Geometric Rounding and Feature Separation in Meshes

Geometric rounding of a mesh is the task of approximating its vertex coordinates by floating point numbers while preserving mesh structure. Geometric rounding allows algorithms of computational geometry to interface with numerical algorithms. We present a practical geometric rounding algorithm for 3D triangle meshes that preserves the topology of the mesh. The basis of the algorithm is a novel strategy: 1) modify the mesh to achieve a feature separation that prevents topology changes when the coordinates change by the rounding unit; and 2) round each vertex coordinate to the closest floating point number. Feature separation is also useful on its own, for example for satisfying minimum separation rules in CAD models. We demonstrate a robust, accurate implementation

Robust free space construction for a polyhedron with planar motion

We present a free space construction algorithm for a polyhedron that translates in the xy plane and rotates around its z axis, relative to a stationary polyhedron. We employ the proven paradigm of constructing the configuration space subdivision defined by patches that comprise the configurations where the boundary features of the polyhedra are in contact. We implement the algorithm robustly and efficiently. The challenge is to detect degenerate predicates efficiently and to handle them correctly. We use our ACP (Adaptive Controlled Perturbation) robustness strategy to prevent degenerate predicates due to input in special position. The remaining cases are predicates that are identical to the zero polynomial because their arguments are derived from overlapping sets of input vertices. We detect and handle these cases with custom logic. We validate the implementation by computing maximum clearance paths. •Kinematics of a polyhedron that moves in a plane relative to a stationary polyhedron.•Both Polyhedrons can be multiply connected and non-convex.•Configuration space construction algorithm is presented.•Algorithm is implemented robustly and efficiently.•Implementation is validated by computing maximal clearance paths