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Bicubic G 1 interpolation of arbitrary quad meshes using a 4-split S. Hahmann G.P. Bonneau B. Caramiaux CAI Hongjie Mar. 20, 2008 Geometric Modeling and Processing 2008

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Authors Stefanie Hahmann Main Posts Professor at Institut National Polytechnique de Grenoble (INPG), France Researcher at Laboratorie Jean Kuntzmann (LJK) Research CAGD Geometry Processing Scientific Visualization

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Authors Georges-Pierre Bonneau Main Posts Professor at Université Joseph Fourier Researcher at LJK Research CAGD Visualization

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Outline Applications of surface modeling Background Subdivision surface Global tensor product surface Locally constructed surface Circulant Matrices Vertex Consistency Problem Surface Construction by Steps

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Applications of Surface Modeling Medical imaging Geological modeling Scientific visualization 3D computer graphic animation

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A peep of HD 3D Animation From Appleseed EX Machina (2007)

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Subdivision Surface Doo-Sabin 细分方法 Catmull-Clark 细分方法 Loop 细分方法 Butterfly 细分方法 From PhD thesis of Zhang Jinqiao

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Locally Constructed Surface From S. Hahmann, G.P. Bonneau. Triangular G 1 interpolation by 4-splitting domain triangles

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Circulant Matrices Definition: A circulant matrix M is of the form Remark: Circulant matrix is a special case of Toeplitz matrix

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Circulant Matrices Property: Let f(x)=a 0 +a 1 x +…+ a n-1 x n-1, then eigenvalues, eigenvectors and determinant of M are Eigenvalues: Eigenvectors: Determinant:

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Examples of Circulant Matrices Determine the singularity of Solution : f(x)= x n-1,

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Examples of Circulant Matrices Compute the determinant of Compute the rank of

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Vertex Consistency Problem For C 2 surface assembling If G 1 continuity at boundary is satisfied, then

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Vertex Consistency Problem Twist compatibility for C 2 surface then

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Vertex Consistency Problem Matrix form It is generally unsolvable when n is even

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Sketch of the Algorithm Given a quad mesh To find 4 interpolated bi-cubic tensor surfaces for each patch with G 1 continuity at boundary

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Preparation: Simplification Simplification of G 1 continuity condition

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Choice of Let be constant, depended only on n (the order of vertex v ) Specialize G 1 continuity condition at u i = 0, then Non-trivial solution require

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Choice of Determine n i is the order of v i

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Step 1:Determine Boundary Curve Differentiate G 1 continuity equation and specialize at u i = 0, then Matrix form

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Examples of Circulant Matrices Determine the singularity of Solution : f(x)= x n-1,

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Step 1:Determine Boundary Curve Differentiate G 1 continuity equation and specialize at u i = 0, then Matrix form

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Step 1:Determine Boundary Curve Notations Selection of d 1, d 2

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Step 2:Twist Computations d 1, d 2 is in the image of T Determine the twist Determine

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Change of G 1 Conditions From To

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Step 3: Edge Computations Determine Determine V i (u i ) where V 0, V 1 are two n × n matrices determined by G 1 condition

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Step 3: Edge Computations Determine

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Step 4: Face Computations C 1 continuity between inner micro faces We choose A 1, A 2, A 3, A 4 as dof.

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Results

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Conclusions Suited to arbitrary topological quad mesh Preserved G 1 continuity at boundary Given explicit formulas Low degrees (bi-cubic) Shape parameters control is available

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Reference S. Hahmann, G.P. Bonneau, B. Caramiaux Bicubic G 1 interpolation of arbitrary quad meshes using a 4-split S. Hahmann, G.P. Bonneau Triangular G1 interpolation by 4-splitting domain triangles Charles Loop A G 1 triangular spline surface of arbitrary topological type S. Mann, C. Loop, M. Lounsbery, et al A survey of parametric scattered data fitting using triangular interpolants

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Thanks! Q&A

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