Page 109 - Geometric Modeling and Algebraic Geometry

6 Classification of Surfaces 107
Then, if u 1 = u 0 =0, one can choose (t 1 ,u 1 )=(1, 0) , (t 2 ,u 2 )=(0, ∞).
Similarly, we obtain the equations of the surface:
⎧
⎪ X = tu 2
⎪
⎨ Y = u 2
(S):
⎪ Z = tu − u
⎪
⎩
T = t
6.6 Detection of the singularities of a patch

Self-intersection curves of a polynomial patch are often computed approximately
(see e.g. [8], [29], [20],...). Here we provide a symbolic method adapted to our
setting.

We write the parametric equations of the surface in the Bernstein’s basis

2
2
2
2
{tu ,t(1 − u) , (1 − t)u , (1 − t)(1 − u) , 2tu(1 − u), 2(1 − t)u(1 − u)}
and consider it in [0, 1] × [0, 1]. The surface depends on the 6 control points, by
changing coordinates we can choose these points to be: (0 :0:0:1), (1 :0:0:1),
(0 :1:0:1), (0 :0:1:1), (a : b : c :1) and (d : e : f :1). Therefore the surface
hasaB´ ezier representation:
⎧
⎨ x = t(1 − u) +2atu(1 − u)+2d(1 − t)u(1 − u)
2
y =(1 − t)u +2btu(1 − u)+2e(1 − t)u(1 − u)
2
z =(1 − t)(1 − u) +2ctu(1 − u)+2f(1 − t)u(1 − u)
⎩
2
We used the software Maple for our computation. We denote by F the implicit equa-
tion of the surface, F x , F y , F z the partial derivatives of F and we denote by M x ,
M y , M z polynomials in (t, u) obtained by replacing the parametric expressions of
x, y, z in F x , F y , F z .As F is of degree 4, M x , M y , M z are of projective bedegree
(3,6) but of affine bidegree (3,5) in (t, u). The implicit equation C(t, u) of the double
locus (also denoted by C(t, u)) in the parameter space of the surface is the gcd of
M x , M y and M z . The curve C(t, u) is a curve of degree (2,2). We have:

M x
=((e + f − c − b − 1)u +(2b − 2e +1)u − b + e)t
2
C(t, u)
+(4ec − 4bf − 2c +2b)u +(1+2c − f − 6ec +6bf − 4b − e)u 2
3
+(2ec − 2bf − 1+2b +2e)u − e

M y
=(c − f + a − d)(u − 1) t+(4af − 2f − 4cd − 2a +1)u +(6cd − 6af
3
2
C(t, u)
+ d − 3+5f +4a)u +(2af +3 − 4f − 2d − 2a − 2cd)u + d + f − 1
2