Cones and conics - pictures

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

Definitions
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.

ellipse parabola hyperbola

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.

point line line pair

main conics page

Definitions 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.
	ellipse	parabola	hyperbola

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.
	point	line	line pair