Page 87 - Geometric Modeling and Algebraic Geometry

5 Canal Surfaces Defined by Quadratic Families of Spheres 85
For a rational curve C(t) ⊂ R define its discriminant by the formula D(t)=
4
1
d (t)(C (t)+ C (t)+ C (t) − C (t)), where d(t) is a common denominator of the
˙ 2
2
˙ 2
˙ 2
˙ 2
1 2 3 4
˙
derivative vector C(t). Let S be a unit sphere in RP . Define the parametrization P S
3
of S by two complex parameters u 0 and u 1

P S (u 0 ,u 1 )= u 0 ¯u 0 + u 1 ¯u 1 , 2Re(u 0 ¯u 1 ), 2Im(u 0 ¯u 1 ),u 0 ¯u 0 − u 1 ¯u 1 . (5.5)
For any 2 × 2-matrix X =(x ij ) with complex entries define the following extended
2 × 4-matrix X and its minors q ij = q ij (X):

X = x 00 x 01 ¯x 00 ¯x 01 , i.e. q 01 =det X, q 02 = x 00 ¯x 00 , etc. (5.6)

x 10 ¯x 10
x 10 x 11 ¯x 10 ¯x 11
Theorem 4. For any given curve C ⊂ R with D(t) ≥ 0 the Gaussian map of the
4
1
canal surface Env(C) has the form
n(t, u)= P S (x 1∗ (t)(1 − u)+ x 2∗ (t)u), (5.7)
where x i∗ are rows of the 2×2 complex polynomial matrix X(t) such that the minors
q ij = q ij (X(t)) (see (5.6)) satisfy the following condition

˙
˙
˙
˙
2Im(q 12 ):2Re(q 12 ): Im(q 13 − q 02 ): Im(q 13 + q 02 )= C 1 : C 2 : C 3 : C 4 . (5.8)
˙
If n(t, u) has minimal degree in t then deg x 0∗ (t)+deg x 1∗ (t)=deg(d(t)C(t)). All
such cases correspond to different factorizations of the discriminant D(t)= q 01 ¯q 01 .
Proof. The proof follows directly from [3, Sec. 4]. In particular the condition (5.8)
follows from the following equation (cf. [3, Eq. (7)])
(q 13 , −q 12 , −q 03 ,q 02 )= di( ˙r +˙s 3 , ˙s 1 − i˙s 2 , ˙s 1 +i ˙s 2 , ˙r − ˙s 3 ).
Now we are ready to consider minimal parametrizations of six cases of quadratic
(++) (+0)
canal surfaces E ++ , P ++ , H ++ , H +0 , H , H . We skip parabolic Dupin cy-
+− +−
[2]
clide case P , since it was already considered earlier ([1, 5, 8]).
+0

Fig. 5.5. Quadratic canal surfaces of types E ++, P ++, H ++.