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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.

Pa*rab"o*loid (?), n. [Parabola + -oid: cf. F. paraboloide.] Geom.

The solid generated by the rotation of a parabola about its axis; any surface of the second order whose sections by planes parallel to a given line are parabolas.

The term paraboloid has sometimes been applied also to the parabolas of the higher orders.

Hutton.

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