arXiv subjects

Jean Michel Menjanahary

Publications and source records attributed to Jean Michel Menjanahary.

5 recordsLinked to original sources

arXiv2025

Formula for Dupin Cyclidic Cube and Miquel Point

Dupin cyclides are surfaces conformally equivalent to a torus, a circular cone, or a cylinder. Their patches admit rational bilinear quaternionic Bézier parametrizations and are used in geometric design and architecture. Dupin cyclidic cubes are a natural trivariate generalization of Dupin cyclide patches. In this article, we derive explicit formulas for control points and weights of rational trilinear quaternionic Bézier parametrizations of Dupin cyclidic cubes and relate them with the classical construction of the Miquel point.

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arXiv2024

Classification of Dupin Cyclidic Cubes by Their Singularities

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.

math.AG↗

arXiv2024

Dupin cyclides passing through a fixed circle

We derive algebraic equations on the coefficients of the implicit equation to characterize all Dupin cyclides passing through a fixed circle. The results are applied to solve the basic problems in CAGD about blending of Dupin cyclides along circles.

math.AG↗

arXiv2023

Dupin Cyclides as a Subspace of Darboux Cyclides

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.

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arXiv2023

Using Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.

math.AG↗