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This paper presents a robust algorithm for finding the parametric intersection of two quadric surfaces, P and Q, with rational coefficients. The research demonstrates that using projective formalism simplifies the implementation, with reduced complexity compared to previous approaches. We illustrate how to compute rational solutions and parameters for the resulting intersection curves in both affine and projective spaces. Key applications include boundary evaluation and the convex hull of quadric patches. Our method offers an optimal number of radicals and manageable sizes for parameterized solutions.
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UNIVERSITE PIERRE &MARIECURIE Ç = Ç Ç = Ç P Q R Q P Q R Q T æ ö x ç ÷ æ ö + a - b - a + b y ç ÷ ut vs us vt ut vs us vt ç ÷ = , , , ç ÷ ç ÷ a b a g b d z è ø ç ÷ ç ÷ w è ø Robust Intersection of Two Quadric Surfaces Laurent Dupont - Daniel Lazard - Sylvain Lazard - Sylvain Petitjean Loria (Univ. Nancy 2, Inria, CNRS) - LIP6 (Univ. Paris 6) Given:P and Q, quadric surfaces (ellipsoids, hyperboloids, paraboloids,...) with rational coefficients. Problem: Find a parametric form for the intersection of P and Q. Applications: - Boundary evaluation (CSG to Brep) - Convex hull of quadric patches Advantages Previous work: J. Levin (1976) Our algorithm The use of a projective formalism simplifies the algorithm and its implementation (fewer types of quadrics, no infinite branches). In affine space, find a simple ruled quadric R in the pencil R() = P- Q, real, such that is a solution of the degree 3 equation det(Ru()) = 0. In projective space, find a ruled quadric R in the pencil R() = P- Q, real, such that det(R()) 0 and is rational. A rational can usually be found after computing an approximation of the roots of det(R()). R has rational coefficients. Using Gauss’ reduction method for quadratic forms, compute the linear transformation that sends R into canonical form: x²+ y²+ z²+ w²=0 Compute the orthogonal transformation that sends R into canonical form: x²+ y²+ z+=0 ( or =0) The coefficients of the linear transformation and , , , are rational. These new parameterizations of canonical projective quadric surfaces are, in general, optimal in the number of radicals. Parameterize R in the local frame. Ex: signature (2,2) (corresponds to a hyperboloid of one sheet or a hyperbolic paraboloid, in affine space) x² + y² - z² - w²=0, , , , >0 (u,v),(t,s) 1 Parameterize R in its canonical frame. Ex: hyperbolic paraboloid x² - y² + z=0, , >0 (u,v)2 Q has rational coefficients in the local frame. T æ ö x æ ö ç ÷ + - u v u v uv ç ÷ = y , , , ç ÷ ç ÷ g a b 2 2 ç ÷ è ø z è ø Main contributions The coefficients of the parametric form ofP Q lie in an extension of which is, in general, of minimal degree. In this extension of , the parametric form does not contain nested radicals. The exact explicit form of the parametric solution has manageable size. Theorem: If P and Q intersect in more than 2 points, then there exists a rational number such that det(R()) 0. Compute the equation of Q in the local frame of R and substitute the parametrization into this equation. We get a degree 2 homogeneous equation in s,t: a(u,v)s²+b(u,v)st+c(u,v)t²=0 where a,b,c are degree 2 homogeneous polynomials in u,v. Solve for (s,t) (s (u,v), t (u,v)) where (u,v) is such that b²-4ac 0. We get a degree 2 equation in v: a(u)v²+b(u)v+c(u)=0 where a,b,c are degree 2 polynomials in u. Solve for v v (u) where u issuch that b²-4ac 0. Substitute the solution of the degree 2 equation into the parametrization. Transform this solution from the local frame to the initial frame.
