Hierarchical Triangular Splines

Authors: Stefanie Hahmann, Georges-Pierre Bonneau, Alex Yvart


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Smooth parametric surfaces interpolating triangular meshes are very useful for modeling surfaces of arbitrary topology. Several interpolants based on this kind of surfaces have been developed over the last fifteen years. However, with current 3D acquisition equipments, models are becoming more and more complex. Since previous interpolating methods lack a local refinement property, there is no way to locally adapt the level of detail. In this paper, we introduce a hierarchical triangular surface model. The surface is overall tangent plane continuous and is defined parametrically as a piecewise quintic polynomial. It can be adaptively refined while preserving the overall tangent plane continuity. This model enables designers to create a complex smooth surface composed of a small number of patches, to which details can be added by locally refining the patches until an arbitrary small size is reached. It is implemented as a hierarchical data structure where the top layer describes a coarse, smooth base surface, and the lower levels encode the details in local frame coordinates.

Local refinement step of hierarchical spline surface

This fgure illustrates one refnement step starting with the base surface, though the method is based on successive refnements as well. Figure (a) shows the base surface with bold curves highlighting the mapping of the two domain triangles, which are shown on the top right. The different colors correspond to the different triangular Bezier patches on the surface. Recall that each domain triangle is mapped to a group of 4 Bezier patches, referred to as a macro-patch.
The refnement procedure subdivides the two domain triangles as shown in (b) on the top right. The gray area maps to the surface region surrounded by vertices P1; P2; P3; P4; P5; P6, see Figure (b). The refnement procedure replaces this region, which is initially composed of 6 triangular Bezier patches, by 6 macro- patches, i.e. 24 triangular Bezier patches interpolating the new editable vertex (see Figure (d, e) as an example). Outside this region, the surface is not modifed. This implies in particular that the new surface portion that is computed has to have the same curve and the same tangent planes along its boundary in order to fill in smoothly.

Modeling system

Different interactive modeling tools are supported by the hierarchical triangular spline. The surface model itself offers some degrees of freedom that can easily be made available to the designer as intuitive design handles. For example, the editable vertices that are interpolated by the surface can be picked on the surface and displaced while the surrounding surface is following continuously. Furthermore at each editable vertex, all the tangent directions of the incoming patch boundary curves are free, but subject to lie in the same plane. They define the surface tangent plane and the normal vector at these points. By offering the designer the possibility to interact directly with these geometric quantities, several design effects can be obtained. Modifying the normal direction gives a new orientation to the tangent plane, while modifying the length of the normal vector has a tension effect influencing the local curvature. A twisting effect is obtained by rotating the tangent plane. Some of these design tools are illustrated together with the modeler in the above Figure.


base surface and successive deformations at level 0,1,2,3,4.

The border of the macro-patches are drawn in blue.

The different color coding corresponds to different refinement levels.

Hierarchicl deformations To finalize this dog’s head, four fangs are added to the mouth. The mouth can later be closed by editing only one vertex high enough in the hierarchy (at the end of the muzzle).

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