Search results for "Space of spheres"
showing 4 items of 4 documents
The non-degenerate Dupin cyclides in the space of spheres using Geometric Algebra
2012
International audience; Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and \textcolor{black}{were} introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a one-parameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the non-degenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Alge…
Iterative construction of Dupin cyclides characteristic circles using non-stationary Iterated Function Systems (IFS)
2012
International audience; A Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by J.C. Fiorot. As we su…
Darboux curves on surfaces I
2017
International audience; In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Mobius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable w…
Blending pieces of Dupin cyclides for 3D modeling and reconstruction : study in the space of spheres
2013
The thesis deals with the blending of canal surfaces in geometric modeling using pieces of Dupin Cyclides. We try to solve a problem of reconstructing real parts manufactured and controlled by the CEA of Valduc. Using the space of spheres in which we can manipulate both points, spheres and canal surfaces, we simplify some problems. This space is represented by a 4-dimensional quadric in a 5-dimensional space, equipped with the Lorentz form, it is the Lorentz space. In the space of spheres, problems of blending canal surfaces by pieces of Dupin cyclides are simplified in linear problems. We give algorithms to make such blends using the space of spheres and after we come back to 3 dimensions …