The Cone

Click for VRML

Introduction:

The cone is a well-known geometric figure that gives rise to the so called conic sections, which include curves like the parabola, circle, hyperboloid, and ellipse. These curves can be obtained from the above cone by slicing it in different ways with a plane.

Parallel to xy plane. A circle. An ellipse.
Parallel to edge of cone. A parabola. Parallel to yz plane. A hyperboloid.

 

Although the circular cone is by far the best known type of cone, it is possible to have a cone over any curve. For example, the cone below is over a figure eight.

Definition:

Properties:

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

Mean curvature of the surface.
Surface colored by Mean curvature.