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A G 2 and shape preserving subdivision scheme for curve interpolation

A G 2 and shape preserving subdivision scheme for curve interpolation. Chongyang Deng 2007-05-16. Outline. 1. Introduction 2. The subdivision scheme 3. Example 4. Smoothness analysis 5. Generating spiral by subdivision scheme 6. Future work.

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A G 2 and shape preserving subdivision scheme for curve interpolation

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  1. A G2 and shape preserving subdivision scheme for curve interpolation Chongyang Deng 2007-05-16

  2. Outline 1. Introduction 2. The subdivision scheme 3. Example 4. Smoothness analysis 5. Generating spiral by subdivision scheme 6. Future work Reference: Deng et al. A G2 and shape preserving subdivision scheme for curve interpolation. Submitted.

  3. 1.Introduction • Subdivision curve and surface Main advantages: 1. Arbitrary topology 2. Efficiency 3. Simplicity …

  4. 1.Introduction • Classification of subdivision 1. Interpolation VS approximation 2. Linear VS nonlinear

  5. 1.Introduction • Linear schemes • Four point subdivision scheme and its extensions • Dyn, N., Levin, D., and Gregory, J.A., 1987. A 4-point interpolatory subdivision scheme for curve design. CAGD, 4, 257-268. • Hassan, M.F., Ivrissimitzis, I.P., Dodgson, N.A., and Sabin, M.A., 2002. An interpolating 4-point C2 ternary stationary subdivision scheme. CAGD, 19(1), 1-18.

  6. 1.Introduction

  7. 1.Introdution • Advantages of linear subdivision schemes 1.Simple to implement 2.Easy to analyze 3.Affine invariance

  8. 1.Introdution • Disadvantages of linear subdivision schemes: Difficult to control the shape of the limit curve (artifacts and undesired inflexions)

  9. 1.Introdution • Example

  10. 1.Introdution • Example

  11. 1.Introdution • Example

  12. 1.Introdution • Nonlinear (geometric driven) subdivision schemes • Yang Xunnian, Normal based subdivision scheme for curve design. CAGD 2006(23):243-260.

  13. 1.Introdution

  14. 1.Introdution • Examples

  15. 2.The subdivision scheme • Outline 2.1 Origin idea 2.2 Preprocess 2.3 Adding new points 2.4 Calculating tangent vectors

  16. 2.The subdivision scheme • Origin idea: C0: Adjacent two points run to equality C1: Adjacent three points run to collinear C2: Adjacent four points run to lie on a circle

  17. 2.The subdivision scheme • Differential geometry: For planar G2 continuous curve, the tangent line and the osculating circle (circle of curvature) at one point are the first and second order approximants of the curve near this point

  18. 2.The subdivision scheme • But it is complex to directly calculate and compare the radii of the circles passing three adjacent vertices! So for each subdivision step, we select the added points as like there is a G1 continuous circular arc spline interpolating the vertices.

  19. 2.The subdivision scheme • Definition 1 (a) Convex edge (b) Inflexion edge (c) Straight edge

  20. 2.The subdivision scheme • Preprocess

  21. 2.The subdivision scheme • Adding new points

  22. 2.The subdivision scheme • The interpolating G1 arc spline

  23. 2.The subdivision scheme • The interpolating G1 arc spline

  24. 2.The subdivision scheme • Calculating tangent vectors

  25. 2.The subdivision scheme • Why? • is used to control the convergence rate.

  26. 2.The subdivision scheme • Inserting a line segment • Analyze the shape of inflexion 1. The curvature is zero 2.The limit tangent vector can be computed explicitly.

  27. 2.The subdivision scheme • Inserting a line segment • By picking the appropriate initial tangent vector of two ends of a edge we can insert a line segment into the limit curve with G2 continuous.

  28. 2.The subdivision scheme • Inserting a line segment

  29. 3.Examples

  30. 3.Examples Ternary four point subdivision scheme

  31. 3.Examples

  32. 3.Examples

  33. 3.Examples

  34. 3.Examples

  35. 4.Smoothness analysis • There step: • 1.the polygon series converge. • 2.the limit curve is G1 continuous. • 3.the limit curve is G2 continuous.

  36. 4.Smoothness analysis • Convergence rate

  37. 5.Generate spiral by subdivision scheme • Spirals are curves of one-signed, monotone increasing or decreasing curvature. They are commonly perceived as high quality profiles.

  38. 5.Generate spiral by subdivision scheme • Aim: Generating spiral which interpolating the given two points and their tangent vectors(G1 Hermite data).

  39. 5.Generate spiral by subdivision scheme • Calculating tangent vectors

  40. 5.Generate spiral by subdivision scheme • Examples

  41. 5.Generate spiral by subdivision scheme • Examples

  42. 5.Generate spiral by subdivision scheme • Examples

  43. 6.Future work • 1. Matching admissable G2 Hermite data. • 2. Interpolating point array by spiral segments.

  44. The end. Thank you.

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