Interpolatory Subdivision Curves via Diffusion of Normals

1. Interpolatory Subdivision Curvesvia Diffusion of Normals Yutaka Ohtake AlexanderBelyaev Hans-PeterSeidel Max-Planck-Institut für Informatik, Germany

2. Subdivision Curves • A smooth curve is obtained as the limit of a sequence of successive refinements of a polyline.

3. Interpolatory Subdivision • Interpolating the given control points. Interpolation Control Polygon Subdivision level Approximation

4. Interpolatory Subdivision Subdivision level Update Update positions Insert vertices Newly inserted fixed

5. The Four-point Scheme 4-pointscheme Vertex to be inserted Averaging vertex positions

6. Purpose • Developing a new interpolatory subdivision scheme. • Round shapes • Curvature-continuous 4-point scheme Our method

7. Basic Idea • Smooth variation of curve normals Our method 4-point scheme angle ofnormal angle ofnormal arclength arclength

8. Contents • Basic algorithm • Non-uniform weighting for Curvature Continuity • Generating Corners

9. Algorithm Overview • Insert odd vertices. • Produce smoothed normals. • Update odd vertex positions.

10. Averaging Normals • Averaging angles between normals. • Vector based averaging does not work well. Averaging Weight of averaging (Chaikin’s weight)

11. Updating Odd Vertex Positions • A straightforward approach is to place the intersection point. • However, edges flipping near inflection vertices. Intersection point Edge flipping

12. Updating Odd Vertex Positions • Squared differences of normals are minimized. • Quick minimization by the conjugate gradient method. Error to be minimized :

13. Our method 4-point scheme 4-point scheme requires more control points to generate circular shapes.

14. Open Polyline • Normals on end-edges are fixed during the averaging. open closed

15. Contents • Basic algorithm • Non-uniform weighting for Curvature Continuity • Generating Corners

16. Curvature discontinuity angle of normal curvature arc-length Curvature Discontinuity Un-even intervals of control points Even intervals of control points

17. Non-uniform Weighting • Taking into account the length of edges. L. Kobbelt and P. Schröder. “A variational approach to subdivision”. ACM TOG, 1998 4-point scheme Undesirable peaks Uniform weight Non-uniform weight

18. Non-uniform Weighting • Non-uniform corner cutting. J. Gregory and R. Qu. “Non-uniform corner cutting”. CAGD, 1996 Averaging Edge lengths

19. angle of normal arc-length angle of normal arc-length Curvature Continuous Curves Non-uniform weight Uniform weight

20. 4-point scheme Our method Uniform weight Non-uniform weight

21. Contents • Basic algorithm • Non-uniform weighting for Curvature Continuity • Generating Corners

22. Sharp Corners • Corner points are considered as end-points. Sharpened Rounded Averaging rules are linearly interpolated.

23. Examples 1 1 1 1 1 1 1 1 0.5 0.5 0.5 0.5 1 1 1 1 1 1 0.5 1 1 0.5 0.5 0.5

24. Conclusion • A new interpolatory subdivision method • Via averaging normals • Round shapes • Curvature-continuous • A rigorous mathematical study is needed.

25. Future Work • Extension to surfaces (Mesh subdivision). Control mesh Our method Butterfly scheme