Normal based subdivision scheme for curve and surface design

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Normal based subdivision scheme for curve and
surface design

• ???
• 2004.12
• http//www.math.zju.edu.cn/yxn

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What is CAGD
Computer science
Engineering
CAGD
mathematics
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Content

• What is subdivision?
• - corner cutting algorithms
• - interpolating subdivision
• Normal based subd. Scheme
• - the scheme
• - for curve design
• - for surface design
• Summary

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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

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Corner cutting algorithms

• Corner cutting Chaikin, B-spline
• Convergence de Boor, Riesenfeld, Gregory, et al

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Subdivision of B-spline

• Uniform cubic B-spline
• Derive the rule by knots insertion

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Arbitrary control mesh

• The topological rule
• The geometric rule

Catmull-clark scheme
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Catmull-clark subdivision surface
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Interpolating subdivision

• Edge split
• Vertex refinement

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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
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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
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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

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Content

• What is subdivision?
• - corner cutting algorithms
• - interpolating subdivision
• Normal based subd. Scheme
• - the scheme
• - for curve design
• - for surface design
• Summary

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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

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The basic scheme
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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

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Normal computation
Curve case
Surface case
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Convergence

• Active chord tangent angles
• - converge to zero
• - within fixed scale
• Fixed chord tangent angles
• - are bounded
• - convergence
• Polygon series
• - converge
• - tangent continuous

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For curve design

• The freedoms
• - subd. ratio of edges
• - scale for displacement vector
• Shape preserving
• - same scheme
• - explicit choices of freedoms

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Shape preserving scheme
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Freeform curve
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Bottle design
Subdivision curve
Control polygon
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For surface design

• Triangular control mesh
• Topology split
• Vertex refinement
• - Normal based scheme

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Topology split
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Head model
Control mesh
Subdivision surface
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Solid star
Control mesh
Subdivision surface
Butterfly subdivision surface
Modified butterfly subd. surface
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Knot surface
Control mesh
Butterfly subd.
Normal based subd.
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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

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• Thank you !