Cones and conics - pictures

In the RP2 model of projective geometry,
cones arise from conics in an embedding plane.

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

See the theorem of Dandelin.

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.

Theorem A plane conic is non-degenerate if

  • it contains of at least two points, and
  • does not contain 3 collinear points.
line pair

main conics page