paraboloid

A paraboloid is a type of quadric surface formed by rotating a parabola about its axis, producing a surface of revolution. It is a geometric surface that can be described by simple Cartesian equations and is distinguished by its focus–directrix property in three dimensions for certain standard forms.

There are several common types of paraboloids. An elliptic paraboloid is described by equations such as z = x^2/a^2 + y^2/b^2, which opens upward (for positive a and b) and, when a = b, becomes a circular paraboloid with circular cross-sections in horizontal planes. A hyperbolic paraboloid is given by z = x^2/a^2 − y^2/b^2, which has a saddle shape and negative Gaussian curvature at all points. A standard form for a circular paraboloid of revolution is x^2 + y^2 = 4 p z, which has axis along the z-direction; its focus lies at (0, 0, p) and its directrix is the plane z = −p.

Cross-sections of paraboloids depend on the cutting plane. Planes parallel to the axis yield parabolic curves, while planes perpendicular to the axis yield ellipses for elliptic paraboloids and hyperbolas for hyperbolic paraboloids (the z = 0 intersection of a hyperbolic paraboloid is a pair of lines). The elliptic paraboloid is a convex surface, whereas the hyperbolic paraboloid is a doubly ruled surface capable of being generated by moving a line.

Applications frequently exploit the focusing properties of paraboloids. Circular and elliptic paraboloids serve as reflective surfaces for telescopes, satellite dishes, and antennas, where parallel incoming rays are reflected to a common focal point.