REMINDER: proper conics

REMINDER: degenerated curves

If a conic has a double point, it is degenerated into a pair of different real lines or a pair of conjugate imaginary lines with a real intersection point. Beside these two degenerated conics, there is also a degenerated conic that has infinitely many double points - one real double line.

A plane through the vertex intersects the curve k in two points, lines connecting these points with the vertex lie in this plane and they are generatrices of the cone. These intersection points can be real and different, real and coincide or conjugate imaginary points - the generatrices can be real and different, coincide or imaginary. Therefore, there are 3 cases of the plane intersection:

Right-click to start the animation.

The type of the conic depends on the points at infinity of this conic. Points of the inetersection curve are intersection points of generatrices of the cone and the plane. Remember that a line intersects a plane at the point at infinity if they are parallel. Therefore, we conclude the following:

A plane intersection with the plane not passing the vertex is

Right-clik to start the animation.

A plane at infinity of the extended euclidean space doesn't contain the vertex
of a cone - it intersects the cone in a real proper conic.
Points of this conic are points at infinity of generatrices of the cone. This conic is not an ellipse, a parabola or a hyperbola; all points on this conic are points at infinity. |

All tangent planes of the cone contain its vertex. All other points of a cone are regular - they have an unique tangent plane. |

This planes can be real or imaginary. A construction of these planes is given in the animation on the right. |
Click on the image to start the animation. |

Circular cone is obtained by rotating one line about another intersection line (an axis of rotation).

Every plane perpendicular to the axis intersect the circular cone in a circle.

The cone will be given by its vertex and one circle intersection called a base of the cone.

A point of a generatrix that lies on the base is called a

Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb