D. Manocha and J. F. Canny, Implicit representation of rational parametric surfaces, J. Symbol. Comput. 13, 1992, 485--510.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....operations can also be greatly simpli ed when both representations are available, like the intersection of two surface patches or the triangulation for curved object display. This is why conversion algorithms between these two representations have been studied extensively over the past few years [59, 109]. We concentrate here on computing the implicit form of a surface given in parametric equations, a process known as implicitisation. A rst used technique is based on elimination theory (see [41] Consider the following parameterisation of a surface S given in homogeneous form: F (u; v) X(u; ....

....curves on the surface (seam curves) so that it is quite important to deal e ciently with them. Some devices have been developed to reparameterise surfaces so as to eliminate some of their base points, but are currently limited to low degree parameterisations. More recent techniques, like [59], deal directly with base points by introducing an ecient perturbation of one of the equations of System (3) The implicit equation of S is then shown to be contained in the lowest degree term of the resultant f (in terms of the perturbing variable) However, this term also contains an extraneous ....

D. Manocha and J.F. Canny. The implicit representation of rational parametric surfaces. Journal of Symbolic Computation, 13:485-510, 1992.

....can be very slow in practice (see [21] 7] The second method for computing the implicit equation consists in computing a classical projective resultant of the polynomials p0 Gamma xp3 ; p1 Gamma yp3 ; p2 Gamma zp3 (see [10] 11] but this method failed in the presence of base points. In [8] and [5] perturbation techniques are used to obtain the implicit equation by a computation of a classical resultant in the presence of base points. In fact the implicit equation is contained in the lowest degree term of the resultant of the perturbed system (expressed in terms of the perturbing ....

Canny, J. F., and Manocha, D. Implicit representation of rational parametric surfaces. Journal of Symbolic Computation 13 (1992), 485510.

.... for instance the case of tubular or conical surfaces (see [18, 15] This result is relevant for applications in CAD CAM (see [2, 16] Related problems are the parametrization problem for algebraic curves (studied in [26, 1, 20, 17, 22, 14, 13, 21, 23] and the implicitization problem studied in [6, 9, 11, 8]. Another related problem is the problem of resolution of singularities (see [25, 10, 12, 24, 3, 5, 4] which is required as a subtask for surface parametrization. ....

Canny, J., and Manocha, D. Implicit representation of rational parametric surfaces. Journal of Symbolic Computation 13, 5 (1992).

....at least in the absence of base points. In 1983, Sederberg [19] resurrected Dixon s and Salmon s work in addressing the problem of how to implicitize surface patches. Other implicitization methods are surveyed in [10] and include ones based on Grobner bases [1] numerical techniques [16], and multivariate resultants [4] To implicitize a tensor product surface of degreem#n, Dixon s resultant produces a 2mn# 2mn matrix whose elements are linear in x; y; z. The determinant of that matrix is the implicit equation. For a biquadratic surface, the matrix is 8 # 8, and for a bicubic ....

....this problem can sometimes be circumvented by introducing some auxiliary polynomials into the Grobner system [12, 10] Nonetheless, Grobner bases are known to be very slow in implicitizing bicubic patches. Several other procedures have been devised to implicitize surfaces with base points [2, 9, 15, 16]. We don t review those methods here, but observe that those methods are generally more complicated than Dixon s method. Furthermore, base points are not arare occurrence; most of the teapot patches have numerous base points. This paper presents a fundamentally new procedure for implicitizing ....

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Manocha, Dinesh and John F. Canny. The Implicit Representation of Rational Parametric Surfaces. Journal of Symbolic Computation, 13:485--510, 1992.

....Both representations are important. But many surface design systems start from parametric representations. Implicitization is the process of converting surfaces from parametric form into implicit form. Resultants are a standard tool for solving the implicitization problem for rational surfaces [1, 6]. The resultant of three bi variate polynomials is a polynomial in the coecients of these three bi variate polynomials; the vanishing of the resultant signals a common root of the three original polynomials. For a generic rational tensor product surface of xed bi degree, the implicit equation can ....

D. Manocha and J. F. Canny, Implicit representation of rational parametric surfaces. Journal of Symbolic Computation, 13:485-510, 1992.

....retrieving the input output equation as a special case of determining the dependency relation of some algebraically dependent polynomials over some field. This is known as implicitization in algebraic geometry, and many constructive approaches to this problem have been described in the literature: [7, 10, 20, 22, 24] to mention a few. Let us here only briefly describe how Grobner bases can be used to solve the implicitization problem. Grobner bases (gb) are a well known algorithmic method in elimination theory that has been implemented in all major computer algebra programs, e.g. Maple, Axiom, Reduce and ....

