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PTOPO is a maple package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second. Comparison of maple parametric plot vs PTOPO

No abstract

Voronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axisaligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly useful scenario in certain application domains. We construct the exact Voronoi diagram inside an orthogonal polyhedron with holes defined by such polyhedra. Our approach avoids creating full-dimensional elements on the Voronoi diagram and yields a skeletal representation of the input object. We introduce a complete algorithm in 2D and 3D that follows the subdivision paradigm relying on a bounding-volume hierarchy; this is an original approach to the problem. The complexity is adaptive and comparable to that of previous methods. Under a mild assumption it is O(n/∆ + 1/∆ 2 ) in 2D or O(nα 2 /∆ 2 + 1/∆ 3 ) in 3D, where n is the number of sites, namely edges or facets resp., ∆ is the maximum cell size for the subdivision to stop, and α bounds vertex cardinality per facet. We also provide a numerically stable, open-source implementation in Julia, illustrating the practical nature of our algorithm.