Page 43 - Geometric Modeling and Algebraic Geometry

40 F. Aries et al.

a 00 a 01 a 02 b 00 b 01 b 02 y 0

a 10 a 11 a 12 b 10 b 11 b 12 y 1

a 20 a 21 a 22 b 20 b 21 b 22 y 2
Φ 4 =6 3 a 30 a 31 a 32 b 30 b 31 b 32 y 3 . (2.18)

300
120
102
111
201
210 0

210
030
012
021
111
120 0

201
021
003
012
102
111 0
Then Φ 4 (f) is an implicit equation of S(f). And Φ 4 is also a covariant. it has degree
12 and type Pol (C ). The attached geometric object is its zero locus, that is merely
4
4
the surface itself.
This covariant has another property: it vanishes if and only if the parameterization
admits a base point (this means that the f i ’s have a common zero in CP ; thus it is
2
revealed to be a resultant).
Formula (2.18) has been proposed in [3]. See [4, 6, 10], for formulas close to this
one, and proofs.
Associated Quadric.
One produces a new covariant by the following contraction (see [11]) of Φ 1 and Φ 2 :
1 3 3
Φ 5 = ∂ Φ 1 ∂ Φ 2 . (2.19)
6 dx i dx j dx k dλ i dλ j dλ k
i,j,k
It has degree 6 and type Pol (C ). One finds (by evaluation on the representative of
4
2
the dense orbit) that Φ 5 (f)=0 is an equation for the Associated Quadric.
Preimage of a point of the Steiner surface.

The map [f] from CP to CP induced by f is birational onto its image S(f): its
3
2
inverse is induced by the rational map [Φ 6 (f)] : CP → CP where
2
3

a 00 a 01 a 02 b 00 b 01 b 02 0

a 10 a 11 a 12 b 10 b 11 b 12 0

a 20 a 21 a 22 b 20 b 21 b 22 0

Φ 6 =6 3 a 30 a 31 a 32 b 30 b 31 b 32 0 . (2.20)

300
120
102
111
201
210 λ 0

210
030
012
021
111
120 λ 1

201
021
003
012
102
111 λ 2
This is a covariant of degree 10 and type Pol (C , C ).
3
4
2
The dual surface.
Consider the quadratic form α 0 f 0 + ··· + α 3 f 3 and take its discriminant (that is the
determinant of its matrix):