Paraboloidi

Paraboloidi, or paraboloids, are quadric surfaces in three-dimensional space. The simplest form is obtained by rotating a parabola about its axis of symmetry, producing a circular paraboloid with rotational symmetry around the z-axis. More generally, paraboloids can have elliptical or hyperbolic cross-sections, giving elliptical and hyperbolic paraboloids.

A circular paraboloid can be described by z = (x^2 + y^2) / (4p), where p > 0 is the distance from the vertex to the focus along the axis. The vertex is at (0,0,0) and the focus at (0,0,p). The elliptical paraboloid has equation z = x^2 / a^2 + y^2 / b^2; if a = b it reduces to the circular case. The hyperbolic paraboloid has equation z = x^2 / a^2 - y^2 / b^2 and has a saddle shape; it is a ruled surface, containing straight lines in two directions.

Paraboloids are important in optics and engineering because of their reflective properties: a parabolic dish concentrates parallel rays to a single focus, which underpins satellite dishes and telescope mirrors. In architecture, hyperbolic paraboloids are valued for structural efficiency and distinctive curvature.