cones and conics

In this page, we show pictures of conics obtained as
sections of a right circular cone. For completeness,
we include the degenerate cases.

Given a point V, a line L through V, and αε(0,π/2),
the cone with vertex V, axis L and angle α consists of
all points on lines through V making angle α with L.
The lines are called the generators of the cone.

As an example, take the origin O as vertex,
the z-axis as axis, and α = π/4. This defines
the cone K0 with equation x2+y2=z2.
Sections of K0 by planes z = k (k ≠ 0) are circles.

Intersections with planes not through the vertex
are the usual plane conics.

Note. It is far from clear that these are conics
as described by the usual focus-directrix definition.

The theorem of Dandelin shows this directly.

Here, we will show that the sections of a cone
have equations quadratic in x and y. Since we
know that such curves are plane conics, this is

The equations of conic sections

Intersections with planes through the vertex
give degenerate plane conics.

Observe that each is either finite (a single point!),
or contains a line (a generator of K0).
On the other hand, it is easy to see that no
non-degenerate conic contains a line.

line pair

main conics page