Paraboloid - WikiHQ

thumb|right|Paraboloid of revolution

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 a similar property of symmetry.

Every plane section of a paraboloid made by a plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.

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; 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 , , and , it can be represented by the equation

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

thumb|right|Hyperbolic paraboloid

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

In this position, the hyperbolic paraboloid opens downward along the -axis and upward along the -axis (that is, the parabola in the plane opens upward and the parabola in the plane opens downward).

Any paraboloid (elliptic or hyperbolic) is a translation surface, as it can be generated by a moving parabola directed by a second parabola.

Properties and applications

Elliptic paraboloid

thumb|right|[[Polygon mesh of a circular paraboloid]]

thumb|right|Circular paraboloid

In a suitable Cartesian coordinate system, an elliptic paraboloid has the equation

If , an elliptic paraboloid is a circular paraboloid or paraboloid of revolution. It is a surface of revolution obtained by revolving a parabola around its axis.

A circular paraboloid contains circles. This is also true in the general case (see Circular section).

From the point of view of projective geometry, an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity.

; Plane sections

The plane sections of an elliptic paraboloid can be:

a parabola, if the plane is parallel to the axis,
a point, if the plane is a tangent plane.
an ellipse or empty, otherwise.

Parabolic reflector

On the axis of a circular paraboloid, there is a point called the focus (or focal point), such that, if the paraboloid is a mirror, light (or other waves) from a point source at the focus is reflected into a parallel beam, parallel to the axis of the paraboloid. This also works the other way around: a parallel beam of light that is parallel to the axis of the paraboloid is concentrated at the focal point. For a proof, see .

Therefore, the shape of a circular paraboloid is widely used in astronomy for parabolic reflectors and parabolic antennas.

The surface of a rotating liquid is also a circular paraboloid. This is used in liquid-mirror telescopes and in making solid telescope mirrors (see rotating furnace).

Parabola with focus and arbitrary line.svg|Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point, , or vice versa

Erdfunkstelle Raisting 2a.jpg|Parabolic reflector

Centrifugal 0.PNG|Rotating water in a glass

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Hyperbolic paraboloid

thumb|A hyperbolic paraboloid with lines contained in it

thumb|[[Pringles fried snacks are in the shape of a hyperbolic paraboloid.]]

The hyperbolic paraboloid is a doubly 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. Hence the hyperbolic paraboloid is a conoid.

These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed skew lines.

This property makes it simple to manufacture a hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods. In particular, Pringles fried snacks resemble a truncated hyperbolic paraboloid.

A hyperbolic paraboloid is a saddle surface, as its Gauss curvature is negative at every point. Therefore, although it is a ruled surface, it is not developable.

From the point of view of projective geometry, a hyperbolic paraboloid is one-sheet hyperboloid that is tangent to the plane at infinity.

A hyperbolic paraboloid of equation <math>z=axy</math> or <math>z=\tfrac a 2(x^2-y^2)</math> (this is the same up to a rotation of axes) may be called a rectangular hyperbolic paraboloid, by analogy with rectangular hyperbolas.

;Plane sections

thumb|A hyperbolic paraboloid with hyperbolas and parabolas

A plane section of a hyperbolic paraboloid with equation

can be

a line, if the plane is parallel to the -axis, and has an equation of the form <math> bx \pm ay+b=0</math>,
a parabola, if the plane is parallel to the -axis, and the section is not a line,
a pair of intersecting lines, if the plane is a tangent plane,
a hyperbola, otherwise.

thumb|[[STL (file format)|STL hyperbolic paraboloid model]]

Examples in architecture

Saddle roofs are often hyperbolic paraboloids as they are easily constructed from straight sections of material. Some examples:

Philips Pavilion Expo '58, Brussels (1958)
IIT Delhi - Dogra Hall Roof
St. Mary's Cathedral, Tokyo, Japan (1964)
St Richard's Church, Ham, in Ham, London, England (1966)
Cathedral of Saint Mary of the Assumption, San Francisco, California, US (1971)
Scotiabank Saddledome in Calgary, Alberta, Canada (1983)
Scandinavium in Gothenburg, Sweden (1971)
L'Oceanogràfic in Valencia, Spain (2003)
London Velopark, England (2011)
Waterworld Leisure & Activity Centre, Wrexham, Wales (1970)
Markham Moor Service Station roof, A1(southbound), Nottinghamshire, England
Cafe "Kometa", Sokol district, Moscow, Russia (1960). Architect V.Volodin, engineer N.Drozdov. Demolished.

W-wa Ochota PKP-WKD.jpg|Warszawa Ochota railway station, an example of a hyperbolic paraboloid structure

Superfície paraboloide hiperbólico - LEMA - UFBA .jpg|Surface illustrating a hyperbolic paraboloid

Restaurante Los Manantiales 07.jpg|Restaurante Los Manantiales, Xochimilco, Mexico

L'Oceanogràfic Valencia 2019 4.jpg|Hyperbolic paraboloid thin-shell roofs at L'Oceanogràfic, Valencia, Spain (taken 2019)

Sam_Scorer%2C_Little_Chef_-_geograph.org.uk_-_173949.jpg|Markham Moor Service Station roof, Nottinghamshire (2009 photo)

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Cylinder between pencils of elliptic and hyperbolic paraboloids

400px|thumb|elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid

The pencil of elliptic paraboloids

and the pencil of hyperbolic paraboloids

approach the same surface

for <math> b \rightarrow \infty</math>,

which is a parabolic cylinder (see image).

Curvature

The elliptic paraboloid, parametrized simply as

<math display="block">\vec \sigma(u,v) = \left(u, v, \frac{u^2}{a^2} + \frac{v^2}{b^2}\right) </math>

has Gaussian curvature

<math display="block">K(u,v) = \frac{4}{a^2 b^2 \left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^2}</math>

and mean curvature

<math display="block">H(u,v) = \frac{a^2 + b^2 + \frac{4u^2}{a^2} + \frac{4v^2}{b^2{a^2 b^2 \sqrt{\left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^3</math>

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,