Computing quadric surface intersections based on an analysis of plane cubic curves

Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves.

[1]  Joshua Z. Levin.  Mathematical models for determining the intersections of quadric surfaces , 1979 .

[2]  Les A. Piegl,et al.  Geometric method of intersecting natural quadrics represented in trimmed surface form , 1989 .

[3]  James R. Miller,et al.  Analysis of quadric-surface-based solid models , 1988, IEEE Computer Graphics and Applications.

[4]  Itzhak Wilf,et al.  Projective splitting of quadric faces , 1990, Comput. Aided Des..

[5]  R. J. Walker.  Algebraic curves , 1950 .

[6]  Chandrajit L. Bajaj,et al.  Automatic parameterization of rational curves and surfaces IV: algebraic space curves , 1988, TOGS.

[7]  Principles of Geometry. , 1927 .

[8]  Joe D. Warren,et al.  On computing the intersection of a pair of algebraic surfaces , 1989, Comput. Aided Geom. Des..

[9]  Ramon F. Sarraga,et al.  Algebraic methods for intersections of quadric surfaces in GMSOLID , 1983, Comput. Vis. Graph. Image Process..

[10]  C. Bajaj,et al.  Automatic parametrization of rational curves and surfaces II: cubics and cubicoids , 1987 .

[11]  Robert Bix,et al.  Conics and Cubics: A Concrete Introduction to Algebraic Curves , 1998 .

[12]  Brown,et al.  PADL-2: A Technical Summary , 1982, IEEE Computer Graphics and Applications.

[13]  Rida T. Farouki,et al.  Automatic parsing of degenerate quadric-surface intersections , 1989, TOGS.

[14]  Gilchrist,et al.  GMSolid: Interactive Modeling for Design and Analysis of Solids , 1982, IEEE Computer Graphics and Applications.

[15]  Joshua Levin,et al.  A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces , 1976, CACM.

[16]  H. F. Baker.  Principles of Geometry , 1934 .

[17]  S. Abhyankar,et al.  Automatic parameterization of rational curves and surfaces 1: conics and conicoids , 1987 .

[18]  Ron Goldman,et al.  Geometric Algorithms for Detecting and Calculating All Conic Sections in the Intersection of Any 2 Natural Quadric Surfaces , 1995, CVGIP Graph. Model. Image Process..

[19]  John K. Johnstone,et al.  On the lower degree intersections of two natural quadrics , 1994, TOGS.

[20]  Ron Goldman,et al.  Combining algebraic rigor with geometric robustness for the detection and calculation of conic sections in the intersection of two natural quadric surfaces , 1991, SMA '91.

[21]  J. G. Semple,et al.  Algebraic Projective Geometry , 1953 .

[22]  Richard R. Patterson,et al.  Parametric cubics as algebraic curves , 1988, Comput. Aided Geom. Des..

[23]  D. M. Y. Sommerville.  Analytical geometry of three dimensions , 1934 .

[24]  R. R. Patterson.  Parametrizing and graphing nonsingular cubic curves , 1988 .

[25]  Malcolm A. Sabin.  A Method for Displaying the Intersection Curve of Two Quadric Surfaces , 1976, Comput. J..

[26]  F. Uhlig,et al.  A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil , 1976 .

[27]  E. Calabi.  Linear systems of real quadratic forms. II , 1964 .

[28]  Itzhak Wilf,et al.  Quadric-surface intersection curves: shape and structure , 1993, Comput. Aided Des..

[29]  Ron Goldman,et al.  Using multivariate resultants to find the intersection of three quadric surfaces , 1991, TOGS.

[30]  James R. Miller,et al.  Geometric approaches to nonplanar quadric surface intersection curves , 1987, TOGS.

[31]  Yong-Ming Li,et al.  Identification of inflection points and cusps on rational curves , 1997, Comput. Aided Geom. Des..

[32]  Micha Sharir,et al.  Planning, geometry, and complexity of robot motion , 1986 .

[33]  Rida T. Farouki,et al.  Singular Points of Algebraic Curves , 1990, J. Symb. Comput..

[34]  Ronald Goldman.  Quadrics of Revolution , 1983, IEEE Computer Graphics and Applications.

[35]  Herbert Freeman,et al.  A Procedure for Generating Visible-line of Solids Bounded by Quadric Surfaces , 1971, IFIP Congress.

[36]  T. A. Brown,et al.  Theory of Equations. , 1950, The Mathematical Gazette.

[37]  John E. Hopcroft,et al.  Geometric ambiguities in boundary representations , 1986 .

[38]  Ron Goldman,et al.  Rational Quadratic Parameterizations of Quadrics , 1997, Int. J. Comput. Geom. Appl..