1 / 16

Some Remarks on Subdivision Curves

Some Remarks on Subdivision Curves. Find new intermediate points S that lie on the implied curve. S. B. C. S. S. A. D. Quadratic Interpolating Subdivision. Cannot generally fit a parabola thru 4 points . B. C. A. D. Quadratic Interpolating Subdivision.

yetta
Download Presentation

Some Remarks on Subdivision Curves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

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

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

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

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

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

  9. B D A C Circle Spline Construction (3) • left circle× bisector  SLright circle × bisector  SR • average btw. SL and SR  S SL S SR

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

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

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

  13. Blending With Intermediate Circles (2) 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. D B C A

  14. Blending With Intermediate Circles (3) 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. Draw n equal-angle-spaced Circles from B to C. D B C A

  15. S Blending With Intermediate Circles (4) 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. Draw n equal-angle-spaced Circles from B to C. D Make n equal segments on each arc andchoose uth point on uth circle. B C A  G1-continuity @ B, C

  16. REFERENCE – TO LEARN MORE: C. H. Séquin, K. Lee, and J. Yen: Fair G2 and C2-Continuous Circle Splinesfor the Interpolation of Sparse Data Points JCAD Vol 37, No 2, pp 201-211, Feb. 2005.

More Related