Page 17 - Geometric Modeling and Algebraic Geometry

1 The GAIA Project 11
Combining the parametric and implicit representation the intersection is de-
scribed by q(p(t)) = 0, a linear equation in the variable t. Using exact arithmetic it
is easy to classify the solution as:

• An empty set, if the lines are parallel.
• The whole line, if the lines coincide.
• One point, if lines are non-parallel.
Next we look at the intersection of two rational parametric curves of degree n
and d, respectively. From algebraic geometry it is known that a rational parametric
curve of degree d is contained in an implicit parametric curve of total degree d,see
[27].

• The first curve is described as a rational parametric curve
n
p n t + p d−1 t n−1
p(t)= + ... + p 0 .
h n t + h n−1 t n−1 + ... + h 0
n
• The second curve is described as an implicit curve of total degree d
d d−i
i j
q(x, y)= c i,j x y =0.
i=0j=0
By combining the parametric and implicit representations, the intersection is de-
scribed by q(p(t)) = 0. This is a degree n × d equation in the variable t.Aseven
the general quintic equation cannot be solved algebraically, a closed expression for
the zeros of q(p(t)) can in general only be given for n × d ≤ 4. Thus, in general, the
intersection of two rational cubic curves cannot be found as a closed expression. In
CAD-systems we are not interested in the whole infinite curve, but only a bounded
portion of the curve. So approaches and representations that can help us to limit the
extent of the curves and the number of possible intersections will be advantageous.
We now turn to intersections of two surfaces. Let p(s, t) be a rational tensor
product surface of bi-degree (n 1 ,n 2 ),

n 1 n 2

i j
p i,j s t
p(s, t)= i=0j=0 .
n 1 n 2

i j
h i,j s t
i=0j=0
From algebraic geometry it is known that the implicit representation of p(s, t) has
total algebraic degree d =2n 1 n 2 . The number of monomials in a polynomial of total

degree d in 3 variables is d+3 = (d+1)(d+2)(d+3) . So a bicubic rational surface has
3 6
an implicit equation of total degree 18. This has 1330 monomials with corresponding
coefficients.
Using this fact we can look at the complexity of the intersection of two rational
bicubic surfaces p 1 (u, v) and p 2 (s, t). Assume that we know the implicit equation