Definitions

Paraboloid

[puh-rab-uh-loid]

In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point. In a suitable coordinate system, it can be represented by the equation


z = frac{x^2}{a^2} + frac{y^2}{b^2}.

This is an elliptical paraboloid which opens upward.

The hyperbolic paraboloid is a doubly ruled surface shaped like a saddle. In a suitable coordinate system, it can be represented by the equation


z = frac{x^2}{a^2} - frac{y^2}{b^2}.

This is a hyperbolic paraboloid that opens up along the x-axis and down along the y-axis.

Properties

With a = b an elliptic paraboloid is a paraboloid of revolution: a surface obtained by revolving a parabola around its axis. It is the shape of the parabolic reflectors used in mirrors, antenna dishes, and the like; and is also the shape of the surface of a rotating liquid, a principle used in liquid mirror telescopes. It is also called a circular paraboloid.

A point light source at the focal point produces a parallel light beam. This also works the other way around: a parallel beam of light incident on the paraboloid is concentrated at the focal point. This applies also for other waves, hence parabolic antennas.

The hyperbolic paraboloid is a ruled surface: it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, but not to each other. The Pringles potato chip gives a good physical approximation to the shape of a hyperbolic paraboloid.

Curvature

The elliptic paraboloid, parametrized simply as

$vec sigma\left(u,v\right) = left\left(u, v, \left\{u^2 over a^2\right\} + \left\{v^2 over b^2\right\}right\right)$
has Gaussian curvature
$K\left(u,v\right) = \left\{4 over a^2 b^2 left\left(1 + \left\{4 u^2 over a^4\right\} + \left\{4 v^2 over b^4\right\}right\right)^2\right\}$
and mean curvature
$H\left(u,v\right) = \left\{a^2 + b^2 + \left\{4 u^2 over a^2\right\} + \left\{4 v^2 over b^2\right\} over a^2 b^2 left\left(1 + \left\{4 u^2 over a^4\right\} + \left\{4 v^2 over b^4\right\}right\right)^\left\{3/2\right\}\right\}$
which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.

The hyperbolic paraboloid, when parametrized as

$vec sigma \left(u,v\right) = left\left(u, v, \left\{u^2 over a^2\right\} - \left\{v^2 over b^2\right\}right\right)$
has Gaussian curvature
$K\left(u,v\right) = \left\{-4 over a^2 b^2 left\left(1 + \left\{4 u^2 over a^4\right\} + \left\{4 v^2 over b^4\right\}right\right)^2\right\}$
and mean curvature
$H\left(u,v\right) = \left\{-a^2 + b^2 - \left\{4 u^2 over a^2\right\} + \left\{4 v^2 over b^2\right\} over a^2 b^2 left\left(1 + \left\{4 u^2 over a^4\right\} + \left\{4 v^2 over b^4\right\}right\right)^\left\{3/2\right\}\right\}.$

Multiplication table

If the hyperbolic paraboloid
$z = \left\{x^2 over a^2\right\} - \left\{y^2 over b^2\right\}$
is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface
$z = \left\{1over 2\right\} \left(x^2 + y^2\right) left\left(\left\{1over a^2\right\} - \left\{1over b^2\right\}right\right) + x y left\left(\left\{1over a^2\right\}+\left\{1over b^2\right\}right\right)$
and if $a=b$ then this simplifies to
$z = \left\{2over a^2\right\} x y$.
Finally, letting $a=sqrt\left\{2\right\}$, we see that the hyperbolic paraboloid
$z = \left\{x^2 - y^2 over 2\right\}$.
is congruent to the surface
$z = x y$
which can be thought of as the geometric representation (a three-dimensional nomograph, as it were) of a multiplication table.

The two paraboloidal $mathbb\left\{R\right\}^2 rarr mathbb\left\{R\right\}$ functions

$z_1 \left(x,y\right) = \left\{x^2 - y^2 over 2\right\}$
and
$z_2 \left(x,y\right) = x y$
are harmonic conjugates, and together form the analytic function
$f\left(z\right) = \left\{1over 2\right\} z^2 = f\left(x + i y\right) = z_1 \left(x,y\right) + i z_2 \left(x,y\right)$
which is the analytic continuation of the $mathbb\left\{R\right\}rarr mathbb\left\{R\right\}$ parabolic function $f\left(x\right) = \left\{1over 2\right\} x^2.$