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Morphing Rational B-spline Curves and Surfaces Using Mass Distributions

Morphing Rational B-spline Curves and Surfaces Using Mass Distributions. Tao Ju, Ron Goldman Department of Computer Science Rice University. Morphing. Transforms one target shape into another Vertex Correspondence Vertex Interpolation Parametric curves and surfaces. Linear Interpolation.

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Morphing Rational B-spline Curves and Surfaces Using Mass Distributions

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  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 t = 0 t = .25 • Uniform transition • Every point moves at same speed • Unsatisfactory artifacts • Flattening, wriggles, etc. t = .5 t = .75 t = 1

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

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

  6. Linear vs. Weighted Averaging

  7. Local Morph Control • Modification of mass distribution changes the morphing behaviorlocally • 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 behaviorlocally • 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 t0 • 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 Control Points Selection Morph View Normalized Distance Surface Time (t)

  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 massmattached to a pointP in affine space. • Notation: mP/m • Operations: • Scalarmultiplication • Addition

  14. Appendix – Auxiliary Masses • P(u) can be rewritten as • Where mp(u) is a new mass distribution function defined by • Here wk 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 wk and vk forms a degree n rational B-spline curve with control points Rk and weights Wk • Conversely, given Wk and Rk at t, we have

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