Normal based subdivision scheme for curve and surface design

杨勋年2004.12

http://www.math.zju.edu.cn/yxn

What is CAGD

Computer science

CAGD

Engineering

mathematics

Content

• What is subdivision?- corner cutting algorithms

- interpolating subdivision• Normal based subd. Scheme

- the scheme- for curve design- for surface design

• Summary

What is subdivision

• Recursive refinementfor the generation of- functions (approx. theory, wavelet)- curves and surfaces (CAGD)

• Classification- Steady vs nonsteady- rational vs nonrational- Linear vs nonlinear

Corner cutting algorithms

• Corner cutting: Chaikin, B-spline

• Convergence: de Boor, Riesenfeld, Gregory, et al

Subdivision of B-spline

• Uniform cubic B-spline

• Derive the rule by knots insertion

Arbitrary control mesh

• The topological rule

• The geometric rule

Catmull-clark scheme

Catmull-clark subdivision surface

Interpolating subdivision

• Edge split

• Vertex refinement

Four-point scheme

• Cubic precision (Dyn, et al 1987)

• Linear subdivision

Add a point by local cubic curve interpolation

A geometric look at four point scheme

Butterfly scheme

• Extension of 4-point scheme (Dyn, et al 1990)• Triangular control mesh (1 to 4)• Local bicubic surface interpolation

Control meshParametric domain

Limitations

• Interpolating or fitting- efficient representation- scanning data processing

• By CC scheme- solve inverse problem

• By butterfly scheme- not fair- not easy for normal control

Content

• What is subdivision?- corner cutting algorithms

- interpolating subdivision• Normal based subd. Scheme

- the scheme- for curve design- for surface design

• Summary

Our approach

• Normal refinement

- for each vertex for each level

• Vertex refinement

- subdivide each edge

- project sub-edges onto normals

- compute displacement vector

- compute new vertex

The basic scheme

Normal refinement

• Fixed normal at selected vertexes

- the normal will be interpolated

• Refine other normal for each subdivision

• The rule for normal computation

- chord tangent angles are close

Normal computation

Curve case Surface case

Convergence

• Active chord tangent angles- converge to zero- within fixed scale

• Fixed chord tangent angles- are bounded- convergence

• Polygon series- converge- tangent continuous

For curve design

• The freedoms

- subd. ratio of edges

- scale for displacement vector

• Shape preserving

- same scheme

- explicit choices of freedoms

Shape preserving scheme

12 1kip

ki k

i

1kip

mp

1kin

kip

kin

Freeform curve

Bottle design

Control polygon Subdivision curve

For surface design

• Triangular control mesh

• Topology split

• Vertex refinement

- Normal based scheme

Topology split

Head model

Control mesh Subdivision surface

Solid star

Control mesh Subdivision surface

Butterfly subdivision surface Modified butterfly subd. surface

Knot surface

Control mesh

Butterfly subd. Normal based subd.

Summary

• Normal based subdivision - a geometric scheme

- tangent continuous- natural shape

• Contributions - normal refinement as well as vertex refinement- geometric dependent instead of parametric dependent

Thank you !