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

The border of the macropatches 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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