On the Parameterization of Catmull-Rom Curves Cem Yuksel Scott - - PowerPoint PPT Presentation

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On the Parameterization of Catmull-Rom Curves Cem Yuksel Scott - - PowerPoint PPT Presentation

Jan 18, 2024 259 likes •891 views

On the Parameterization of Catmull-Rom Curves Cem Yuksel Scott Schaefer John Keyser Texas A&M University Catmull-Rom Curves P 3 P 1 P 0 P 2 Catmull-Rom Curves P 3 P 1 P 0 P 2 Catmull-Rom Curves Important Properties

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On the Parameterization of Catmull-Rom Curves

Cem Yuksel Scott Schaefer John Keyser Texas A&M University

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Catmull-Rom Curves

P0 P1 P2 P3

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SLIDE 3

Catmull-Rom Curves

P0 P1 P2 P3

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Catmull-Rom Curves

  • Important Properties

– Interpolate control points – Local support – Piecewise polynomial representation

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SLIDE 5

Catmull-Rom Curves

  • Important Properties

– Interpolate control points – Local support – Piecewise polynomial representation

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SLIDE 6

Catmull-Rom Curves

P0 P1 P2 P3

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Catmull-Rom Curves

P0 P1 P2 P3 t0 t1 t2 t3 C12 ( P0,1,2,3, t0,1,2,3 )

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SLIDE 8

Catmull-Rom Curves

t0 t1 t2 t3 Uniform ti+1 = ti + 1 P0 P1 P2 P3 t0 t1 t2 t3

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SLIDE 9

Catmull-Rom Curves

P0 P1 P2 P3 t0 t1 t2 t3 t0 t1 t2 t3 Chordal ti+1 = ti + | Pi+1 – Pi |

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Catmull-Rom Curves

  • Parameterization

– Uniform: ti+1 = ti + 1 – Chordal: ti+1 = ti + | Pi+1 – Pi |

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Catmull-Rom Curves

  • Parameterization

– Uniform: ti+1 = ti + | Pi+1 – Pi |0 – Chordal: ti+1 = ti + | Pi+1 – Pi |1

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Catmull-Rom Curves

  • Parameterization

ti+1 = ti + | Pi+1 – Pi |

a

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Catmull-Rom Curves

  • Parameterization

ti+1 = ti + | Pi+1 – Pi |

a a = 0 a = 1 Uniform Chordal

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Catmull-Rom Curves

  • Parameterization

ti+1 = ti + | Pi+1 – Pi |

a a = 0 a = 1 a = Uniform Chordal Centripetal

1 2

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DEMO

On the Parameterization of Catmull-Rom Curves

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CUSPS & SELF-INTERSECTIONS

On the Parameterization of Catmull-Rom Curves

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Cusps & Self-Intersections

P0 P1 P2 P3

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Cusps & Self-Intersections

P0 P1 P2 P3

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Cusps & Self-Intersections

P0 P1 P2 P3

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Cusps & Self-Intersections

P0 P1 P2 P3

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SLIDE 21

Cusps & Self-Intersections

P0 P1 P2 P3

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Cusps & Self-Intersections

P0 P1 P2 P3

a < 1

2

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Cusps & Self-Intersections

P0 P1 P2 P3

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Cusps & Self-Intersections

P0 P1 P2 P3

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Cusps & Self-Intersections

P0 P1 P2 P3

a = 1

2

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Cusps & Self-Intersections a = 1

2 P0 P1 P2 P3

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Cusps & Self-Intersections a = 1

2 P0 P1 P2 P3

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DISTANCE BOUND

On the Parameterization of Catmull-Rom Curves

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Distance Bound

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  • Distance to the infinite line
  • Distance to end points

