Bézier guarding: precise higher-order meshing of curved 2D domains

We present a mesh generation algorithm for the curvilinear triangulation of planar domains with piecewise polynomial boundary. The resulting mesh consists of regular, injective higher-order triangular elements and precisely conforms with the domain's curved boundary. No smoothness requirements are imposed on the boundary. Prescribed piecewise polynomial curves in the interior, like material interfaces or feature curves, can be taken into account for precise interpolation by the resulting mesh's edges as well. In its core, the algorithm is based on a novel explicit construction of guaranteed injective Bézier triangles with certain edge curves and edge parametrizations prescribed. Due to the use of only rational arithmetic, the algorithm can optionally be performed using exact number types in practice, so as to provide robustness guarantees.