Paraboloid
Template:Short description Template:More citations needed
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 Template:Math, Template:Math, and Template:Math, it can be represented by the equation<ref>Template:Cite book</ref> <math display="block">z = \frac{x^2}{a^2} + \frac{y^2}{b^2}.</math> where Template:Math and Template:Math are constants that dictate the level of curvature in the Template:Math and Template:Math 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<ref name="Weisstein">Weisstein, Eric W. "Hyperbolic Paraboloid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicParaboloid.html</ref><ref>Template:Cite book</ref> <math display="block">z = \frac{y^2}{b^2} - \frac{x^2}{a^2}.</math> In this position, the hyperbolic paraboloid opens downward along the Template:Math-axis and upward along the Template:Math-axis (that is, the parabola in the plane Template:Math opens upward and the parabola in the plane Template:Math 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 applicationsEdit
Elliptic paraboloidEdit
In a suitable Cartesian coordinate system, an elliptic paraboloid has the equation <math display="block">z = \frac{x^2}{a^2}+\frac{y^2}{b^2}.</math>
If Template:Math, 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 reflectorEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Template:Slink.
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, Template:Math, or vice versa
- Erdfunkstelle Raisting 2a.jpg
Parabolic reflector
- Centrifugal 0.PNG
Rotating water in a glass
Hyperbolic paraboloidEdit
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.<ref>Template:Citation.</ref>
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
A plane section of a hyperbolic paraboloid with equation <math display="block">z = \frac{x^2}{a^2} - \frac{y^2}{b^2}</math> can be
- a line, if the plane is parallel to the Template:Mvar-axis, and has an equation of the form <math> bx \pm ay+b=0</math>,
- a parabola, if the plane is parallel to the Template:Mvar-axis, and the section is not a line,
- a pair of intersecting lines, if the plane is a tangent plane,
- a hyperbola, otherwise.
Examples in architectureEdit
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.
- 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, Little Chef - geograph.org.uk - 173949.jpg
Markham Moor Service Station roof, Nottinghamshire (2009 photo)
Cylinder between pencils of elliptic and hyperbolic paraboloidsEdit
The pencil of elliptic paraboloids <math display="block">z=x^2 + \frac{y^2}{b^2}, \ b>0, </math> and the pencil of hyperbolic paraboloids <math display="block">z=x^2 - \frac{y^2}{b^2}, \ b>0, </math> approach the same surface <math display="block"> z=x^2</math> for <math> b \rightarrow \infty</math>, which is a parabolic cylinder (see image).
CurvatureEdit
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,<ref name="Weisstein" /> when parametrized 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>
Geometric representation of multiplication tableEdit
If the hyperbolic paraboloid <math display="block">z = \frac{x^2}{a^2} - \frac{y^2}{b^2}</math> is rotated by an angle of Template:Math in the Template:Math direction (according to the right hand rule), the result is the surface <math display="block">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)</math> and if Template:Math then this simplifies to <math display="block">z = \frac{2xy}{a^2}.</math> Finally, letting Template:Math, we see that the hyperbolic paraboloid <math display="block">z = \frac{x^2 - y^2}{2}.</math> is congruent to the surface <math display="block">z = xy</math> which can be thought of as the geometric representation (a three-dimensional nomograph, as it were) of a multiplication table.
The two paraboloidal Template:Math functions <math display="block">z_1 (x,y) = \frac{x^2 - y^2}{2}</math> and <math display="block">z_2 (x,y) = xy</math> are harmonic conjugates, and together form the analytic function <math display="block">f(z) = \frac{z^2}{2} = f(x + yi) = z_1 (x,y) + i z_2 (x,y)</math> which is the analytic continuation of the Template:Math parabolic function Template:Math.
Dimensions of a paraboloidal dishEdit
The dimensions of a symmetrical paraboloidal dish are related by the equation <math display="block">4FD = R^2,</math> where Template:Math is the focal length, Template:Math is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and Template:Math is the radius of the rim. They must all be in the same unit 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 dish measured 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: Template:Math (or the equivalent: Template:Math) and Template:Math, where Template:Math, Template:Math, and Template:Math are defined as above. The diameter of the dish, measured along the surface, is then given by <math display="block">\frac{RQ}{P} + P \ln\left(\frac{R+Q}{P}\right),</math> where Template:Math means the natural logarithm of Template:Math, i.e. its logarithm to base Template:Math.
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 paraboloidal wok), is given by <math display="block">\frac{\pi}{2} R^2 D,</math> where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder (Template:Math), a hemisphere (Template:Math, where Template:Math), and a cone (Template:Math). Template:Math 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 a surface of revolution which gives <math display="block">A = \frac{\pi R\left(\sqrt{(R^2+4D^2)^3}-R^3\right)}{6D^2}.</math>