Dr' Scott Schaefer

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Analysis of Subdivision Surfaces at
Extraordinary Vertices

• Dr. Scott Schaefer

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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces
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Structure of Subdivision Surfaces

• If ordinary case is smooth, then obviously entire
  surface is smooth except possibly at
  extraordinary vertices

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Smoothness of Surfaces

• A surface is a Ck manifold if locally the surface
  is the graph of a Ck function
• Must develop a local parameterization around
  extraordinary vertices to analyze smoothness

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Subdivision Matrices

• Encode local subdivision rules around
  extraordinary vertex

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Subdivision Matrix Example
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Subdivision Matrix Example

• Repeated multiplication by S performs subdivision
  locally
• Only need to analyze S to determine smoothness of
  the subdivision surface

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Smoothness at Extraordinary Vertices

• Reif showed that it is necessary for the
  subdivision matrix S to have eigenvalues of the
  form where for the
  surface to be C1 at the extraordinary vertex
• A sufficient condition for C1 smoothness is that
  the characteristic map must be regular and
  injective

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The Characteristic Map

• Let the eigenvalues of S be of the form
• where .
• The eigenvectors associated with provide a
  local parameterization around the extraordinary
  vertex

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The Characteristic Map
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The Characteristic Map
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The Characteristic Map
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The Characteristic Map
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Analyzing Arbitrary Valence

• Matrices become very large, very quickly
• Must analyze every valence independently
• Need tools for somehow analyzing
  eigenvalues/vectors of arbitrary valence easily

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Structure of Subdivision Matrices
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Structure of Subdivision Matrices
Circulant matrix
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Circulant Matrices

• Matrix whose rows are horizontal shifts of a
  single row

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Eigenvalues/vectors of Circulant Matrices

• Given an circulant matrix with rows
  associated with c(x), its eigenvalues are of the
  form and has eigenvectors
• where and

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Eigenvalues/vectors of Circulant Matrices

• Given an circulant matrix with rows
  associated with c(x), its eigenvalues are of the
  form and has eigenvectors
• where and

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Eigenvalues/vectors of Circulant Matrices

• Given an circulant matrix with rows
  associated with c(x), its eigenvalues are of the
  form and has eigenvectors
• where and

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Block-Circulant Matrices

• Matrix composed of circulant matrices

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Block-Circulant Matrices

• Matrix composed of circulant matrices

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Block-Circulant Matrices

• Matrix composed of circulant matrices

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Eigenvalues/vectors ofBlock-Circulant Matrices

• Find eigenvalues/vectors of block matrix

eigenvectors
eigenvalues
inverse of eigenvectors
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Eigenvalues/vectors ofBlock-Circulant Matrices

• Find eigenvalues/vectors of block matrix
• Eigenvalues of block matrix are eigenvalues of
  expanded matrix evaluated at

eigenvectors
eigenvalues
inverse of eigenvectors
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Eigenvalues/vectors ofBlock-Circulant Matrices

• Find eigenvalues/vectors of block matrix
• Eigenvalues of block matrix are eigenvalues of
  expanded matrix evaluated at
• Eigenvectors of block matrix are multiples of
  
• times eigenvectors of block
  matrix

eigenvectors
eigenvalues
inverse of eigenvectors
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Eigenvalues/vectors ofBlock-Circulant Matrices
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Eigenvalues/vectors ofBlock-Circulant Matrices
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Example Loop Subdivision
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Example Loop Subdivision
Some parts of the matrix are not circulant
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Example Loop Subdivision

• Eigenvectors/values for block-circulant portion
  are eigenvectors/values for entire matrix except
  at j0

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Example Loop Subdivision
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Example Loop Subdivision
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Example Loop Subdivision
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Example Loop Subdivision

• Subdominant eigenvalue is
• Corresponding eigenvector is

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Example Loop Subdivision

• Subdominant eigenvalue is
• Corresponding eigenvector is
• Plot real/imaginary parts to create char map

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ExampleLoop Subdivision
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Application Exact Evaluation
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Application Exact Evaluation

• Subdivide until x is in
• ordinary region

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Application Exact Evaluation

• Subdivide until x is in
• ordinary region

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Application Exact Evaluation

• Subdivide until x is in
• ordinary region

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Application Exact Evaluation

• Subdivide until x is in
• ordinary region
• Extract B-spline control
• points and evaluate at x

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Application Exact Evaluation

• Subdivide until x is in
• ordinary region
• Extract B-spline control
• points and evaluate at x