Paraboloid

In geometry, a paraboloid is a quadric surface that has (exactly) one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has the same property of symmetry.

There are two kinds of paraboloids, elliptic and hyperbolic, depending on the nature of the planar cross sections: A paraboloid is elliptic if almost all cross sections are ellipses; it is hyperbolic if almost all cross sections are hyperbolas.

Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors. The paraboloid is hyperbolic if the factors are real and elliptic if the factors are complex conjugate

An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation

where a and b are constants that dictate the level of curvature in the xz and yz planes respectively. In this position, the elliptic paraboloid opens upward.

A hyperbolic paraboloid (not to be confused with a hyperboloid) is a doubly ruled surface shaped like a saddle. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation

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