Bicubic G1 interpolation of arbitrary quad meshes using a 4-split Geometric Modeling and Processing 2008 Bicubic G1 interpolation of arbitrary quad meshes using a 4-split S. Hahmann G.P. Bonneau B. Caramiaux CAI Hongjie Mar. 20, 2008

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

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

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

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

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

Subdivision Surface From PhD thesis of Zhang Jinqiao 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

Locally Constructed Surface From S. Hahmann, G.P. Bonneau. Triangular G1 interpolation by 4-splitting domain triangles

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

Circulant Matrices Property: Let f(x)=a0+a1x +…+ an-1xn-1, then eigenvalues, eigenvectors and determinant of M are Eigenvalues: Eigenvectors: Determinant:

Examples of Circulant Matrices Determine the singularity of Solution: f(x)=0.5+0.5xn-1,

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

Vertex Consistency Problem For C2 surface assembling If G1 continuity at boundary is satisfied, then

Vertex Consistency Problem Twist compatibility for C2 surface then

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

Sketch of the Algorithm Given a quad mesh To find 4 interpolated bi-cubic tensor surfaces for each patch with G1 continuity at boundary

Preparation: Simplification Simplification of G1 continuity condition

Choice of Let be constant, depended only on n (the order of vertex v) Specialize G1 continuity condition at ui=0, then Non-trivial solution require

Choice of Determine ni is the order of vi

Step 1:Determine Boundary Curve Differentiate G1 continuity equation and specialize at ui=0, then Matrix form

Examples of Circulant Matrices Determine the singularity of Solution: f(x)=0.5+0.5xn-1,

Step 1:Determine Boundary Curve Differentiate G1 continuity equation and specialize at ui=0, then Matrix form

Step 1:Determine Boundary Curve Notations Selection of d1,d2

Step 2:Twist Computations d1,d2 is in the image of T Determine the twist Determine

Change of G1 Conditions From To

Step 3: Edge Computations Determine Determine Vi(ui) where V0,V1 are two n×n matrices determined by G1 condition

Step 3: Edge Computations Determine

Step 4: Face Computations C1 continuity between inner micro faces We choose A1,A2,A3,A4 as dof.

Results

Results

Conclusions Suited to arbitrary topological quad mesh Preserved G1 continuity at boundary Given explicit formulas Low degrees (bi-cubic) Shape parameters control is available

Reference S. Hahmann, G.P. Bonneau, B. Caramiaux Bicubic G1 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 G1 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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