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A paraboloid is a three-dimensional surface that is a quadratic analogue of a parabola. It is generated by rotating a parabola around an axis that is perpendicular to its directrix. Alternatively, it can be formed by moving a parabola along a line perpendicular to its plane, such that the vertex of the parabola traces out a straight line. The most common type is the elliptic paraboloid, which has the equation z = ax^2 + by^2, where a and b have the same sign. If a and b are both positive, the paraboloid opens upwards, resembling a bowl. If they are both negative, it opens downwards.

A hyperbolic paraboloid is another type of paraboloid, characterized by the equation z = ax^2 - by^2, where a and b are positive. This surface has a saddle shape, curving upwards in one direction and downwards in another. It is a doubly ruled surface, meaning it can be generated by two distinct families of straight lines.

Paraboloids have numerous applications in science and engineering. The reflective properties of paraboloids are utilized in parabolic antennas, which collect and focus electromagnetic waves, such as radio signals or light. This is because parallel rays entering the paraboloid parallel to its axis of symmetry are reflected to a single point, the focus, and vice versa. Similarly, parabolic mirrors are used in telescopes and searchlights. In architecture, paraboloid shapes can be found in roofs and domes, offering structural stability and aesthetic appeal. The study of paraboloids is a fundamental part of multivariable calculus and analytical geometry.