A bisection method for systems of nonlinear equations
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Computer Aided Geometric Design
A fast parallel Horner algorithm
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Rasterization of nonparametric curves
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Contour tracing by piecewise linear approximations
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An accurate algorithm for rasterizing algebraic curves
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Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting
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Comparison of interval methods for plotting algebraic curves
Computer Aided Geometric Design
Novel techniques for robust voxelization and visualization of implicit surfaces
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Context-based surface completion
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Resolution independent curve rendering using programmable graphics hardware
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Hierarchical error-driven approximation of implicit surfaces from polygonal meshes
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2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Non-local isotopic approximation of nonsingular surfaces
Computer-Aided Design
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In this article we present new algorithms for rasterizing implicit curves, i.e., curves represented as level sets of functions of two variables. Considering the pixels as square regions of the plane, a “correct” algorithm should paint those pixels whose centers lie at less than half the desired line width from the curve. A straightforward implementation, scanning the display array evaluating the Euclidean distance from the center of each pixel to the curve, is impractical, and a standard quad-tree-like recursive subdivision scheme is used instead. Then we attack the problem of testing whether or not the Euclidean distance from a point to an implicit curve is less than a given threshold. For the most general case, when the implicit function is only required to have continuous first-order derivatives, we show how to reformulate the test as an unconstrained global root-finding problem in a circular domain. For implicit functions with continuous derivatives up to order k we introduce an approximate distance of order k. The approximate distance of order k from a point to an implicit curve is asymptotically equivalent to the Euclidean distance and provides a sufficient test for a polynomial of degree k not to have roots inside a circle. This is the main contribution of the article. By replacing the Euclidean distance test with one of these approximate distance tests, we obtain a practical rendering algorithm, proven to be correct for algebraic curves. To speed up the computation we also introduce heuristics, which used in conjunction with low-order approximate distances almost always produce equivalent results. The behavior of the algorithms is analyzed, both near regular and singular points, and several possible extensions and applications are discussed.