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Theses

Compact and efficient implicit representations

Clément Laroche 1, 2
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
Abstract : In the perspective of manipulating geometric objects, there exists two main representations of curves and surfaces: parametric and implicit representations. Both are useful for different purposes and thus complement each other. Parametric representations are efficient in sampling points on an object; implicit representations are efficient in determining whether a point belongs to an object or not. Because of that, having both representations of the same objects at the same time maximizes the range of operations one can do with geometric objects. Switching from one representation to another is not an easy task. It usually requires the use of algebraic properties. Thus, there is a strong link between algebra and geometry, symbolised by the algebraic varieties: they are geometric objects described by an algebraic structure. This thesis explores new kinds of implicit representations and algorithms for computing implicit representations. We show that different methods are adapted to different situations even when it comes to the choice of an implicit representation amongst several possibilities. Space curves can thus be described implicitly by conical surfaces, moving lines and/or moving quadrics... each description having different geometrical properties and practical usage. As there is not one implicit representation or implicitization algorithm that would be the best in any situation, we develop methods that fit to different kinds of informations known about the object we want to represent. As we show, objects constructed by sweeping a rigid body can be represented using the knowledge of that nature. Similarly, very particular curves may have a complicated algebraic structure. Depending on our tolerance to approximation, such curves can thus be perturbed to simplify greatly their algebraic structure or, on the contrary, be represented by a rich implicit representation format.
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https://hal.inria.fr/tel-03117752
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Clément Laroche. Compact and efficient implicit representations. Algebraic Geometry [math.AG]. Université d'Athènes, 2020. English. ⟨tel-03117752⟩

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