Normal based subdivision scheme for curve and surface design 杨勋年

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 refinement for 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 mesh Parametric 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 caseSurface 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

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 meshSubdivision surface Butterfly subdivision surfaceModified 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 !