On the Parameterization of Catmull-Rom Curves. Cem Yuksel Scott Schaefer John Keyser Texas A&M University

Transcription

1 On the Parameterization of Catmull-Rom Curves Cem Yuksel Scott Schaefer John Keyser Texas A&M University

2 Catmull-Rom Curves P 3 P 1 P 0 P 2

3 Catmull-Rom Curves P 3 P 1 P 0 P 2

4 Catmull-Rom Curves Important Properties Interpolate control points Local support Piecewise polynomial representation

5 Catmull-Rom Curves Important Properties Interpolate control points Local support Piecewise polynomial representation

6 Catmull-Rom Curves P 3 P 1 P 0 P 2

7 Catmull-Rom Curves P 3 P 1 t 3 t 1 P 0 t 0 C 12 ( P 0,1,2,3, t 0,1,2,3 ) t 2 P 2

8 Catmull-Rom Curves P 3 P 1 t 3 t 1 P 0 t 0 Uniform t 2 P 2 t 0 t 1 t 2 t 3 t i+1 = t i + 1

9 Catmull-Rom Curves P 3 P 1 t 3 t 1 P 0 t 0 Chordal t 2 P 2 t 0 t 1 t 2 t 3 t i+1 = t i + P i+1 P i

10 Catmull-Rom Curves Parameterization Uniform: t i+1 = t i + 1 Chordal: t i+1 = t i + P i+1 P i

11 Catmull-Rom Curves Parameterization Uniform: t i+1 = t i + P i+1 P i 0 Chordal: t i+1 = t i + P i+1 P i 1

12 Catmull-Rom Curves Parameterization t i+1 = t i + P i+1 P i a

13 Catmull-Rom Curves Parameterization t i+1 = t i + P i+1 P i a a = 0 a = 1 Uniform Chordal

14 Catmull-Rom Curves Parameterization t i+1 = t i + P i+1 P i a a = 0 a = 2 a = 1 Uniform Centripetal Chordal 1

15 On the Parameterization of Catmull-Rom Curves DEMO

16 On the Parameterization of Catmull-Rom Curves CUSPS & SELF-INTERSECTIONS

17 Cusps & Self-Intersections P 0 P 1 P 2 P 3

18 Cusps & Self-Intersections P 0 P 1 P 2 P 3

19 Cusps & Self-Intersections P 0 P 1 P 2 P 3

20 Cusps & Self-Intersections P 0 P 1 P 2 P 3

21 Cusps & Self-Intersections P 0 P 1 P 2 P 3

22 Cusps & Self-Intersections P 0 P 1 P 2 P 3 a < 1 2

23 Cusps & Self-Intersections P 0 P 1 P 2 P 3

24 Cusps & Self-Intersections P 0 P 1 P 2 P 3

25 Cusps & Self-Intersections P 0 P 1 P 2 P 3 a = 1 2

26 Cusps & Self-Intersections P 0 P 1 P 2 P 3 a = 1 2

27 Cusps & Self-Intersections P 0 P 1 P 2 P 3 a = 1 2

28 On the Parameterization of Catmull-Rom Curves DISTANCE BOUND

29 Distance Bound

30 Distance Bound Distance to the infinite line h Distance to end points l

31 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3

32 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3 h d 2 x A

33 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3 r = d 1 d 2 h d 2 r 1-a 4 (1 + r a )

34 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3 r = d 1 d 2 h r 1-a a < 1/2 a = 1/2 d 2 4 (1 + r a ) h 8 h d 2 /4 a = 2/3 h d 2 /8 a = 1 h d 2 /4

35 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3 r = d 1 d 2 h r 1-a a < 1/2 a = 1/2 d 2 4 (1 + r a ) h 8 h d 2 /4 a = 2/3 h d 2 /8 a = 1 h d 2 /4

36 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3 r = d 1 d 2 h r 1-a a < 1/2 a = 1/2 d 2 4 (1 + r a ) h 8 h d 2 /4 a = 2/3 h d 2 /8 a = 1 h d 2 /4

37 Distance Bound Distance to the infinite line h P 0 P 1 P 2 d 1 d 2 P 3 r = d 1 d 2 h r 1-a a < 1/2 a = 1/2 d 2 4 (1 + r a ) h 8 h d 2 /4 a = 2/3 h d 2 /8 a = 1 h d 2 /4

38 Distance Bound Distance to the end points l l d 2 r 2 r 4a 3 r a (1 + r a )

39 Distance Bound a only a = 1/2 a = 2/3 a = 1

40 Distance Bound a and r a = 1/2 a = 2/3 a = 1

41 On the Parameterization of Catmull-Rom Curves INTERSECTION-FREE CURVES

42 Intersection-Free Curves

43 Intersection-Free Curves

44 Intersection-Free Curves γ γ > π / 3

45 Intersection-Free Curves γ γ > π / 3

46 Intersection-Free Curves Avoid self-intersections Centripetal parameterization Avoid adjacent segment intersections Control polygon angle > π / 3 Avoid non-adjacent segment intersections Bounding box

47 On the Parameterization of Catmull-Rom Curves DISCUSSION

48 Discussion Distance to Control Polygon Uniform is closer for longer segments Chordal is closer for shorter segments

49

50 Discussion Edge Direction Chordal has extreme sensitivity to short edge directions

51 Discussion Edge Direction Chordal has extreme sensitivity to short edge directions

52 Discussion Edge Direction Chordal has extreme sensitivity to short edge directions

53 Discussion Edge Direction Chordal has extreme sensitivity to short edge directions

54 Discussion Curvature Centripetal tends to have higher curvature at control points.

55 Discussion Curvature Centripetal tends to have higher curvature at control points.

56 Discussion Curvature Centripetal tends to have higher curvature at control points.

57 Discussion Curvature Centripetal tends to have higher curvature at control points.

58 Catmull-Rom Curves Lee Perry-Smith Lee Perry-Smith Alexander Tomchuk Cem Yuksel, Scott Schaefer, John Keyser, Hair Meshes, Siggraph Asia 2009

59 Catmull-Rom Curves Cem Yuksel, Scott Schaefer, John Keyser, Hair Meshes, Siggraph Asia 2009

60 Catmull-Rom Curves Uniform Chordal Centripetal Cem Yuksel, Scott Schaefer, John Keyser, Hair Meshes, Siggraph Asia 2009

61 Summary Parameterization of Catmull-Rom curves 0 a 1 Cusps and self-intersections Distance bound Intersection-free curves C 1 Catmull-Rom curves only!

62 On the Parameterization of Catmull-Rom Curves QUESTIONS?