A G2 and shape preserving subdivision scheme for curve interpolation

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.