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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. Engineering. CAGD. mathematics. Content. What is subdivision? - corner cutting algorithms - interpolating subdivision Normal based subd. Scheme
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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 Engineering CAGD 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
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 Parametric domain Control mesh
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
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
Bottle design Subdivision curve Control polygon
For surface design • Triangular control mesh • Topology split • Vertex refinement - Normal based scheme
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