"Discrete conformal transform"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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The general idea of discrete differential geometry is to find and investigate discrete models that exibit properties and structures characterisitic of the corresponding smooth geometric objects. Several structure preserving definitions of discrete holomorphic functions and Riemann surfaces are known today. Linear theories are based on discrete Cauchy-Riemann equations. Nonlinear theories are based on patterns of circles or on a discrete notion of conformal equivalence for triangulated surfaces. In these lectures we introduce discrete versions of conformal structure, holomorphic functions, period matrix, discrete conformal metrics and other notions from the classical theory. We focus on proving discrete versions of the Riemann mapping theorem, classical uniformization theorems and on computation of periods of Riemann surfaces. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps and to convex variational principles. We establish a connection between conformal geometry for triangulated surfaces and the geometry of ideal hyperbolic polyhedra. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also allows to merge the theories of discretely conformally equivalent triangulated surfaces and of circle packings. Applications in geometry processing and computer graphics will be discussed. | The general idea of discrete differential geometry is to find and investigate discrete models that exibit properties and structures characterisitic of the corresponding smooth geometric objects. Several structure preserving definitions of discrete holomorphic functions and Riemann surfaces are known today. Linear theories are based on discrete Cauchy-Riemann equations. Nonlinear theories are based on patterns of circles or on a discrete notion of conformal equivalence for triangulated surfaces. In these lectures we introduce discrete versions of conformal structure, holomorphic functions, period matrix, discrete conformal metrics and other notions from the classical theory. We focus on proving discrete versions of the Riemann mapping theorem, classical uniformization theorems and on computation of periods of Riemann surfaces. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps and to convex variational principles. We establish a connection between conformal geometry for triangulated surfaces and the geometry of ideal hyperbolic polyhedra. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also allows to merge the theories of discretely conformally equivalent triangulated surfaces and of circle packings. Applications in geometry processing and computer graphics will be discussed. | ||
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| + | ==expositions== | ||
| + | * Stephenson, Kenneth. "Circle packing: a mathematical tale." Notices of the AMS 50.11 (2003): 1376-1388. | ||
| + | * http://www.math.fsu.edu/~aluffi/archive/paper356.pdf | ||
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| + | ==encyclopedia== | ||
| + | * http://en.wikipedia.org/wiki/Circle_packing_theorem | ||
[http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2698v1.pdf Discrete conformal maps and ideal hyperbolic polyhedra] | [http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2698v1.pdf Discrete conformal maps and ideal hyperbolic polyhedra] | ||
[[분류:integrable systems]] | [[분류:integrable systems]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:dimer model]] | [[분류:dimer model]] | ||
2014년 1월 6일 (월) 15:02 판
The general idea of discrete differential geometry is to find and investigate discrete models that exibit properties and structures characterisitic of the corresponding smooth geometric objects. Several structure preserving definitions of discrete holomorphic functions and Riemann surfaces are known today. Linear theories are based on discrete Cauchy-Riemann equations. Nonlinear theories are based on patterns of circles or on a discrete notion of conformal equivalence for triangulated surfaces. In these lectures we introduce discrete versions of conformal structure, holomorphic functions, period matrix, discrete conformal metrics and other notions from the classical theory. We focus on proving discrete versions of the Riemann mapping theorem, classical uniformization theorems and on computation of periods of Riemann surfaces. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps and to convex variational principles. We establish a connection between conformal geometry for triangulated surfaces and the geometry of ideal hyperbolic polyhedra. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also allows to merge the theories of discretely conformally equivalent triangulated surfaces and of circle packings. Applications in geometry processing and computer graphics will be discussed.
expositions
- Stephenson, Kenneth. "Circle packing: a mathematical tale." Notices of the AMS 50.11 (2003): 1376-1388.
- http://www.math.fsu.edu/~aluffi/archive/paper356.pdf