Automatic ballooning an assembly drawing of a computer aided design

• US 20050288909A1
• Filed: 06/25/2004
• Published: 12/29/2005
• Est. Priority Date: 06/25/2004
• Status: Active Grant

• Current Assignee: Autodesk, Inc.
• Original Assignee: Autodesk, Inc.
• Inventors: Mikulecky, Martin
• Granted Patent: US 7,636,096 B2
• Time in Patent Office: Days
• US Class Current: 703/1
• CPC Class Codes: G06F 30/00 Computer-aided design [CAD]

Abstract

Methods and apparatus, including computer program products, are described for ballooning an assembly drawing of a computer aided design. In one implementation, a set of anchor points is received and a set of balloons is arranged along a polygon enclosing a region including the set of anchor points. The set of balloons is iterated through, and the balloons in the set of balloons are attached to anchor points in the set of anchor points by leaders. Upon completion of one iteration no two leaders intersect.

Claims

1. A method for ballooning, comprising:

receiving a set of anchor points;

arranging a set of balloons along a polygon enclosing a region including the set of anchor points; and

iterating through the set of balloons and attaching balloons in the set of balloons to anchor points in the set of anchor points by leaders, wherein upon completion of one iteration no two leaders intersect.

2. The method of claim 1, wherein iterating through the set of balloons and attaching balloons in the set of balloons to anchor points in the set of anchor points by leaders, wherein upon completion of one iteration no two leaders intersect, does not require checking for conflicting leaders.

3. The method of claim 1, wherein:

arranging a set of balloons along a polygon comprises dividing the region including the set of anchor points into at least two sub-regions and arranging the set of balloons along the polygon such that an amount of balloons along a boundary of each sub-region is at least equal to an amount of anchor points included in the sub-region; and

iterating through the set of balloons and attaching each balloon in the set of balloons to an anchor point comprises iterating through the amount of balloons along the boundary of each sub-region and attaching each balloon to an anchor point included in the sub-region by a leader without creating intersections among the leaders;

where a boundary of a sub-region includes at least a portion of one side of the polygon.

4. The method of claim 3, wherein dividing the region including the set of anchor points comprises dividing the region such that each of the sub-regions includes an approximately equal amount of anchor points.

5. The method of claim 3, wherein dividing the region including the set of anchor points comprises dividing the region such that each of the sub-regions is approximately equal in area.

6. The method of claim 3, wherein iterating through the amount of balloons along the boundary of each sub-region comprises iterating through the amount of balloons in either a clockwise or a counterclockwise direction of iteration, wherein:

a first balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region; and

a last balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region.

7. The method of claim 6, wherein iterating through the amount of balloons along the boundary of each sub-region comprises, for each balloon in the amount of balloons:

iterating through the anchor points included in the sub-region that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the side of the polygon along which the balloon is positioned;

comparing the calculated values for each such anchor point; and

attaching the balloon to an anchor point using a leader based on the comparison of the calculated values.

8. The method of claim 7, wherein:

calculating the value of an angle comprises calculating a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

comparing the calculated values comprises determining the smallest non-reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the smallest non-reflex angle.

9. The method of claim 7, wherein:

calculating the value of an angle comprises calculating a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

comparing the calculated values comprises determining the largest reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the largest reflex angle.

10. The method of claim 7, wherein:

calculating the value of an angle comprises calculating a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in the direction of iteration;

comparing the calculated values comprises determining the largest non-reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the largest non-reflex angle.

11. The method of claim 7, wherein:

calculating the value of an angle comprises calculating a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in the direction of iteration;

comparing the calculated values comprises determining the smallest reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the smallest reflex angle.

12. The method of claim 1, wherein there are an equal number of balloons and anchor points and upon completion of one iteration each balloon in the set of balloons is attached to one anchor point in the set of anchor points by a leader.

