Page 39 - Geometric Modeling and Algebraic Geometry

36 F. Aries et al.
Also of interest are some facts connected to the quadratic parameterization [f] (rather
than to the Steiner surface S(f) itself):
2
• It is defined on the whole CP .
2
• The direct image of each line of CP is a conic on S(f).
• The preimage of each conic drawn on S(f) is a straight line of CP . As a conse-
2
quence, the preimage of any tangent plane is a pair of lines. The lines are distinct,
unless the plane is a trope.
• The four lines obtained as preimages of the four tropes (equivalently: of the tor-
sal conics; yet equivalently: of the Associated Quadric) form a non–degenerate
quadrilateral.
• The preimage of each of the singular lines of S(f) is a straight line of CP .
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The 3 lines obtained this way are non concurrent: they form a (non–degenerate)
triangle, that we call the Exceptional Triangle.
• The preimage of the triple point is the union of the vertices of the Exceptional
Triangle.
• The parameterization is faithful (i.e. generically injective). Precisely, it is injec-
tive on the complement of the Exceptional Triangle in CP .
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2.5 A collection of covariants

2.5.1 Preliminaries

This section presents the new contribution of the paper: a collection of homogeneous
covariants for the action of G on F, with a simple geometric interpretation for each
of them.
Let us start with some notations. Denote the canonical basis of C with λ 0 , λ 1 ,
3
λ 2 and its dual basis with x 0 , x 1 , x 2 . Denote also the canonical basis of C with α 0 ,
4
α 1 , α 2 , α 3 and its dual basis with y 0 , y 1 , y 2 , y 3 . Given two complex vector spaces W
and V , denote with Pol (W, V ) the space of homogeneous polynomial maps from
n
W to V of degree n. Denote also Pol (W) the space of polynomial homogeneous
n
functions of degree n over W. Otherwise stated,
Pol (W)= Pol (W, C). (2.9)
n n
For f =(f 0 ,f 1 ,f 2 ,f 3 ) ∈F, denote the coefficients of f i with a ij and b ij ,as
follows:

f i = a i0 x + a i1 x + a i2 x +2 b i0 x 1 x 2 +2 b i1 x 0 x 2 +2 b i2 x 0 x 1 . (2.10)
2
2
2
0 1 2
Each of the homogeneous covariants we will present, considered up to a scalar,
represents some geometric object associated to the parameterization [f], according