paraboloide

Paraboloide, or paraboloid in English, is a quadric surface in three-dimensional space that includes two principal families: the elliptic paraboloid and the hyperbolic paraboloid. These surfaces arise in analytic geometry as second-degree surfaces with distinct geometric properties and cross-sections described by conic sections.

The elliptic paraboloid is typically given by an equation of the form z = x^2/a^2 + y^2/b^2, where a and b are real constants. Its vertex is at the origin and its axis is the z-axis. Planes parallel to the x– or y–axes intersect the surface in parabolas, while planes z = constant cut the surface in ellipses (circles when a = b). The surface opens upward if the squared terms have positive coefficients.

The hyperbolic paraboloid has an equation such as z = x^2/a^2 − y^2/b^2. It has a saddle shape, curving upward in one direction and downward in the other. Cross-sections with planes z = constant yield hyperbolas (or degenerate cases). The hyperbolic paraboloid is a doubly ruled surface, meaning it can be generated by moving a straight line along two different directions, which makes it notable in architecture and design.

Properties and applications: paraboloids are unbounded quadric surfaces with constant-curvature cross-sections along certain directions. Elliptic paraboloids are used in reflectors and focusing devices, while hyperbolic paraboloids are valued for their structural strength and aesthetic, lightweight roofs and architectural shells.

See also: quadric surface, conic sections, parabola.