13. A method for attaching balloons to anchor points, comprising:

receiving a set of balloons arranged along a substantially straight line;

receiving a set of anchors points positioned on either or both sides of the substantially straight line;

iterating through the balloons in the set of balloons, selecting an outermost balloon for a first iteration and progressively selecting a next adjacent balloon for each successive iteration until all balloons in the set of balloons have been iterated through, and for each balloon;

iterating through the anchor points that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the substantially straight line;

comparing the calculated values for each such anchor point; and

attaching the balloon to an anchor point using a leader based on the comparison of the calculated values;

wherein upon completion of one iteration through the balloons no two leaders intersect.

14. The method of claim 13, wherein:

calculating the value of an angle comprises calculating a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction away from a second outermost balloon situated on the substantially straight line;

comparing the calculated values for each such anchor point comprises determining the smallest non-reflex angle; and

attaching the balloon to an anchor point based on the comparison of the calculated values comprises attaching the balloon to an anchor point corresponding to the smallest non-reflex angle.

15. The method of claim 13, wherein:

calculating the value of an angle comprises calculating a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction away from a second outermost balloon situated on the substantially straight line;

comparing the calculated values for each such anchor point comprises determining the largest reflex angle; and

attaching the balloon to an anchor point based on the comparison of the calculated values comprises attaching the balloon to an anchor point corresponding to the largest reflex angle.

16. The method of claim 13, wherein:

calculating the value of an angle comprises calculating a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction toward a second outermost balloon situated on the substantially straight line;

comparing the calculated values for each such anchor point comprises determining the largest non-reflex angle; and

attaching the balloon to an anchor point based on the comparison of the calculated values comprises attaching the balloon to an anchor point corresponding to the largest non-reflex angle.

17. The method of claim 13, wherein:

calculating the value of an angle comprises calculating a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction toward a second outermost balloon situated on the substantially straight line;

comparing the calculated values for each such anchor point comprises determining the smallest reflex angle; and

attaching the balloon to an anchor point based on the comparison of the calculated values comprises attaching the balloon to an anchor point corresponding to the smallest reflex angle.

18. The method of claim 13, wherein the substantially straight line is a substantially horizontal line.

19. The method of claim 13, wherein the substantially straight line is a substantially vertical line.

20. The method of claim 13, wherein there are an equal number of balloons and anchor points and upon completion of one iteration each balloon in the set of balloons is attached to one anchor point in the set of anchor points by a leader.

21. A method for ballooning, comprising:

receiving a set of anchor points;

arranging a set of balloons along a polygon enclosing a region including the set of anchor points, comprising dividing the region including the set of anchor points into at least two sub-regions and arranging the set of balloons along the polygon such that an amount of balloons along a boundary of each sub-region is at least equal to an amount of anchor points included in the sub-region; and

iterating through the set of balloons and attaching balloons in the set of balloons to anchor points in the set of anchor points by leaders, comprising iterating through the amount of balloons along the boundary of each sub-region and attaching each balloon to an anchor point included in the sub-region by a leader;

where a boundary of a sub-region includes at least a portion of a side of the polygon and if the boundary of a sub-region includes portions of two or more sides of the polygon, no two sides of the polygon included in the boundary intersect at an angle of less than 90 degrees or greater than 180 degrees.

22. The method of claim 21, wherein iterating through the amount of balloons along the boundary of each sub-region comprises iterating through the amount of balloons in either a clockwise or a counterclockwise direction of iteration, wherein:

a first balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region; and

a last balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region.

23. The method of claim 22, wherein iterating through the amount of balloons along the boundary of each sub-region comprises, for each balloon in the amount of balloons:

iterating through the anchor points included in the sub-region that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the side of the polygon along which the balloon is positioned;

comparing the calculated values for each such anchor point; and

attaching the balloon to an anchor point using a leader based on the comparison of the calculated values.

