Paraboloids

Paraboloids are a family of quadric surfaces that can be described as surfaces of revolution generated by rotating a parabola about its axis, or by stretching that circular form along horizontal directions to produce elliptic paraboloids. They are unbounded surfaces that open in the direction of their axis and have a vertex at the origin in standard forms.

The most common examples are circular and elliptic paraboloids. A circular paraboloid is given by the equation z = (x^2 + y^2) / (4p), with p ≠ 0. It has its vertex at (0,0,0) and its axis along the z-axis. If p > 0, it opens upward; if p < 0, it opens downward. For this circular case, the focus is at (0,0,p) and the directrix is the plane z = -p.

An elliptic paraboloid results from anisotropic scaling of the circular form and is described by z = x^2/a^2 + y^2/b^2, with a, b > 0. Its vertex is at the origin and the axis is the z-axis. Horizontal cross-sections z = constant are ellipses with semi-axes a√(z) and b√(z). The general form z = x^2/(4p_x) + y^2/(4p_y) captures differing curvatures along x and y while preserving a paraboloid shape.

Cross-sections offer characteristic contours: vertical planes containing the axis cut parabolas, while planes perpendicular to the axis cut ellipses (circles in the circular case). Applications include parabolic reflectors in satellite dishes and telescopes, where the focusing property of circular paraboloids is exploited.