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Paraboloid

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Quadric surface with one axis of symmetry and no center of symmetry

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Ingeometry, aparaboloid is aquadric surface that has exactly oneaxis of symmetry and nocenter of symmetry. The term "paraboloid" is derived fromparabola, which refers to aconic section that has a similar property of symmetry.

Everyplane section of a paraboloid made by a planeparallel to the axis of symmetry is a parabola. The paraboloid ishyperbolic if every other plane section is either ahyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid iselliptic if every other nonempty plane section is either anellipse, 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 acylinder, and has animplicit equation whose part of degree two may be factored over thecomplex numbers into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors arecomplex conjugate.

An elliptic paraboloid is shaped like an oval cup and has amaximum or minimum point when its axis is vertical. In a suitablecoordinate system with three axesx,y, andz, it can be represented by the equation^[1] $z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.$ wherea andb are constants that dictate the level of curvature in thexz andyz planes respectively. In this position, the elliptic paraboloid opens upward.

A hyperbolic paraboloid (not to be confused with ahyperboloid) is adoubly ruled surface shaped like asaddle. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation^[2]^[3] $z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}.$ In this position, the hyperbolic paraboloid opens downward along thex-axis and upward along they-axis (that is, the parabola in the planex = 0 opens upward and the parabola in the planey = 0 opens downward).

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

Properties and applications

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Elliptic paraboloid

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In a suitableCartesian coordinate system, an elliptic paraboloid has the equation $z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.$

Ifa =b, an elliptic paraboloid is acircular paraboloid orparaboloid of revolution. It is asurface of revolution obtained by revolving aparabola around its axis.

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

From the point of view ofprojective geometry, an elliptic paraboloid is anellipsoid that istangent to theplane at infinity.

Plane sections

The plane sections of an elliptic paraboloid can be:

aparabola, if the plane is parallel to the axis,
apoint, if the plane is atangent plane.
anellipse orempty, otherwise.

Parabolic reflector

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Main articles:Parabolic reflector andparabolic antenna

On the axis of a circular paraboloid, there is a point called thefocus (orfocal 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, seeParabola § Proof of the reflective property.

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

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

Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point,F, orvice versa
Parabolic reflector
Rotating water in a glass

Hyperbolic paraboloid

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Pringles fried snacks are in the shape of a hyperbolic paraboloid.

The hyperbolic paraboloid is adoubly ruled surface: it contains two families of mutuallyskew lines. The lines in each family are parallel to a common plane, but not to each other. Hence the hyperbolic paraboloid is aconoid.

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 fixedskew 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.^[4]

A hyperbolic paraboloid is asaddle surface, as itsGauss curvature is negative at every point. Therefore, although it is a ruled surface, it is notdevelopable.

From the point of view ofprojective geometry, a hyperbolic paraboloid isone-sheet hyperboloid that istangent to theplane at infinity.

A hyperbolic paraboloid of equation $z=axy$ or $z={\tfrac {a}{2}}(x^{2}-y^{2})$ (this is the sameup to arotation of axes) may be called arectangular hyperbolic paraboloid, by analogy withrectangular hyperbolas.

Plane sections

A hyperbolic paraboloid with hyperbolas and parabolas

A plane section of a hyperbolic paraboloid with equation $z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}$ can be

aline, if the plane is parallel to thez-axis, and has an equation of the form $bx\pm ay+b=0$ ,
aparabola, if the plane is parallel to thez-axis, and the section is not a line,
a pair ofintersecting lines, if the plane is atangent plane,
ahyperbola, otherwise.

Examples in architecture

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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)
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.

