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CSCE 441: Keyframe Animation/Smooth Curves (Cont.)

CSCE 441: Keyframe Animation/Smooth Curves (Cont.). Jinxiang Chai. Key-frame Interpolation. Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ. t. Key-frame Interpolation.

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CSCE 441: Keyframe Animation/Smooth Curves (Cont.)

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  1. CSCE 441: Keyframe Animation/Smooth Curves (Cont.) Jinxiang Chai

  2. Key-frame Interpolation • Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ t

  3. Key-frame Interpolation • Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ t

  4. Key-frame Interpolation • Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ t Nonlinear interpolation

  5. Review: Natural cubic cruves

  6. Review: Natural cubic cruves

  7. Review: Natural cubic cruves

  8. Review: Natural cubic curves

  9. R1 P4 R4 P1 Review: Hermite Curves P1: start position P4: end position R1: start derivative R4: end derivative

  10. R1 P4 R4 P1 Review: Hermite Curves

  11. R1 P4 R4 P1 Review: Hermite Curves Herminte basis matrix

  12. R1 P4 R4 P1 Review: Hermite Curves Herminte basis matrix

  13. R1 P4 R4 P1 Review: Hermite Curves

  14. R1 P4 R4 P1 Review: Hermite Curves

  15. R1 P4 R4 P1 Review: Hermite Curves

  16. R1 P4 R4 P1 Review: Hermite Curves

  17. R1 P4 R4 P1 Review: Hermite Curves Hermite basis functions

  18. R1 P4 R4 P1 Review: Hermite Curves basis function 1 basis function 1 basis function 1 basis function 1

  19. R1 P4 R4 P1 Review: Hermite Curves *R1 *R4 *P1 + *P4 + + =

  20. Review: Bezier Curves

  21. Review: Bezier Curves

  22. Review: Bezier Curves

  23. Review: Bezier Curves

  24. Review: Bezier Curves *v2 *v3 *v1 *v0 + + + =

  25. Review: Different basis functions • Cubic curves: • Hermite curves: • Bezier curves:

  26. Complex curves Suppose we want to draw or interpolate a more complex curve

  27. Complex curves • Suppose we want to draw or interpolate a more complex curve How can we represent this curve?

  28. Complex curves • Suppose we want to draw a more complex curve • Idea: we’ll splice together a curve from individual segments that are cubic Béziers

  29. Complex curves • Suppose we want to draw or interpolate a more complex curve • Idea: we’ll splice together a curve from individual segments that are cubic Béziers

  30. Splines • A piecewise polynomial that has a locally very simple form, yet be globally flexible and smooth

  31. Splines • There are three nice properties of splines we’d like to have - Continuity - Local control - Interpolation

  32. Continuity • C0: points coincide, velocities don’t • C1: points and velocities coincide • What’s C2? - points, velocities and accelerations coincide

  33. Continuity • Cubic curves are continuous and differentiable • We only need to worry about the derivatives at the endpoints when two curves meet

  34. Local control • We’d like our spline to have local control - that is, have each control point affect some well-defined neighborhood around that point

  35. Local control • We’d like our spline to have local control - that is, have each control point affect some well-defined neighborhood around that point

  36. Local control • We’d like our spline to have local control - that is, have each control point affect some well-defined neighborhood around that point

  37. Interpolation • Bézier curves are approximating - The curve does not (necessarily) pass through all the control points - Each point pulls the curve toward it, but other points are pulling as well • Instead, we may prefer a spline that is interpolating - That is, that always passes through every control point

  38. B-splines • We can join multiple Bezier curves to create B-splines • Ensure C2 continuity when two curves meet

  39. Derivatives at end points

  40. Derivatives at end points

  41. Derivatives at end points

  42. Derivatives at end points

  43. Derivatives at end points

  44. Derivatives at end points

  45. Continuity in B splines • Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

  46. Continuity in B splines • Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

  47. Continuity in B splines • Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

  48. Continuity in B splines • Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

  49. Continuity in B splines • Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

  50. Continuity in B splines • Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

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