Cubic Bézier Triangular Patch with Shape Parameters

The tensor product Bézier surfaces are successfully used by many commercial CAD systems to model complicated surfaces. But the theory requires that all data have a rectangular geometry. This is indeed the case for some surfaces (e.g. the trunk lid of a car), but not for others (e.g. interior car body panels). Every surface can be covered with a triangular network instead of rectangular networks. So in computer aided geometric design, Bézier triangular surfaces have now become one of the major tools in outer shape design. In order to control the shape of triangular surfaces in geometric modeling, a set of cubic polynomial basis functions with shape parameters are constructed in this paper, which are the extensions of the cubic Bernstein bases over the triangular domain. A polynomial surface with shape parameters over triangular domain is defined by using the basis functions. We then show that such basis and surfaces share the same properties as the Bernstein basis and the Bézier surfaces in polynomial spaces respectively. And the cubic Bézier triangular surface is its special case. Because of the adjustable shape parameters, modification or deformation of the surface is more flexible. The larger the parameters are, the more the surface approaches to the control net. By changing the value of the shape parameters, we can get surfaces with different shapes in invariable control net. Examples show that this technique is effective in CAGD.