Implicit is a package for implicitizing rational planar curves and rational tensor product surfaces, developed on Maplesoft based on the state-of-the-art implicitization techniques. The main functions ImpCurve and ImpSurface return the implicit equation of a rational planar curve or a rational tensor product surface. Other popularly used functions, such as ImpDegree, ImpMatrix and ImpRuled that are used for deciding the implicit degree and the implicit matrix of a general rational surface, and computing the implicit equation of a rational ruled surface in a more efficient way, are also proposed.
Singularity computation is a fundamental problem in Computer Graphics and Computer Aided Geometric Design, since it is closely related to topology determination, intersection, mesh generation, rendering, simulation and modeling of curves and surfaces. In this paper, we present an efficient and robust algorithm for computing all the singularities (including their orders) of rational parametric surfaces using the technique of moving planes. The main approach is first to construct a representation matrix whose columns correspond to moving planes following the parametric surface. Then by substituting the parametric equation of the rational surface into this representation matrix, one can extract the singularity information from the corresponding matrix and return all the singular loci including self-intersection curves, cusp curves and isolated singular points of the rational surface, together with the order of each singular locus. We present some examples to compare our algorithm with state-of-the-art methods from different perspectives including robustness, efficiency, order computation and numerical stability, and the experimental results show that our method outperforms existing methods in all these aspects. Furthermore, applications of our algorithm in surface rendering, mesh generation and surface/surface intersections are provided to demonstrate that correctly computing the self-intersection curves of a surface is essential to generate high quality results for these applications.
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