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Loic Puech

Bio: Loic Puech is an academic researcher from University of Burgundy. The author has contributed to research in topics: Right triangle & Villarceau circles. The author has an hindex of 1, co-authored 1 publications receiving 2 citations.

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TL;DR: This work proposes an indirect algorithm which constructs a one-parameter family of 3D circular edge triangles lying on Dupin cyclides, as the image of a circle by a carefully chosen inversion is a circle, and by constructing different images of a right triangle on a ring torus.
Abstract: Ring Dupin cyclides are non-spherical algebraic surfaces of degree four that can be defined as the image by inversion of a ring torus. They are interesting in geometric modeling because: (1) they have several families of circles embedded on them: parallel, meridian, and Yvon-Villarceau circles, and (2) they are characterized by one parametric equation and two equivalent implicit ones, allowing for better flexibility and easiness of use by adopting one representation or the other, according to the best suitability for a particular application. These facts motivate the construction of circular edge triangles lying on Dupin cyclides and exhibiting the aforementioned properties. Our first contribution consists in an analytic method for the computation of Yvon-Villarceau circles on a given ring Dupin cyclide, by computing an adequate Dupin cyclide-torus inversion and applying it to the torus-based equations of Yvon-Villarceau circles. Our second contribution is an algorithm which, starting from three arbitrary 3D points, constructs a triangle on a ring torus such that each of its edges belongs to one of the three families of circles on a ring torus: meridian, parallel, and Yvon-Villarceau circles. Since the same task of constructing right triangles is far from being easy to accomplish when directly dealing with cyclides, our third contribution is an indirect algorithm which proceeds in two steps and relies on the previous one. As the image of a circle by a carefully chosen inversion is a circle, and by constructing different images of a right triangle on a ring torus, the indirect algorithm constructs a one-parameter family of 3D circular edge triangles lying on Dupin cyclides.

4 citations


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Journal ArticleDOI
TL;DR: This article shows that a new re-parameterization for reducing and unlocking irreducible geometric systems is done at the lowest level, at the linear algebra routines level, so that numerous solvers widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications.
Abstract: You recklessly told your boss that solving a non-linear system of size n ( n unknowns and n equations) requires a time proportional to n , as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton-Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k ? n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton-Raphson, homotopy, and also p -adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k ? n key unknowns, the cost of a Newton iteration becomes O ( k n 2 ) instead of O ( n 3 ) . Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems. A new re-parameterization for reducing and unlocking irreducible geometric systems.No need for the values of the key unknowns and no limit on their number.Enabling the usage of decomposition methods on irreducible re-parameterized systems.Usage at the lowest linear Algebra level and significant performance improvement.Benefits for numerous solvers (Newton-Raphson, homotopy, p -adic methods, etc.)

5 citations

29 Jul 2015
TL;DR: In this article, the authors applied operations that are known to create effective artworks on torus to Dupin cyclides, and proved them to be feasible, which can be generalized to explore transformations of other mathematical objects under sphere inversion.
Abstract: A torus contains four families of circles: parallels, meridians and two sets of Yvon-Villarceau circles. Craftworks and artworks based on Yvon-Villarceau circles can be very attractive. Dupin cyclides are images of tori under sphere inversion, so they contain the images of the torus circles families. I applied operations that are known to create effective artworks on tori to Dupin cyclides, and proved them to be feasible. The regularity and the hidden complexity of the objects I obtained make them very attractive. Reviving the 19th century's tradition of mathematical models making, I printed several models, which can help in understanding their geometry. The tools I developed can be generalized to explore transformations of other mathematical objects under sphere inversion. This exploration is just at its beginning, but has already produced interesting new objects.

1 citations

Journal ArticleDOI
01 Sep 2022
TL;DR: In this article , the topology of the intersection curve is reduced to the arrangements of the main circle of the torus and another pair of circles, which can be characterized by a simple algebraic sequence.
Abstract: The surface–surface intersection is a fundamental task in CAD/CAM. We present the classification and a full enumeration of the topology of all non-degenerate intersections of two Dupin cyclides. Inversion geometry is first used to transform the intersection of two Dupin cyclides to that of a cyclide and a torus. Then the topology of the intersection curve is reduced to the arrangements of the main circle of the torus and another pair of circles, which can be characterized by a simple algebraic sequence. Based on the classification and enumeration results, an efficient determination algorithm for the topology of two Dupin cyclides is also provided.
28 Dec 2022
TL;DR: In this article , the algebraic conditions for recognition of Dupin cyclides among the general implicit form of Darboux cyclides are derived and a set of algebraic equations on the coefficients of the implicit equation, each such set defining a complete intersection (of codimension 4) locally.
Abstract: Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and to construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather general surfaces in $\mathbb R^3$ of degree 3 or 4. This article derives the algebraic conditions for recognition of Dupin cyclides among the general implicit form of Darboux cyclides. We aim at practicable sets of algebraic equations on the coefficients of the implicit equation, each such set defining a complete intersection (of codimension 4) locally. Additionally, the article classifies all real surfaces and lower dimensional degenerations defined by the implicit equation for Dupin cyclides.