D. Manocha and J.F. Canny. Implicit representation of rational parametric surfaces. J. Symbolic Computation, 13(5):485--510, May 1992.

....operations can also be greatly simplified when both representations are available, like the intersection of two surface patches or the triangulation for curved object display. This is why conversion algorithms between these two representations have been studied extensively over the past few years [59, 109]. We concentrate here on computing the implicit form of a surface given in parametric equations, a process known as implicitisation. A first used technique is based on elimination theory (see [41] Consider the following parameterisation of a surface S given in homogeneous form: F (u; v) ....

....curves on the surface (seam curves) so that it is quite important to deal efficiently with them. Some devices have been developed to reparameterise surfaces so as to eliminate some of their base points, but are currently limited to low degree parameterisations. More recent techniques, like [59], deal directly with base points by introducing an efficient perturbation of one of the equations of System (3) The implicit equation of S is then shown to be contained in the lowest degree term of the resultant f (in terms of the perturbing variable) However, this term also contains an ....

D. Manocha and J.F. Canny. The implicit representation of rational parametric surfaces. Journal of Symbolic Computation, 13:485--510, 1992.

....during the algorithm. Rational parametric surfaces in our algorithms are represented with parametric functions and a corresponding domain. We make use of an implicit form of these surfaces when intersecting them. The implicit form can be computed efficiently using implicitization algorithms [MC92] The rational surfaces may be trimmed by algebraic curves in the domain. A rational surface is then represented by a rational function X: 2 3 , together with a collection of trimmed regions in 2 . Each trimmed region is represented by a circular sequence of algebraic curves and ....

D. Manocha and J.F. Canny. The implicit representation of rational parametric surfaces. Journal of Symbolic Computation, 13:485--510, 1992.

.... t) Z(s; t) W (s; t) we formulate the parametric equations wX(s; t) Gamma xW (s; t) 0 wY (s; t) Gamma yW (s; t) 0 wZ(s; t) Gamma zW (s; t) 0 and the problem of implicitization corresponds to computing the resultant of these equations by considering them as polynomials in s and t [38]. Other algorithms for implicitization include Grobner bases and Ritt Wu s algorithm. Hoffmann has surveyed these techniques in [22] A particular benchmark for implicitization has been a bicubic parametric surface given by Hoffmann, 20] x = Gamma3t(t Gamma 1) 2 (s Gamma 1) 3 3s y = ....

....is the common root of X(s; t) Y (s; t) Z(s; t) W (s; t) They also include the points at infinity. Direct applications of resultants or Grobner bases fail to implicitize such parametrizations. Modified algorithms using resultants and Grobner basis are presented by Manocha and Canny, [38] and Kalkbrener, 25] respectively. In particular, the algorithm in [38] considers a perturbed system of the form: wX(s; t) Gamma xW (s; t) 0 wY (s; t) Gamma yW (s; t) 0 wZ(s; t) Gamma zW (s; t) G(s; t) 0; where G(s; t) is a random polynomial. The resultant is a polynomial of the ....

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D. Manocha and J. Canny. The implicit representation of rational parametric surfaces. Journal of Symbolic Computation, 13:485--510, 1992.

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Manocha, D. and Canny, J.F. (1991a) "Implicit representation of rational parametric surfaces ", to appear in Journal of Symbolic Computation. Also available as Technical Report UCB/CSD 90/592, Computer Science Division, University of California, Berkeley.

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D. Manocha and J. F. Canny, Implicit representation of rational parametric surfaces, J. Symbol. Comput. 13, 1992, 485--510.

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D. Manocha and J.F. Canny, "Implicit Representation of Rational Parametric Surfaces", Journal of Symbolic Computation, Vol. 13, pp. 485-510, 1992.

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D. Manocha and J.F. Canny, "Implicit Representation of Rational Parametric Surfaces", Journal of Symbolic Computation, Vol. 13, pp. 485-510, 1992.

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D. Manocha and J.F. Canny, "Implicit Representation of Rational Parametric Surfaces", Journal of Symbolic Computation, Vol. 13, pp. 485-510, 1992.

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Canny, J. F., and Manocha, D. Implicit representation of rational parametric surfaces. Journal of Symbolic Computation 13 (1992), 485--510.

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D. Manocha, J. F. Canny. Implicit Representation of Rational Parametric Surfaces. J. Symbolic Computation 13 (1992), 485-510.

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