Parameterization of planar curves immersed in triangulations with application to finite elements

摘要

We construct a method for the parameterization of a class of planar piecewise C2-curves over a collection of edges in an ambient triangulation. The map from the collection of edges to the curve is the closest-point projection. A distinguishing feature of the method is that edges in the ambient triangulation need not interpolate the curve. We formulate conditions on the ambient triangulations so that the resulting parameterization over its selected edges is (i) bijective, (ii) maps simple, connected collection of edges to simple, connected components of the curve, and (iii) is C1 within each edge of the collection. These properties of the parameterization make it particularly useful in the construction of high-order finite element approximation spaces on planar curves immersed in triangulations. We discuss this application and illustrate it with numerical examples. The parameterization method applies to a large class of planar curves, including most ones of interest in engineering and computer graphics applications, and to a large family of triangulations, including acute-angled triangulations.