Distance Bound

h l

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Distance Bound

  • Distance to the infinite line

h P0 P1 P2 P3 d2 d1

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Distance Bound

  • Distance to the infinite line

P0 P1 P2 P3 d2 d1 h ≤ d2 A

x

h

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Distance Bound

  • Distance to the infinite line

P0 P1 P2 P3 d2 d1 r = d1 d2 h ≤ r1-a 4 (1 + ra) d2 h

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Distance Bound

  • Distance to the infinite line

P0 P1 P2 P3 d2 d1 r = d1 d2 h ≤ r1-a 4 (1 + ra) d2

a < 1/2 8 a = 1/2 h ≤ h ≤ d2 /4 a = 2/3 h ≤ d2 /8 a = 1 h ≤ d2 /4

h

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Distance Bound

  • Distance to the infinite line

P0 P1 P2 P3 d2 d1 r = d1 d2 h ≤ r1-a 4 (1 + ra) d2

a < 1/2 8 a = 1/2 h ≤ h ≤ d2 /4 a = 2/3 h ≤ d2 /8 a = 1 h ≤ d2 /4

h

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SLIDE 36

Distance Bound

  • Distance to the infinite line

P0 P1 P2 P3 d2 d1 r = d1 d2 h ≤ r1-a 4 (1 + ra) d2

a < 1/2 8 a = 1/2 h ≤ h ≤ d2 /4 a = 2/3 h ≤ d2 /8 a = 1 h ≤ d2 /4

h

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SLIDE 37

Distance Bound

  • Distance to the infinite line

P0 P1 P2 P3 d2 d1 r = d1 d2 h ≤ r1-a 4 (1 + ra) d2

a < 1/2 8 a = 1/2 h ≤ h ≤ d2 /4 a = 2/3 h ≤ d2 /8 a = 1 h ≤ d2 /4

h

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SLIDE 38

Distance Bound

  • Distance to the end points

l ≤ r 2 – r 4a 3 ra (1 + ra) d2 l

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SLIDE 39

a only

Distance Bound

a = 1/2 a = 2/3 a = 1

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a and r

Distance Bound

a = 1/2 a = 2/3 a = 1

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INTERSECTION-FREE CURVES

On the Parameterization of Catmull-Rom Curves

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Intersection-Free Curves

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Intersection-Free Curves

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Intersection-Free Curves γ > π / 3 γ

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Intersection-Free Curves γ > π / 3 γ

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Intersection-Free Curves

  • Avoid self-intersections

– Centripetal parameterization

  • Avoid adjacent segment intersections

– Control polygon angle > π / 3

  • Avoid non-adjacent segment intersections

– Bounding box

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DISCUSSION

On the Parameterization of Catmull-Rom Curves

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Discussion

  • Distance to Control Polygon

– Uniform is closer for longer segments – Chordal is closer for shorter segments

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SLIDE 49
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Discussion

  • Edge Direction

– Chordal has extreme sensitivity to short edge directions

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SLIDE 51

Discussion

  • Edge Direction

– Chordal has extreme sensitivity to short edge directions

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SLIDE 52

Discussion

  • Edge Direction

– Chordal has extreme sensitivity to short edge directions

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Discussion

  • Edge Direction

– Chordal has extreme sensitivity to short edge directions

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Discussion

  • Curvature

– Centripetal tends to have higher curvature at control points.

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Discussion

  • Curvature

– Centripetal tends to have higher curvature at control points.

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Discussion

  • Curvature

– Centripetal tends to have higher curvature at control points.

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SLIDE 57

Discussion

  • Curvature

– Centripetal tends to have higher curvature at control points.

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Catmull-Rom Curves

Cem Yuksel, Scott Schaefer, John Keyser, “Hair Meshes,” Siggraph Asia 2009

Lee Perry-Smith Alexander Tomchuk Lee Perry-Smith

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Catmull-Rom Curves

Cem Yuksel, Scott Schaefer, John Keyser, “Hair Meshes,” Siggraph Asia 2009

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Catmull-Rom Curves

Cem Yuksel, Scott Schaefer, John Keyser, “Hair Meshes,” Siggraph Asia 2009

Centripetal Uniform Chordal

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Summary

  • Parameterization of Catmull-Rom curves
  • Cusps and self-intersections
  • Distance bound
  • Intersection-free curves
  • C1 Catmull-Rom curves only!

0 ≤ a ≤ 1

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QUESTIONS?

On the Parameterization of Catmull-Rom Curves