Page 89 - Geometric Modeling and Algebraic Geometry

5 Canal Surfaces Defined by Quadratic Families of Spheres 87

(++)
Fig. 5.6. Quadratic canal surfaces of type H .
+−

Example 9. The case H +0 : c(t)=(1 − t , 2at, 0,b(1 + t ),b(1 + t )), a> 0.
2
2
2
Then D(t)=4a b (1 + t ) , and there are two bi-degree (4, 2) parametrizations of
2 2
2 2
Env(C) (Fig. 5.7(left)) defined by the following matrices X(t):

2ib(t +i) 0 2ib(t − i) 0
, .
2b a(t +i) 2b a(t − i)
Example 10. The case H [1] : c(t)=(1 − t , 0, 0, 2at, a(1 + t) /2). Then D(t)=
2
2
+−
a (3 + 2t +3t )(1 − t) , and there are two bi-degree (3, 2) parametrizations of
2
2
2
Env(C) (Fig. 5.7(right)) defined by the following matrices X(t):

√
√ √
3a(t + µ) −ia(t − 1) 3a(t +¯µ) −ia(t − 1) 1+2 2i
√ , √ , µ = .
−i 3(t + µ) (t − 1) −i 3(t +¯µ) (t − 1) 3

(++) [1]
Fig. 5.7. Quadratic canal surfaces of type H and H .
+0 +−

5.5 B´ ezier representations

Consider the case E ++ (see Fig. 5.5(left)) of canal surface generated by the ellipse
C: x /a + x /b = x on the 2-plane x 3 = x 4 =0. In order to parameterize
2
2
2
2
2
1 2 0
Env(C) and its PE transforms we find a rational B´ ezier representation of Γ(C) first.