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We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of <i>discrete conformal equivalence</i> for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing… (More)

We define a discrete Laplace-Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called " cotan formula ") except that it is based on the intrinsic Delaunay triangula-tion of… (More)

We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on <i>circle patterns</i>, that is, arrangements of circles---one for each face---with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex… (More)

The discrete Laplace-Beltrami operator plays a prominent role in many Digital Geometry Processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate use of the intrinsic… (More)

We consider the quasiconformal distortion of projective transformations of the real projective plane. For non-affine transformations, the contour lines of quasiconformal distortion form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the quasiconformal distortion of the circumcircle… (More)

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