Page 97 - Geometric Modeling and Algebraic Geometry

6 Classification of Surfaces 95
6.2 Rational ruled surfaces

6.2.1 Ruled surfaces

Definition 1. A ruled surface in projective space is a surface formed by a “singly
infinite system” of straight lines. The lines are called the generators of the surface.
A normal ruled surface is a ruled surface which can not be obtained by projec-
tion from another ruled surface of the same degree in a space of higher dimension.

Proposition 2. A rational ruled surface of degree n spanning P n+1 is normal in
P n+1 .
All rational ruled surfaces of degree n can be obtained as projections of these
normal surfaces.

Proof. See [12], pp 34-36.

Hence, a quartic rational ruled surface S in P can be obtained as projection of
3
a rational normal quartic ruled surface F in P . The center of projection is a line L.
5
See also [9] for a classification of the relative positions of F and L, while in [30] the
projection is decomposed into a projection on P followed by a projection on P in
3
4
order to better describe the provided classification.
6.2.2 Directrices of a surface

We assume that F is not a cone (this case is simple).
Definition 3. A directrix curve of a ruled surface is a curve on the surface meeting
every generator in one point.
A minimum directrix is a directrix curve which is of minimum degree on the
surface.
Remark 4. The image of a directrix (respectively generator) of F by projection is a
directrix (respectively generator) of S. Moreover, the degree of a directrix of F is
the same as the one of its image.
n n n − 1 n
Proposition 5. Let denote if n is even and if n is odd. There are
2 2 2 2
projectively distinct types of rational normal ruled surfaces of degree n in P n+1 ,
n
each one has a directrix of minimum degree m, where m =1, 2,..., .
2
Proof. See [12], pp 38-39.

For n =4 there are only two types, either with minimum directrix conics or with
minimum directrix lines.