Short Description

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. In this tutorial we first explain the mathematics necessary to construct a G1 joint of an arbitrary number of polynomial patches around a common vertex. We then show how to constuct a complete G1 surface of arbitrary topology which interpolates a given surface mesh. Since previous interpolating methods lack a local refinement property, there is no way to locally adapt the level of detail. We therefore 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.

Course info

Authors: Stefanie Hahmann, Georges-Pierre Bonneau

Prerequisites: Geometric Modeling, Basic CAGD knowledge.

Description: The course treats specialized shape modeling techniques:

G1 continuity condiditons
parametric triangular Bezier patches
interpolation of triangular surface meshes


Literature:
"Hierarchical Triangular Splines", Alex Yvart, Stefanie Hahmann, Georges-Pierre Bonneau, ACM Transactions on Graphics 24 (4), 1374-1391, (2005).

"Polynomial Surfaces interpolating arbitrary triangulations", Stefanie Hahmann, Bonneau G.-P., IEEE Transactions on Visualisation and Computer Graphics 9 (1), p. 99-109, (2003).

"Polyhedral Modeling", Bonneau G-P., Hahmann S., IEEE Proceedings Visualization'00, pp. 381-387 (2000).

"Triangular G1 interpolation by 4-splitting domain triangles", Stefanie Hahmann, Bonneau G.-P., CAGD 17 (8), 731-757 (2000).

Format: Slides in pdf-format.

Course material
Tutorial