Types of Cones and Baselines

We will find it convenient to make a crude classification of cones based on their carrier sets.

If C(V,S) is a cone, then we will refer to it as a flat cone if either it is the empty cone or the points of S are collinear.

A cone will be called a thin cone if the points of S are contained in some set of fewer than [q+2/2] concurrent lines. (The function [..] is the greatest integer function, also known as the floor function.) Thus, every flat cone is thin. A non-thin cone will be called a wide cone, i.e., the points of S are not contained in any set of fewer than [q+2/2] concurrent lines. Finally, a wide cone containing at least q + 1 points will be called a thick cone.

Note that oval cones (i.e., cones whose carrier S is an oval), are thick cones.[Forward reference to oval cones] In fact, the definition was chosen to provide a reasonable super-class for this very important class of cones. Flat cones do not present any formidable problems, but their theory does not seem to be very interesting. Flocks of properly (non-flat) thin cones are "wild" in the sense that their variety and ubiquity put them beyond any hope of classification at present. We will therefore concentrate our efforts on the study of thick cones and their flocks.


Let F be a flock of a cone C = C(V,S). Letbe one of the planes of F. Since C is determined by V and any one of the intersections of C with a plane not through V, we may, without loss of generality, assume thatis the carrier plane of C, i.e., we may take S to be in.

The lines ofwhich are the projections from V of the lines of intersection of pairs of planes of F intoare called the baselines of F (in). We distinguish two types of baselines. Those which are the intersections ofwith the other planes of F are called primary baselines, while the others are called secondary baselines.

We have immediately that:

Proposition 5.1: S is contained in the complement of the union of the baselines of F.

In fact, since the definition of the baselines of F did not involve the cone, only its vertex, we have the stronger statement:

Theorem 5.1: Let be one of a set F of q planes, none passing through the point V. If T is any subset of the complement of the union of the baselines of F in , then F is a flock of the cone C(V, T).

Given a set F of q planes, as in the theorem, if S' is the complement of the union of the baselines of F, then the cone C(V,S') is called the critical cone of F.

We will leave open for the moment the interesting geometrical question of which sets of q planes not passing through a given point, have a non-empty critical cone, i.e., when is a set of q planes a flock of some non-empty cone?

We now look at two results which put limits on the possible configurations of baselines of a flock.

Proposition 5.2: At the intersection of two distinct primary baselines there is a secondary baseline which can not coincide with either of these primary baselines.

Note that if there are more than two primary baselines at a point, a secondary baseline can coincide with one of the other primary baselines or with one of the other secondary baselines through this point.

Proposition 5.3: The three secondary baselines at the points of intersection of three non-concurrent primary baselines are concurrent.