Main contributions

1. 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.