1. Morphing Rational B-spline Curves and Surfaces Using Mass Distributions Tao Ju, Ron Goldman Department of Computer Science Rice University

2. Morphing Transforms one target shape into another Vertex Correspondence Vertex Interpolation Parametric curves and surfaces

3. Linear Interpolation Averaging in affine space Uniform transition Every point moves at same speed Unsatisfactory artifacts Flattening, wriggles, etc. t = 0 t =.25 t =.5 t =.75 t = 1

4. Weighted Averaging Interpolation using masses and geometric positions Influence of relative mass Larger mass has more impact Different points morph at different speeds Less flattening and wriggles t = 0 t =.25 t =.5 t =.75 t = 1

5. Rational B-splines A rational B-spline curve of degree n Mass

6. Linear vs. Weighted Averaging

7. Local Morph Control Modification of mass distribution changes the morphing behavior locally Re-formulate rational B-splines to permit assignment of auxiliary mass for morphing Customizable morphing between fixed targets

8. Local Morph Control Modification of mass distribution changes the morphing behavior locally Re-formulate rational B-splines to permit assignment of auxiliary mass for morphing Customizable morphing between fixed targets

9. Mass Modification Transition curve Normalized Distance curve

10. Customize Morphing Two easy steps (can be repeated) Select time frame t 0 Edit the normalized distance curve (surface) Real-time Morph editing environment Fast computation Calculations only involve simple algebra Easy to use User needs no knowledge of B-spline or mass

11. Morph Editing GUI Time (t) Normalized Distance Surface Control Points Selection Morph View

12. Conclusion Contributions Smooth, non-uniform morphing of rational B- spline curves and surfaces Local morph control by modification of the associated mass distribution User interface for real-time morph editing with no knowledge of B-spline required Applications Computer Animation Model design

13. Appendix - Mass Point Definition: a non-zero mass m attached to a point P in affine space. Notation: mP/m Operations: Scalar multiplication Addition

14. Appendix – Auxiliary Masses P(u) can be rewritten as Where m p (u) is a new mass distribution function defined by Here w k are auxiliary positive masses attached to each control point of P(u)

15. Appendix – Compute Mass Normalized distance between two curves P(u) and Q(u) with auxiliary masses w k and v k forms a degree n rational B-spline curve with control points R k and weights W k Conversely, given W k and R k at t, we have