24. A method for ballooning, comprising:

receiving a set of anchor points;

defining a balloon-placement polygon, the balloon-placement polygon enclosing a region including some or none of the anchor points;

defining an outer-polygon, the outer-polygon enclosing a region including the set of anchor points and the balloon-placement polygon;

dividing the region enclosed by the outer-polygon into two or more sub-regions, where each sub-region includes at least a portion of one side of the balloon-placement polygon;

arranging a set of balloons along the balloon-placement polygon such that an amount of balloons included in a sub-region is at least equal to an amount of anchor points included in the sub-region; and

for each sub-region, iterating through the amount of balloons included in the sub-region and attaching the balloons to the anchor points included in the sub-region using leaders;

wherein upon completion of one iteration through the set of balloons, no two leaders intersect.

25. The method of claim 24, wherein defining a balloon-placement polygon comprises receiving a user input specifying a shape of a polygon.

26. The method of claim 24, wherein for each sub-region, iterating through the amount of balloons included in the sub-region and attaching the balloons to the anchor points comprises:

iterating through the amount of balloons in either a clockwise or a counter-clockwise direction of iteration, where a first balloon in the iteration has at most one neighboring balloon within the sub-region and a last balloon in the iteration has at most one neighboring balloon within the sub-region; and

for each balloon in the amount of balloons, beginning with the first balloon,

iterating through the anchor points included in the sub-region that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the side of the balloon-placement polygon along which the balloon is positioned;

comparing the calculated values for each such anchor point; and

attaching the balloon to an anchor point based on the comparison of the calculated values.

27. The method of claim 26, wherein:

calculating the value of an angle comprises calculating a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

comparing the calculated values comprises determining the smallest non-reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the smallest non-reflex angle.

28. The method of claim 26, wherein:

calculating the value of an angle comprises calculating a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

comparing the calculated values comprises determining the largest reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the largest reflex angle.

29. The method of claim 26, wherein:

calculating the value of an angle comprises calculating a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in the direction of iteration;

comparing the calculated values comprises determining the largest non-reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the largest non-reflex angle.

30. The method of claim 26, wherein:

calculating the value of an angle comprises calculating a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in the direction of iteration;

comparing the calculated values comprises determining the smallest reflex angle calculated; and

attaching the balloon to an anchor point comprises attaching the balloon to an anchor point corresponding to the smallest reflex angle.

31. The method of claim 24, wherein no conflict checking for conflicting leaders is required.

32. A computer program product, tangibly stored on a computer-readable medium, for ballooning, comprising instructions operable to cause a programmable processor to:

receive a set of anchor points;

arrange a set of balloons along a polygon enclosing a region including the set of anchor points; and

iterate through the set of balloons and attach balloons in the set of balloons to anchor points in the set of anchor points by leaders, wherein upon completion of one iteration no two leaders intersect.

33. The computer program product of claim 32, wherein instructions operable to iterate through the set of balloons and attach balloons in the set of balloons to anchor points in the set of anchor points by leaders, wherein upon completion of one iteration no two leaders intersect, do not require instructions operable to check for conflicting leaders.

34. The computer program product of claim 32, wherein:

instructions operable to arrange a set of balloons along a polygon comprise instructions operable to divide the region including the set of anchor points into at least two sub-regions and arrange the set of balloons along the polygon such that an amount of balloons along a boundary of each sub-region is at least equal to an amount of anchor points included in the sub-region; and

instructions operable to iterate through the set of balloons and attach each balloon in the set of balloons to an anchor point comprise instructions operable to iterate through the amount of balloons along the boundary of each sub-region and attach each balloon to an anchor point included in the sub-region by a leader without creating intersections among the leaders;

where a boundary of a sub-region includes at least a portion of one side of the polygon.

35. The computer program product of claim 34, wherein instructions operable to divide the region including the set of anchor points comprise instructions operable to divide the region such that each of the sub-regions includes an approximately equal amount of anchor points.

36. The computer program product of claim 34, wherein instructions operable to divide the region including the set of anchor points comprise instructions operable to divide the region such that each of the sub-regions is approximately equal in area.

37. The computer program product of claim 34, wherein instructions operable to iterate through the amount of balloons along the boundary of each sub-region comprise instructions operable to iterate through the amount of balloons in either a clockwise or a counterclockwise direction of iteration, wherein:

a first balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region; and

a last balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region.

