Triangulations with Circular and Parabolic Arcs

In this paper, we expand circular arc triangulations with parabolic arcs, which are triangulations whose edges are circular or parabolic arcs. Creating triangulations this way bears several advantages if angles are to be optimized, especially for finite element method. A typical reason for the occurrence of small angles in a straight line triangulation is that the geometry of the vertex set forces slim triangles in the vicinity of the domain boundary. Such angles can be enlarged, by optimizing the arc curvatures for the given triangulation. With arc triangulation we can improve straight line triangulation such that the result of finite element method with the same account of bivariate functions is better. We compare parabolic and circular arc triangulations. We represent circular arc triangles as rational Bézier triangles and parabolic arc triangles as polynomial Bézier triangles. Clearly, the computing of finite elements with rational Bézier triangles is computationally more demanding than with parabolic arc triangles represented with polynomial Bézier triangles. Maximizing the smallest angle in a combinatorially fixed arc triangulation of a point set can be formulated as a linear program for circular arc and as a nonlinear program for parabolic arcs. This guarantees a fast solution of the optimization problem. We believe that arc triangulations constitute a useful tool especially in finite element methods.