The

paraboloid is the

3-dimensional extension of the

2-D parabola. The two

generic forms of paraboloids are

circular and

elliptic. The generating

equation for a paraboloid

centered at the

origin is:

`
ax`^{2}+by^{2}=z.

When a=b, the paraboloid is

circular, and can be generated as a

surface of rotation. Otherwise, it is

elliptic. Just as the parabola is defined to be the set of points

equidistant from a line and a point not on the line, the paraboloid for a=b is the set of points

equidistant from a

plane and a point not on the

plane. Therefore, each point of a paraboloid where a=b is the center of a

sphere which is tangential to the

focus and the

directrix plane of the paraboloid.

*When a parabolic mirror is mentioned, it is usually the case that the mirror is actually a paraboloid in shape.*