The AROMATH seminar will usually happen on Tuesday at 10h30-11h30 every two weeks, except for a few deviations.
The presentations will typically take place at Inria Sophia Antipolis, Byron Blanc 106, and also online.
To join online, at https://cutt.ly/aromath or with a web browser at https://cutt.ly/aromath-web
use meeting ID: 828 5859 7791, passcode: 123
Clément Laroche - PhD defense - Compact and Efficient Implicit Algebraic Representations
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30 April 2020
Compact and Efficient Implicit Algebraic Representations
Clément Laroche
Thu. April 30th, 2020, 14:00-16:00(GR)=13:00-15:00(FR), twitch.tv and meet.jit.si
Abstract.
In 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. Hence, having both representations of the same objects at the same time maximizes the range of geometric operations. Switching from one representation to another is not easy and usually requires the use of algebraic properties. Thus, there is a strong link between algebra and geometry, captured by algebraic varieties: they are geometric objects described by an algebraic structure, the basis of Hilbert's "dictionary" and Bourbaki's works.
This thesis explores new kinds of implicit representations and algorithms for computing implicit them. We show that different methods are preferred in different situations, such as when it comes to the choice of an implicit representation among 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 always be best, we develop methods that fit to different kinds of information available about the object we wish to represent. For instance, 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. Similarly, objects constructed by sweeping a rigid body can be represented using that information.
