Surfaces from Curves on Triangular Surfaces in Barycentric Coordinates

Abstract

Barycentric coordinates are coordinates in which a position is provided by a blending of a weighted point set where the weights sum up to 1. Bezier-triangles and ERBS-triangles are typical examples of use of Barycentric coordinates.

We look at the framework for the description of curves on surfaces that are described in Barycentric coordinates and how we define surfaces in a Coons Patch like framework with the use of these curves on surfaces. The framework also includes pre-evaluation and other optimization technics for evaluation.

The background is to construct large complex surfaces. Given a surface constructed by a connected set of non-planar triangular surfaces. If the triangular surfaces are generalized expo-rational B-spline based, constructed by blending of triangular sub-surfaces from Bezier-patches, then the surface is smooth at the vertices but only continuous over the edges between the triangular surfaces. If we introduce a second set of vertices defined by the midpoint of each triangular surface, we can introduce a new set of edges constructed by straight lines from a vertex to the midpoint in the parameter plane of the respective triangular surface. In addition we also have information about the derivatives across these edges. This gives us the data to make a connected and smooth set of surfaces that are strongly connected to the set of triangular surfaces. The triangle based surface is easy to manipulate and reshape and then the smooth dual set of squared surfaces will automatically be updated.