38. The computer program product of claim 37, wherein instructions operable to iterate through the amount of balloons along the boundary of each sub-region comprise, for each balloon in the amount of balloons, instructions operable to:

iterate through the anchor points included in the sub-region that are not already attached to a balloon by a leader, and for each such anchor point calculate a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the side of the polygon along which the balloon is positioned;

compare the calculated values for each such anchor point; and

attach the balloon to an anchor point using a leader based on the comparison of the calculated values.

39. The computer program product of claim 38, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the smallest non-reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the smallest non-reflex angle.

40. The computer program product of claim 38, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the largest reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the largest reflex angle.

41. The computer program product of claim 38, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the largest non-reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the largest non-reflex angle.

42. The computer program product of claim 38, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the polygon along which the balloon is positioned in the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the smallest reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the smallest reflex angle.

43. The computer program product of claim 32, wherein there are an equal number of balloons and anchor points and upon completion of one iteration each balloon in the set of balloons is attached to one anchor point in the set of anchor points by a leader.

44. A computer program product, tangibly stored on a computer-readable medium, for attaching balloons to anchor points, comprising instructions operable to cause a programmable processor to:

receive a set of balloons arranged along a substantially straight line;

receive a set of anchors points positioned on either or both sides of the substantially straight line;

iterate through the balloons in the set of balloons, selecting an outermost balloon for a first iteration and progressively selecting a next adjacent balloon for each successive iteration until all balloons in the set of balloons have been iterated through, and for each balloon;

iterate through the anchor points that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the substantially straight line;

compare the calculated values for each such anchor point; and

attach the balloon to an anchor point using a leader based on the comparison of the calculated values;

wherein upon completion of one iteration through the balloons no two leaders intersect.

45. The computer program product of claim 44, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction away from a second outermost balloon situated on the substantially straight line;

instructions operable to compare the calculated values for each such anchor point comprise instructions operable to determine the smallest non-reflex angle; and

instructions operable to attach the balloon to an anchor point based on the comparison of the calculated values comprise instructions operable to attach the balloon to an anchor point corresponding to the smallest non-reflex angle.

46. The computer program product of claim 44, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction away from a second outermost balloon situated on the substantially straight line;

instructions operable to compare the calculated values for each such anchor point comprise instructions operable to determine the largest reflex angle; and

instructions operable to attach the balloon to an anchor point based on the comparison of the calculated values comprise instructions operable to attach the balloon to an anchor point corresponding to the largest reflex angle.

47. The computer program product of claim 44, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction toward a second outermost balloon situated on the substantially straight line;

instructions operable to compare the calculated values for each such anchor point comprise instructions operable to determine the largest non-reflex angle; and

instructions operable to attach the balloon to an anchor point based on the comparison of the calculated values comprise instructions operable to attach the balloon to an anchor point corresponding to the largest non-reflex angle.

48. The computer program product of claim 44, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that originates at the balloon and extends along the straight line in a direction toward a second outermost balloon situated on the substantially straight line;

instructions operable to compare the calculated values for each such anchor point comprise instructions operable to determine the smallest reflex angle; and

instructions operable to attach the balloon to an anchor point based on the comparison of the calculated values comprise instructions operable to attach the balloon to an anchor point corresponding to the smallest reflex angle.

49. The computer program product of claim 44, wherein the substantially straight line is a substantially horizontal line.

50. The computer program product of claim 44, wherein the substantially straight line is a substantially vertical line.

51. The computer program product of claim 44, wherein there are an equal number of balloons and anchor points and upon completion of one iteration each balloon in the set of balloons is attached to one anchor point in the set of anchor points by a leader.

