Polynomial surfaces interpolating arbitrary triangulations

Stefanie Hahmann 1, * Georges-Pierre Bonneau 2
* Corresponding author
2 iMAGIS - Models, Algorithms and Geometry for Computer Generated Image Graphics
GRAVIR - IMAG - Graphisme, Vision et Robotique, Inria Grenoble - Rhône-Alpes
Abstract : Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occur with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: the tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.
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Stefanie Hahmann, Georges-Pierre Bonneau. Polynomial surfaces interpolating arbitrary triangulations. IEEE Transactions on Visualization and Computer Graphics, Institute of Electrical and Electronics Engineers, 2003, 9 (1), pp.99-109. ⟨hal-00319652⟩

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