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:
ax2+by2=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.