52. A computer program product, tangibly stored on a computer-readable medium, for ballooning, comprising instructions operable to cause a programmable processor to:

receive a set of anchor points;

arrange a set of balloons along a polygon enclosing a region including the set of anchor points, comprising instructions operable to divide the region including the set of anchor points into at least two sub-regions and arrange the set of balloons along the polygon such that an amount of balloons along a boundary of each sub-region is at least equal to an amount of anchor points included in the sub-region; and

iterate through the set of balloons and attaching balloons in the set of balloons to anchor points in the set of anchor points by leaders, comprising instructions operable to iterate through the amount of balloons along the boundary of each sub-region and attach each balloon to an anchor point included in the sub-region by a leader;

where a boundary of a sub-region includes at least a portion of a side of the polygon and if the boundary of a sub-region includes portions of two or more sides of the polygon, no two sides of the polygon included in the boundary intersect at an angle of less than 90 degrees or greater than 180 degrees.

53. The computer program product of claim 52, wherein instructions operable to iterate through the amount of balloons along the boundary of each sub-region comprise instructions operable to iterate through the amount of balloons in either a clockwise or a counterclockwise direction of iteration, wherein:

a first balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region; and

a last balloon in the iteration has at most one neighboring balloon on the boundary of the sub-region.

54. The computer program product of claim 53, wherein instructions operable to iterate through the amount of balloons along the boundary of each sub-region comprise, for each balloon in the amount of balloons, instructions operable to:

iterate through in the direction of iteration the anchor points included in the sub-region that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the side of the polygon along which the balloon is positioned;

compare the calculated values for each such anchor point; and

attach the balloon to an anchor point using a leader based on the comparison of the calculated values.

55. A computer program product, tangibly stored on a computer-readable medium, for ballooning, comprising instructions operable to cause a programmable processor to:

receive a set of anchor points;

define a balloon-placement polygon, the balloon-placement polygon enclosing a region including some or none of the anchor points;

define an outer-polygon, the outer-polygon enclosing a region including the set of anchor points and the balloon-placement polygon;

divide the region enclosed by the outer-polygon into two or more sub-regions, where each sub-region includes at least a portion of one side of the balloon-placement polygon;

arrange a set of balloons along the balloon-placement polygon such that an amount of balloons included in a sub-region is at least equal to an amount of anchor points included in the sub-region; and

for each sub-region, iterate through the amount of balloons included in the sub-region and attach the balloons to the anchor points included in the sub-region using leaders;

wherein upon completion of one iteration through the set of balloons, no two leaders intersect.

56. The computer program product of claim 55, wherein instructions operable to define a balloon-placement polygon comprise instructions operable to receive a user input specifying a shape of a polygon.

57. The computer program product of claim 55, wherein for each sub-region, instructions operable to iterate through the amount of balloons included in the sub-region and attach the balloons to the anchor points comprise instructions operable to:

iterate through the amount of balloons in either a clockwise or a counter-clockwise direction of iteration, where a first balloon in the iteration has at most one neighboring balloon within the sub-region and a last balloon in the iteration has at most one neighboring balloon within the sub-region; and

for each balloon in the amount of balloons, beginning with the first balloon,

iterate through the anchor points included in the sub-region that are not already attached to a balloon by a leader, and for each such anchor point calculating a value of an angle formed by an intersection of a line extending between the balloon and the anchor point and the side of the balloon-placement polygon along which the balloon is positioned;

compare the calculated values for each such anchor point; and

attach the balloon to an anchor point based on the comparison of the calculated values.

58. The computer program product of claim 57, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the smallest non-reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the smallest non-reflex angle.

59. The computer program product of claim 57, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in a direction opposite to the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the largest reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the largest reflex angle.

60. The computer program product of claim 57, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a non-reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the largest non-reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the largest non-reflex angle.

61. The computer program product of claim 57, wherein:

instructions operable to calculate the value of an angle comprise instructions operable to calculate a value of a reflex angle formed between a segment connecting the balloon to the anchor point and a ray that extends along a side of the balloon-placement polygon along which the balloon is positioned in the direction of iteration;

instructions operable to compare the calculated values comprise instructions operable to determine the smallest reflex angle calculated; and

instructions operable to attach the balloon to an anchor point comprise instructions operable to attach the balloon to an anchor point corresponding to the smallest reflex angle.

62. The computer program product of claim 61, wherein instructions operable to conflict check for conflicting leaders are not required.