Search results for "Conformal"
showing 10 items of 234 documents
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…
Hom-Lie quadratic and Pinczon Algebras
2017
ABSTRACTPresenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.
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…
Geodesic flow of the averaged controlled Kepler equation
2008
A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to $\SS^2$ is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controll…
An Optimized Architecture for CGA Operations and Its Application to a Simulated Robotic Arm
2022
Conformal geometric algebra (CGA) is a new geometric computation tool that is attracting growing attention in many research fields, such as computer graphics, robotics, and computer vision. Regarding the robotic applications, new approaches based on CGA have been proposed to efficiently solve problems as the inverse kinematics and grasping of a robotic arm. The hardware acceleration of CGA operations is required to meet real-time performance requirements in embedded robotic platforms. In this paper, we present a novel embedded coprocessor for accelerating CGA operations in robotic tasks. Two robotic algorithms, namely, inverse kinematics and grasping of a human-arm-like kinematics chain, ar…
Implementation and evaluation of medical imaging techniques based on conformal geometric algebra
2020
Medical imaging tasks, such as segmentation, 3D modeling, and registration of medical images, involve complex geometric problems, usually solved by standard linear algebra and matrix calculations. In the last few decades, conformal geometric algebra (CGA) has emerged as a new approach to geometric computing that offers a simple and efficient representation of geometric objects and transformations. However, the practical use of CGA-based methods for big data image processing in medical imaging requires fast and efficient implementations of CGA operations to meet both real-time processing constraints and accuracy requirements. The purpose of this study is to present a novel implementation of …
Canal foliations of S 3
2012
The goal of the article is to classify foliations of S3 by regular canal surfaces, that is envelopes of one-parameter families of spheres which are immersed surfaces. We will add some extra information when the leaves are “surfaces of revolution” in a conformal sense.
Analysis of cardiac and pulmonary complication probabilities after radiation therapy for patients with early-stage breast cancer
2009
Objective. The purpose of this study was to evaluate the radiobiological implications of clinical use of respiratory-gated techniques for postoperative radiation therapy of early-stage left-sided breast cancer after breast-conserving surgery. Material and methods. Radiation therapy treatment plans of 80 patients with early-stage breast cancer (stage I–II), receiving whole breast irradiation after breast-conserving therapy, were analyzed. The control group consisting of 47 patients received standard radiation therapy, and the respiratory-gated group consisting of 33 patients received deep inspiration-gated radiation therapy. Normal tissue complication probabilities (NTCP) for cardiac mortali…
Analysis and evaluation of periodic physiological organ motion in radiotherapy treatments.
2003
Background and purpose: A system for the detection, measurement and analysis of the periodic physiological organ motion during radiotherapy treatment is proposed and clinically tested in this paper. Material and methods: The procedure is based on the acquisition of fluoroscopic sequences, followed by an automatic detection of the movement using cross-correlations with matched filters. Results: The system generates a probability density function (PDF) of finding a mobile organ in a position at a certain time. The maximum path of the mobile structures can be determined to define the planning target volume (PTV) without ambiguities. Conclusions: Physiological movements can be accurately includ…
Osculating spheres to a family of curves.
2021
The authors study the extrinsic conformal geometry of space forms involving pencils of circles or spheres. They consider curves orthogonal to a foliation of an open set of a 3-sphere by spheres and prove that the osculating spheres to the curves at points of a leaf form a pencil. They first prove the analogous result in a lower-dimensional case, that is, foliations of the 2-dimensional sphere and their orthogonal foliations. The 3-dimensional result, that is, the result for a foliation of (an open subset of) the 3-dimensional sphere by 2-dimensional spheres, is obtained using the de Sitter space, which is a model for the set of oriented spheres of the 3-dimensional sphere.