Computing Discrete Minimal Surfaces and Their Conjugates

Pinkall, Ulrich; Polthier, Konrad

doi:10.1080/10586458.1993.10504266

Cited by 1,074 publications

(829 citation statements)

References 12 publications

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“…These boundary points are not different from other boundary points. Our boundary conditions are much closer to the Riemann theorem than those of other classical discrete conformal algorithms: [7,16] fix all the boundary points and [5,12] fix two boundary points (that accumulate conformal distortion) but the other ones are not mapped on the circle.…”

Section: Connection To Real Continuous Theorymentioning

confidence: 99%

“…In this section, we will show that our method is a generalization of parameterizations with the cotan conformal coordinates [5,16]. We remind the reader that this method applies to triangular meshes and consists in 1. fixing the images of the boundary points, often on a convex boundary, 2. solving the following system: for each vertex v i which is not on the boundary of the mesh…”

Section: Connection To the Cotan Conformal Coordinatesmentioning

confidence: 99%

“…Many parameterizations methods, including [5,7,12,16], consist in solving sparse linear systems. As the system of equations (2) is also sparse, we could think of using similar techniques.…”

Section: Energy Minimizationmentioning

confidence: 99%

“…Although, it was first used for meshes, with floating point coordinates, it has the main advantage of being easily adaptable to digital surfaces, with integer coordinates. We show that, in the case of triangular meshes, it is a generalization of the cotan conformal coordinates method [16] and establish a discrete version of the Riemann mapping theorem. The boundary conditions introduced are closer to the real continuous theorem than those of classical conformal methods.…”

Section: Introductionmentioning

confidence: 99%

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Smooth 2D Coordinate Systems on Discrete Surfaces

Cartade

Malgouyres

Mercat

et al. 2011

Discrete Geometry for Computer Imagery

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Abstract. We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.

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Section: Connection To Real Continuous Theorymentioning

confidence: 99%

Section: Connection To the Cotan Conformal Coordinatesmentioning

confidence: 99%

“…Many parameterizations methods, including [5,7,12,16], consist in solving sparse linear systems. As the system of equations (2) is also sparse, we could think of using similar techniques.…”

Section: Energy Minimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Smooth 2D Coordinate Systems on Discrete Surfaces

Cartade

Malgouyres

Mercat

et al. 2011

Discrete Geometry for Computer Imagery

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show abstract

“…Now let us take the domain D to be the domain enclosed by our cage P , and let u(η) = η, that is the coordinate functions, in (4). Note that here we take u as the vector function u = ξ :…”

Section: Derivation Of Green Coordinatesmentioning

confidence: 99%

Derivation and Analysis of Green Coordinates

Lipman

Levin

2010

Comput. Methods Funct. Theory

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Abstract. Green coordinates define a special representation of a point inside a closed polygon in terms of its vertices and the normals to its edges (faces). This representation has been found to be very useful for object manipulation in computer graphics. The mapping defined by Green coordinates is shown to be analytic. It has a closed form formula in 2D and 3D, and it can be extended analytically through a face of the polygon. In 2D the mapping is proved to be conformal.

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Mesh Deformation

Liu¹,

Bao²

2009

Wiley Encyclopedia of Computer Science and Engineering

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Generating and manipulating surfaces is one of the central issues in various applications of computer graphics and digital entertainment. As one of these operations for surface generating, deformations allow the user to treat an object as if it were constructed from a special type of topological putty or clay, which may be bent, twisted, tapered, compressed, expanded, and otherwise transformed repeatedly into a final shape. They are highly intuitive and easily visualized operations that simulate some important manufacturing processes for fabricating objects, such as the bending of bar stock and sheet metal. Recent progress of deformation techniques focus on mesh deformation. Although for different computing tasks, different surface representations are employed to reveal different geometric, physical, plausible, or perceptual properties of the shape to be handled. The polygonal meshes become fundamental digital media through the whole geometric processing pipeline because of its convenience of acquisition and rendering. However, a pure mesh representation with its vertex positions defined in global coordinate system is challenging for deforming shapes while caring details. Hence researchers have developed several efficient ways to deform mesh surfaces.

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Computing Discrete Minimal Surfaces and Their Conjugates

Cited by 1,074 publications

References 12 publications

Smooth 2D Coordinate Systems on Discrete Surfaces

Smooth 2D Coordinate Systems on Discrete Surfaces

Derivation and Analysis of Green Coordinates

Mesh Deformation

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