1. Some Remarks on Subdivision Curves A B D C S S S u Find new intermediate points S that lie on the implied curve.

2. Quadratic Interpolating Subdivision u Cannot generally fit a parabola thru 4 points A B D C

3. Quadratic Interpolating Subdivision u Cannot generally fit a parabola thru 4 points  Interpolate between two separate parabolas A B D C S

4. Cubic Interpolating Subdivision u 4-point cubic interpolation in the plane: S = 9B/16 + 9C/16 – A/16 – D/16 S = M + (B – A)/16 + (C – D)/16 A B D CM S

5. Application of Subdivision Step Original data points and control polygon A D C B Focus on 4 consecutive points: A, B, C, D Create a corresponding subdivision point S S

6. Yet Another Conceptual Approach Original data points and control polygon A D C B Focus on 4 consecutive points: A, B, C, D  Blend between two circular arcs !

7. Circle Spline Construction (1) Original data points and control polygon LEFT CIRCLE thru A, B, C A D C B Focus on 4 consecutive points: A, B, C, D

8. Circle Spline Construction (2) Original data points and control polygon LEFT CIRCLE thru A, B, C RIGHT CIRCLE thru B, C, D A D C B Focus on 4 consecutive points: A, B, C, D

9. Circle Spline Construction (3) A D C B u left circle × bisector  S L right circle × bisector  S R u average btw. S L and S R  S S SLSL SRSR

10. Circle Spline Construction (4) RECURSE ! A D C B S Cannot guarantee convergence behavior !

11. A Better Circle Spline Not based on subdivision, but on iterated interpolation. A D C B How should this blending be done ?...

12. Blending With Intermediate Circles (1) A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw Tangent Vectors for both circles at B and C.

13. Blending With Intermediate Circles (2) A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw Tangent Vectors for both circles at B and C. Draw a bundle of regularly spaced Tangent Vectors.

14. Blending With Intermediate Circles (3) A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw n equal-angle-spaced Circles from B to C. Draw Tangent Vectors for both circles at B and C. Draw a bundle of regularly spaced Tangent Vectors.

15. Blending With Intermediate Circles (4) A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. S Draw n equal-angle-spaced Circles from B to C. Draw Tangent Vectors for both circles at B and C. Make n equal segments on each arc and choose u th point on u th circle. Draw a bundle of regularly spaced Tangent Vectors.  G 1 -continuity @ B, C

16. REFERENCE – TO LEARN MORE: C. H. Séquin, K. Lee, and J. Yen: Fair G 2 and C 2 -Continuous Circle Splines for the Interpolation of Sparse Data Points JCAD Vol 37, No 2, pp 201-211, Feb. 2005.