Efficient topology determination of implicitly defined algebraic plane curves

González-Vega, Laureano; Necula, Ioana

doi:10.1016/s0167-8396(02)00167-x

Cited by 135 publications

(141 citation statements)

References 14 publications

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“…They involve the exact computation of critical points and genericity condition tests and adjacency tests. The approach has been applied successfully to curves in 2D, and even in 3D, 4D [25,32,24,23,4] and extended to surfaces [11,42].…”

Section: Previous Workmentioning

confidence: 99%

Topology and arrangement computation of semi-algebraic planar curves

Alberti

Mourrain

Wintz

2008

Computer Aided Geometric Design

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We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties.The method isolates singular points from regular parts and deals with them independently. The topology near singular points is guaranteed through topological degree computation. In either case the topology inside regions is recovered from information on the boundary of a cell of the subdivision.Obtained regions are segmented to provide an efficient insertion operation while dynamically maintaining an arrangement structure.We use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. It is finally shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation.

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Section: Previous Workmentioning

confidence: 99%

Topology and arrangement computation of semi-algebraic planar curves

Alberti

Mourrain

Wintz

2008

Computer Aided Geometric Design

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“…In the case of a polynomial parametric surface, we recast the problem of approximating ridges into the field of algebraic geometry. We recall that the standard tool to compute a graph encoding the topology of a 2-D or 3-D curve is the Cylindrical Algebraic Decomposition (CAD) [GVN02,GLMT04].…”

Section: Previous Workmentioning

confidence: 99%

Ridges and Umbilics of Polynomial Parametric Surfaces

Cazals

Faugère

Pouget

et al. 2008

Geometric Modeling and Algebraic Geometry

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Abstract. Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately -in which case a local orientation procedure of principal directions is needed, but no method developed so far topologically certifies the curves reported. In this context, we make two contributions.First, for any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges of the surface (blue and red), we analyze its singularities and we explain how colors can be recovered. Second, we instantiate to the algebraic setting the implicit equation P = 0. For a polynomial surface, this equation defines an algebraic curve, and we develop the first certified algorithm to produce a topologically certified approximation of it. The algorithm exploits the singular structure of P -umbilics and purple points, and reduces the problem to solving zero dimensional systems using Rational Univariate Representations and isolate roots of univariate rational polynomials. An experimental section illustrates the efficiency of the algorithm on a Bezier patch.

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“…Marching or tracing methods generate point sequences along the connected components of the curve. They necessarily use some topological information to find starting, turning and singular points [11][12][13][14]. Subdivision algorithms are based on the "divide and conquer" paradigm.…”

Section: Introductionmentioning

confidence: 99%

Approximating Algebraic Space Curves by Circular Arcs

Béla

Jüttler

2012

Curves and Surfaces

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Abstract. We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating circles of the algebraic curve.

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Efficient topology determination of implicitly defined algebraic plane curves

Cited by 135 publications

References 14 publications

Topology and arrangement computation of semi-algebraic planar curves

Topology and arrangement computation of semi-algebraic planar curves

Ridges and Umbilics of Polynomial Parametric Surfaces

Approximating Algebraic Space Curves by Circular Arcs

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