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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? w²0
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 - ? w²0, ?, ?, ?, ? gt0
(u,v),(t,s)? 1
Parameterize R in its canonical frame. Ex
hyperbolic paraboloid ? x² - ? y² ? z0, ?,
? gt0
(u,v)? 2
Q has rational coefficients in the local frame.
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Main contributions ? The coefficients of the
parametric form of P ? 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)s²b(u,v)stc(u,v)t²0 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)v²b(u)vc(u)0 where a,b,c are degree 2
polynomials in u. Solve for v ? v? (u) where u
is such 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.