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Wikipedia: paraboloid

Paraboloid of revolution

Hyperbolic paraboloid

The surface of a liquid rotating around a vertical axis is an upward-opening circular paraboloid.

In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation:

$\left( \frac{x}{a} \right) ^2 + \left( \frac{y}{b} \right) ^2 - z = 0$ (elliptical paraboloid, opens upward),

$\left( \frac{x}{a} \right) ^2 - \left( \frac{y}{b} \right) ^2 - z = 0$ (hyperbolic paraboloid, opens up on x-axis and down on y-axis).

There are two kinds of paraboloid: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point. The hyperbolic paraboloid is shaped like a saddle and can have a critical point called a saddle point. It is a doubly ruled surface.

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.

A daily life example of a hyperbolic paraboloid is the shape of a Pringles potato chip.^{[citation needed]}

The roof of the dining hall at St Dunstan's College in South-east London has the shape of a hyperbolic paraboloid.^[1]

Curvature

The elliptic paraboloid, parametrized simply as

$\vec \sigma(u,v) = \left(u, v, {u^2 \over a^2} + {v^2 \over b^2}\right)$

has Gaussian curvature

$K(u,v) = {4 \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^2}$

and mean curvature

$H(u,v) = {a^2 + b^2 + {4 u^2 \over a^2} + {4 v^2 \over b^2} \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^{3/2}}$

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

$\vec \sigma (u,v) = \left(u, v, {u^2 \over a^2} - {v^2 \over b^2}\right)$

has Gaussian curvature

$K(u,v) = {-4 \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^2}$

and mean curvature

$H(u,v) = {-a^2 + b^2 - {4 u^2 \over a^2} + {4 v^2 \over b^2} \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^{3/2}}.$

Multiplication table

If the hyperbolic paraboloid

$z = {x^2 \over a^2} - {y^2 \over b^2}$

is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface

$z = {1\over 2} (x^2 + y^2) \left({1\over a^2} - {1\over b^2}\right) + x y \left({1\over a^2}+{1\over b^2}\right)$

and if $\ a=b$ then this simplifies to

$z = {2\over a^2} x y$ .

Finally, letting $a=\sqrt{2}$ , we see that the hyperbolic paraboloid

$z = {x^2 - y^2 \over 2}$ .

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 $\mathbb{R}^2 \rarr \mathbb{R}$ functions

$z_1 (x,y) = {x^2 - y^2 \over 2}$

and

$\ z_2 (x,y) = x y$

are harmonic conjugates, and together form the analytic function

$f(z) = {1\over 2} z^2 = f(x + i y) = z_1 (x,y) + i z_2 (x,y)$

which is the analytic continuation of the $\mathbb{R}\rarr \mathbb{R}$ parabolic function $\ f(x) = {1\over 2} x^2.$

References

^
Cherry, Bridget (1983). London 2: South: The Buildings of England. Yale University Press, 418. ISBN 0300096518.

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Curvature

Multiplication table

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
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