Page 45 - Geometric Modeling and Algebraic Geometry

42 F. Aries et al.
Then the polar of S(f) and the triple point τ(f) has equation ϕ 5 (f)=0, where

3

Φ 10 = τ i (f) ∂Φ 4 . (2.27)
dy i
i=0
(The τ i ’s are defined in Equation (2.23).) This way we get a covariant of degree 21
with type Pol (C ). One checks that its zero locus in CP is a union of three planes:
3
4
3
they are the faces of the trihedron drawn by the singular lines of S(f).
Exceptional Triangle.

Consider the discriminant of Φ 3 , quadratic form on (C ) :
3 ∗

2 2 2
∂ Φ 3
∂ Φ 3
∂ Φ 3
dλ 2
1 2 0 dλ 0 λ 1 dλ 0 λ 2
2
2
Φ 11 = ∂ Φ 3 ∂ Φ 3 ∂ Φ 3 . (2.28)
dλ
8 dλ 0 λ 1 2 1 dλ 1 λ 2
2 2 2
∂ Φ 3 ∂ Φ 3 ∂ Φ 3
2
dλ
dλ 0 λ 2 dλ 1 λ 2
2
This is a covariant of degree 12 and type Pol (C ). The zero locus of Φ 11 (f) in CP 2
3
3
12
is the Exceptional Triangle .
Polar plane Π of the Associated Quadric and the Triple Point.
The polar surface of the Associated Quadric and the Triple Point is a plane, call it
Π. It has equation Φ 12 (f)=0, where
3

Φ 12 = τ i ∂Φ 5 . (2.29)
dy i
i=0
4 ∗
This is a covariant of degree 15 and type (C ) = Pol (C ).
4
1
Conic, preimage of Π.
By merely substituting y i with f i (x) in Φ 12 , one finds a new covariant Φ 13 :
Φ 13 (f)(x)= Φ 12 (f)(f(x)). (2.30)
The covariant Φ 13 has degree 16 and type Pol (C ). Naturally, Φ 13 (f)=0 is the
3
2
equation of the conic that is the preimage by [f] of the section of S(f) by Π(f).
The equation obtained this way is of smaller degree than the one obtained by simply sub-
12
stituting the y i’s with the f i’s in Φ 10. Actually, this latter is proportional to the square of
Φ 11.