Hierarchical Triangular Splines
Authors: Stefanie Hahmann, Georges-Pierre Bonneau, Alex Yvart
Hierarchical Triangular Splines
Alex Yvart, Stefanie Hahmann, Georges-Pierre Bonneau,,
ACM Transactions on Graphics 24(4), pp. 1374-1391 (2005)
Smooth Adaptive Fitting of 3D models using hierarchical triangular splines
Alex Yvart, Stefanie Hahmann, Georges-Pierre Bonneau ,
International Conference on Shape Modeling and Applications, SMI'05, 13-22 (2005)
MIT Boston, IEEE Computer Society Press
Previous related works:
Polynomial Surfaces interpolating arbitrary triangulations
Stefanie Hahmann, Georges-Pierre Bonneau ,
IEEE Transactions on Visualisation and Computer Graphics 9 (1), p. 99-109, (2003)
Triangular G1 interpolation by 4-splitting domain triangles
Stefanie Hahmann, Georges-Pierre Bonneau,
CAGD 17 (8), 731-757 (2000)
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
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
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
| 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).
Stefanie Hahmann's homepage.