Browsing by Author "Gotsman, Craig"

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    Approximating Planar Conformal Maps Using Regular Polygonal Meshes
    (© 2017 The Eurographics Association and John Wiley & Sons Ltd., 2017) Chen, Renjie; Gotsman, Craig; Chen, Min and Zhang, Hao (Richard)
    Continuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user‐provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non‐triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We provide a formulation of the problem and its solution using complex number algebra, significantly simplifying the notation. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.Continuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user‐provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non‐triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.
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    On Landmark Distances in Polygons
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Gotsman, Craig; Hormann, Kai; Digne, Julie and Crane, Keenan
    We study the landmark distance function between two points in a simply connected planar polygon. We show that if the polygon vertices are used as landmarks, then the resulting landmark distance function to any given point in the polygon has a maximum principle and also does not contain local minima. The latter implies that a path between any two points in the polygon may be generated by steepest descent on this distance without getting ''stuck'' at a local minimum. Furthermore, if landmarks are increasingly added along polygon edges, the steepest descent path converges to the minimal geodesic path. Therefore, the landmark distance can be used, on the one hand in robotic navigation for routing autonomous agents along close-to-shortest paths and on the other for efficiently computing approximate geodesic distances between any two domain points, a property which may be useful in an extension of our work to surfaces in 3D. In the discrete setting, the steepest descent strategy becomes a greedy routing algorithm along the edges of a triangulation of the interior of the polygon, and our experiments indicate that this discrete landmark routing always delivers (i.e., does not get stuck) on ''nice'' triangulations.