Warszawa Ochota railway station, an example of a hyperbolic paraboloid structure
Surface illustrating a hyperbolic paraboloid
Restaurante Los Manantiales, Xochimilco, Mexico
Hyperbolic paraboloid thin-shell roofs atL'Oceanogràfic, Valencia, Spain (taken 2019)
Markham Moor Service Station roof, Nottinghamshire (2009 photo)

Cylinder between pencils of elliptic and hyperbolic paraboloids

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elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid

Thepencil of elliptic paraboloids $z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,$ and the pencil of hyperbolic paraboloids $z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,$ approach the same surface $z=x^{2}$ for $b\rightarrow \infty$ ,which is aparabolic cylinder (see image).

Curvature

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The elliptic paraboloid, parametrized simply as ${\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)$ hasGaussian curvature $K(u,v)={\frac {4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}$ andmean curvature $H(u,v)={\frac {a^{2}+b^{2}+{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}$ 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,^[2] when parametrized as ${\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}-{\frac {v^{2}}{b^{2}}}\right)$ has Gaussian curvature $K(u,v)={\frac {-4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}$ and mean curvature $H(u,v)={\frac {-a^{2}+b^{2}-{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}.$

Geometric representation of multiplication table

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If the hyperbolic paraboloid $z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}$ is rotated by an angle of⁠π/4⁠ in the+z direction (according to theright hand rule), the result is the surface $z=\left({\frac {x^{2}+y^{2}}{2}}\right)\left({\frac {1}{a^{2}}}-{\frac {1}{b^{2}}}\right)+xy\left({\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}\right)$ and ifa =b then this simplifies to $z={\frac {2xy}{a^{2}}}.$ Finally, lettinga =√2, we see that the hyperbolic paraboloid $z={\frac {x^{2}-y^{2}}{2}}.$ is congruent to the surface $z=xy$ which can be thought of as the geometric representation (a three-dimensionalnomograph, as it were) of amultiplication table.

The two paraboloidalR² →R functions $z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}$ and $z_{2}(x,y)=xy$ areharmonic conjugates, and together form theanalytic function $f(z)={\frac {z^{2}}{2}}=f(x+yi)=z_{1}(x,y)+iz_{2}(x,y)$ which is theanalytic continuation of theR →R parabolic functionf(x) =⁠x²/2⁠.

Dimensions of a paraboloidal dish

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The dimensions of a symmetrical paraboloidal dish are related by the equation $4FD=R^{2},$ whereF is the focal length,D is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), andR is the radius of the rim. They must all be in the sameunit of length. If two of these three lengths are known, this equation can be used to calculate the third.

A more complex calculation is needed to find the diameter of the dishmeasured along its surface. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation:P = 2F (or the equivalent:P =⁠R²/2D⁠) andQ =√P² +R², whereF,D, andR are defined as above. The diameter of the dish, measured along the surface, is then given by ${\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right),$ wherelnx means thenatural logarithm ofx, i.e. its logarithm to basee.

The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidalwok), is given by ${\frac {\pi }{2}}R^{2}D,$ where the symbols are defined as above. This can be compared with the formulae for the volumes of acylinder (πR²D), ahemisphere (⁠2π/3⁠R²D, whereD =R), and acone (⁠π/3⁠R²D).πR² is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept. The surface area of a parabolic dish can be found using the area formula for asurface of revolution which gives $A={\frac {\pi R\left({\sqrt {(R^{2}+4D^{2})^{3}}}-R^{3}\right)}{6D^{2}}}.$

References

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^Thomas, George B.; Maurice D. Weir;Joel Hass; Frank R. Giordiano (2005).Thomas' Calculus 11th ed. Pearson Education, Inc. p. 892.ISBN 0-321-18558-7.
^^a ^bWeisstein, Eric W. "Hyperbolic Paraboloid." From MathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/HyperbolicParaboloid.html
^Thomas, George B.; Maurice D. Weir; Joel Hass; Frank R. Giordiano (2005).Thomas' Calculus 11th ed. Pearson Education, Inc. p. 896.ISBN 0-321-18558-7.
^Zill, Dennis G.; Wright, Warren S. (2011),Calculus: Early Transcendentals, Jones & Bartlett Publishers, p. 649,ISBN 9781449644482.