

the iteration'"'"'s respective position being displaced both radially and laterally from the next preceding iteration'"'"'s respective position; and

presenting a respective image on the display;

each respective image including a respective object that is perceptible as having the automatically obtained respective position;

the respective object of each following iteration being perceptible as a continuation of the respective object of the next preceding iteration, concurrently displaced along the path both radially and laterally from the next preceding iteration'"'"'s respective position;

the automatically obtained respective position of each of the following iterations being on the path toward the destination position indicated by the iteration'"'"'s motion requesting signal;

the respective position of each following iteration being displaced toward the destination position by a respective displacement from the respective position of the next preceding iteration;

in each following iteration, the act of using the destination position comprising;

operating the processor to automatically perform an asymptotic function to obtain the respective displacement;

the following iterations together obtaining a sequence of respective displacements so that the respective positions of the following iterations approach the destination position asymptotically along the path.

31. A method of operating a system that includes a display, a user input device set, and a processor connected for receiving signals from the user input device set and for presenting images on the display;

the user input device set providing motion requesting signals from a user viewing images presented on the display that include objects in a three-dimensional workspace, the motion requesting signals requesting presentation of a sequence of images in which objects move along paths within the three-dimensional workspace;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, the number of following iterations being two or more;

each following iteration having a next preceding iteration;

the first iteration comprising;

presenting a respective image on the display;

the respective image of the first iteration including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal from the user input device set requesting motion of the respective object of the next preceding iteration toward a destination position within the three-dimensional workspace;

the motion requesting signal indicating the destination position;

the motion requesting signal requesting concurrent radial and lateral motion from the next preceding iteration'"'"'s respective position toward the destination position;

the requested radial motion and the requested lateral motion in combination being equivalent to motion along a path toward the destination position within the three-dimensional workspace;

using the destination position indicated by the motion requesting signal to automatically obtain a respective position within the three-dimensional workspace;

the iteration'"'"'s respective position being displaced both radially and laterally from the next preceding iteration'"'"'s respective position; and

presenting a respective image on the display;

each respective image including a respective object that is perceptible as having the automatically obtained respective position;

the respective object of each following iteration being perceptible as a continuation of the respective object of the next preceding iteration, concurrently displaced along the path both radially and laterally from the next preceding iteration'"'"'s respective position;

the following iterations including first phase iterations and second phase iterations;

the first phase iterations preceding the second phase iterations;

the automatically obtained respective position of each first phase iteration and the automatically obtained respective position of each second phase iteration being on the path toward the destination position indicated by the iteration'"'"'s motion requesting signal;

the respective position of each following iteration being displaced by a respective displacement from the respective position of the next preceding iteration;

in each second phase iteration, the act of using the destination position comprising;

operating the processor to automatically perform an asymptotic function to obtain the respective displacement;

the following iterations together obtaining a sequence of respective displacements so that the respective positions of the first phase iterations accelerate toward the destination position along the path and so that the respective positions of the second phase iterations approach the destination position asymptotically along the path.

32. A method of operating a system that includes a display, a user input device set, and a processor connected for receiving signals from the user input device set and for presenting images on the display;

the user input device set providing motion requesting signals from a user viewing images presented on the display that include objects in a three-dimensional workspace, the motion requesting signals requesting presentation of a sequence of images in which objects move along paths within the three-dimensional workspace;

the method comprising a sequence of iterations, the sequence of iterations including a first iteration and a number of following iterations, the number of following iterations being two or more;

each following iteration having a next preceding iteration;

the first iteration comprising;

presenting a respective image on the display;

the respective image of the first iteration including a respective object that is perceptible as having a respective position within a three-dimensional workspace;

each following iteration comprising;

receiving a respective motion requesting signal from the user input device set requesting motion of the respective object of the next preceding iteration toward a destination position within the three-dimensional workspace;

the motion requesting signal indicating the destination position;

the motion requesting signal requesting concurrent radial and lateral motion from the next preceding iteration'"'"'s respective position toward the destination position;

the requested radial motion and the requested lateral motion in combination being equivalent to motion along a path toward the destination position within the three-dimensional workspace;

using the destination position indicated by the motion requesting signal to automatically obtain a respective position within the three-dimensional workspace;

the iteration'"'"'s respective position being displaced both radially and laterally from the next preceding iteration'"'"'s respective position; and

presenting a respective image on the display;

each respective image including a respective object that is perceptible as having the automatically obtained respective position;

the respective object of each following iteration being perceptible as a continuation of the respective object of the next preceding iteration, concurrently displaced along the path both radially and laterally from the next preceding iteration'"'"'s respective position;

the automatically obtained respective position of each of the following iterations being displaced from the respective position of the next preceding iteration by a respective displacement;

in each of the following iterations, the act of using the destination position to automatically obtain a respective position comprising;

determining the respective displacement by operating the processor to automatically perform a clipping operation on first and second functions, the first function being an acceleration function defining a first displacement to a first position that is on the path toward the destination position indicated by the motion requesting signal;

the second function being an asymptotic function defining a second displacement to a second position that is on the path toward the destination position indicated by the motion requesting signal;

the clipping operation choosing one of the first displacement and the second displacement;

the respective object of each following iteration being perceptible as accelerating along the path when the clipping operation chooses the first displacement and as approaching the destination position asymptotically along the path when the clipping operation chooses the second displacement.