Page 90 - Geometric Modeling and Algebraic Geometry

88 R. Krasauskas and S. Zube
We use homogeneous control points, denoting by e 0 ,...,e 4 the standard frame in

RP . Consider the usual parametrization of a half of C: c(t)= 2 p i B (t) with
2
4
i
i=0
control points
(p 0 ,p 1 ,p 2 )=(e 0 − be 2 ,ae 1 ,e 0 + be 2 ).

Define a biquadratic parametrization h(t, u)= 2 j=0 ij B (t)B (u), where
r
2
2
2
j
i
i=0
⎛ ⎞
ab(e 3 + e 4 ) ab(−e 3 + e 4 )
√ ae 0 √
(r ij )= ⎝ −b a − b (e 3 + e 4 )(a − b )e 1 b a − b (−e 3 + e 4 ) ⎠ . (5.9)
2
2
2
2
2
2
ab(e 3 + e 4 ) ae 0 ab(−e 3 + e 4 )
This parametrization is derived from Example 5: take the parametrization F(t, u) of
Γ(C) and intersect with the hyperplane x 2 =0.
Then another rational parametrization of Γ(C) can be defined by drawing lines
through the ellipse points c(t) and the hyperboloid points h(t, u). The parametriza-
tion of the intersection of Γ(C) with a hyperplane Π(x)=0
f(t, u)= Π(h(t, u))c(t) − Π(c(t))h(t, u). (5.10)
Now it is easy to calculate B´ ezier control points of this parametrization of bi-degree
(4, 2). Indeed, just use conversion formulas from products B (t)B (t) to B 4 i+j (t).
2
2
j
i
If we choose a hyperplane Π(x)= x 4 − rx 0 then the section is a pipe surface
shown in Fig. 5.5(left).
5.6 Implicit equations and double points
It will be convenient to use affine coordinates X =(x, y, z, r)=(x 1 /x 0 ,...,x 4 /x 0 )
in R . We start from the case E ++ of ellipse. By the definition Γ(E) is the envelope
4
1
of the family of isotropic cones Γ(E(t)) = X − E(t),X − E(t) , where E(t)=
(a(1 − t )/(1 + t ), 2bt/(1 + t ), 0, 0). The equation of the envelope is obtained by
2
2
2
elimination of parameter t from the system

f 1 (X, t)= X − E(t),X − E(t) =0, (5.11)
˙
f 2 (X, t)= E(t),X − E(t) =0.
Here f 1 ,f 2 are rational functions in the variable t, i.e. f 1 = g 1 (X, t)/p(t),f 2 =
g 2 (X, t)/q(t). Let

F(X)=Res g 1 (X, t),g 2 (X, t),t (5.12)
be the resultant of two polynomials g 1 ,g 2 (here we assume that degree of poly-
nomials p, q are minimal). A priori F may be a reducible polynomial, i.e. F =
F 1 F 2 ··· F n then one factor, assume F 1 , is the equation of Γ(E). After easy compu-
tation with MAPLE package we see that degree of F is 8.
√
For example, if a = 2 and b =1 the equation of Γ(E) is