Conformal Geometry Computational Algorithms And Engineering Applications Pdf

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Conformal geometric algebra

Use a web browser to open the live streaming link online. Public to general audience for free Abstract: This course will cover fundamental concepts and theorems in algebraic topology, surface differential geometry, Riemann surface theory and geometric partial differential equations; it also covers the computational methods for surface fundamental group, homology group, harmonic maps, meromorphic differentials, foliation, conformal mapping, quasi-conformal mapping and Ricci flow. Assignments: This lecture series will offer elementary library, the students are encouraged to implement some of the fundamental algorithms, such as computational topology, harmonic map and optimal transport map. Teaching assistants will answer the equestions and offer some helps for coding. Online Demo: The online demo is written using WebGL, you can observe the conformal mappings of 3D surfaces interactively. More videos, images can be accessed by scanning the bar codes in the text book. Lecture Slides The lecture slides of the previous year can be found here.


Nina Amenta Title: Rigidity and Deformation Abstract: A collection of related shapes - the same bone from different species, the same organ from different individuals, a person in different poses - can tell a story about evolution, or disease, or emotion. Defining shape deformation in a way that is amenable to statistical analysis is an old challenge and still an active research are. The opposite of deformation is rigidity, and in this talk, we explore the relationship of rigidity and deformation. Herman Gluck proved in that almost all triangulated surface meshes in three-dimensional space are rigid; that is, only in very special cases is there a motion where the edge lengths remain fixed but the dihedral angles between adjacent triangles can vary smoothly. We show that motions fixing the dihedrals and allowing edge lengths to change smoothly are similarly rare. This implies at least a local mapping between configurations of the triangle mesh and vectors of dihedrals, and it raises the interesting question of what other quantities we could assign to mesh edges to produce a parameterization of the shape space achievable by a given mesh topology.

Computational Conformal Geometry Applied in Engineering Fields

Jin, X. Gu, Y. He, and Y.

This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations. The effect of the mapping is that generalized i. In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D , which combine very efficiently. A consequence of rotors representing transformations is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly.


Conformal Geometry

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 CGA-based medical imaging techniques that makes them effective and practically usable. The paper exploits a new simplified formulation of CGA operators that allows significantly reduced execution times while maintaining the needed result precision.

It seems that you're in Germany. We have a dedicated site for Germany. Authors: Jin , M. This book offers an essential overview of computational conformal geometry applied to fundamental problems in specific engineering fields. It introduces readers to conformal geometry theory and discusses implementation issues from an engineering perspective. The respective chapters explore fundamental problems in specific fields of application, and detail how computational conformal geometric methods can be used to solve them in a theoretically elegant and computationally efficient way. The fields covered include computer graphics, computer vision, geometric modeling, medical imaging, and wireless sensor networks.

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    Conformal geometry studies conformal structure of general surfaces.

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    This book offers an essential overview of computational conformal geometry applied to fundamental problems in specific engineering fields.