Paraboloid

In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic.

The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation

where a and b are constants that dictate the level of curvature in the xz and yz planes respectively. This is an elliptic paraboloid which opens upward for c > 0 and downward for c < 0.

The hyperbolic paraboloid (not to be confused with a hyperboloid) is a doubly ruled surface shaped like a saddle. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation

For c > 0, this is a hyperbolic paraboloid that opens down along the x-axis and up along the y-axis (i.e., the parabola in the plane x = 0 opens upward and the parabola in the plane y=0 opens downward).

Properties

With a = b an elliptic paraboloid is a paraboloid of revolution: a surface obtained by revolving a parabola around its axis. It is the shape of the parabolic reflectors used in mirrors, antenna dishes, and the like; and is also the shape of the surface of a rotating liquid, a principle used in liquid mirror telescopes and in making solid telescope mirrors (see Rotating furnace). This shape is also called a circular paraboloid.