Computer aided geometric design with Powell-Sabin splines

Powell-Sabin splines are C-continuous quadratic splines defined on an arbitrary triangulation. Their construction is based on a particular split of each triangle in the triangulation into six smaller triangles. In this article we give an overview of the properties of Powell-Sabin splines in the context of computer aided geometric design. These splines can be represented in a compact normalized B-spline basis with an intuitive geometric interpretation involving control triangles. Using these triangles one can interactively change the shape of the splines in a predictable way. We describe the simple subdivision rules for Powell-Sabin splines, and discuss some applications. We consider a new efficient spline visualization technique based on subdivision. We also look at two useful generalizations of the Powell-Sabin splines, i.e., QHPS splines and NURPS surfaces. The QHPS splines are a hierarchical variant of Powell-Sabin splines. They have very similar properties as the Powell-Sabin splines, and their hierarchical nature allows a local refinement of the spline in a very straightforward way. The NURPS surface is the rational extension of the Powell-Sabin spline. By means of weights they give extra degrees of freedom to the designer for the modelling of surfaces.