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Reliable location with respect to the projection of a smooth space curve

Rémi Imbach 1 Guillaume Moroz 1 Marc Pouget 1
1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry, Inria Nancy - Grand Est
Abstract : Consider a plane curve B defined as the projection of the intersection of two analytic surfaces in R 3 or as the apparent contour of a surface. In general, B has node or cusp singular points and thus is a singular curve. Our main contribution is the computation of a data structure answering point location queries with respect to the subdivision of the plane induced by B. This data structure is composed of an approximation of the space curve together with a topological representation of its projection B. Since B is a singular curve, it is challenging to design a method only based on reliable numerical algorithms. In a recent work, the authors show how to describe the set of singularities of B as regular solutions of a so-called ball system suitable for a numerical subdivision solver. Here, the space curve is first enclosed in a set of boxes with a certified path-tracker to restrict the domain where the ball system is solved. Boxes around singular points are then computed such that the correct topology of the curve inside these boxes can be deduced from the intersections of the curve with their boundaries. The tracking of the space curve is then used to connect the smooth branches to the singular points. The subdivision of the plane induced by B is encoded as an extended planar combinatorial map allowing point location. We experimented our method and show that our reliable numerical approach can handle classes of examples that are not reachable by symbolic methods.
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Rémi Imbach, Guillaume Moroz, Marc Pouget. Reliable location with respect to the projection of a smooth space curve. [Research Report] INRIA. 2017. ⟨hal-01632344⟩

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