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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 • Properties: • Go through four control points • not good for local control

  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 2 basis function 3 basis function 4

  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 - the curve is always located within the convex hull based on control points. • 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 t=1 t=0

  40. Derivatives at end points t=1 t=0

  41. Derivatives at end points t=1 t=0

  42. Derivatives at end points t=1 t=0

  43. Derivatives at end points t=1 t=0

  44. 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

  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 What does this derived equation mean geometrically? - What is the relationship between a, b and c, if a = 2b - c? b is the middle point of a and c. w2=v1+4v3-4v2

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