Paraboloid

Elliptic paraboloid of a circular paraboloid In a suitable Cartesian coordinate system, an elliptic paraboloid has the equation z = \frac{x^2}{a^2}+\frac{y^2}{b^2}. 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 Hyperbolic paraboloid 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 z=axy or z=\tfrac a 2(x^2-y^2) (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 z = \frac{x^2}{a^2} - \frac{y^2}{b^2} can be • a line, if the plane is parallel to the -axis, and has an equation of the form bx \pm ay+b=0, • 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. 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) == Cylinder between pencils of elliptic and hyperbolic paraboloids ==

The pencil 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 a parabolic cylinder (see image). == Curvature ==

The elliptic paraboloid, parametrized simply 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}} 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, \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 ==

If the hyperbolic paraboloid z = \frac{x^2}{a^2} - \frac{y^2}{b^2} is rotated by an angle of in the direction (according to the right 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 if then this simplifies to z = \frac{2xy}{a^2}. Finally, letting , 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-dimensional nomograph, as it were) of a multiplication table. The two paraboloidal functions z_1 (x,y) = \frac{x^2 - y^2}{2} and z_2 (x,y) = xy are harmonic conjugates, and together form the analytic function f(z) = \frac{z^2}{2} = f(x + yi) = z_1 (x,y) + i z_2 (x,y) which is the analytic continuation of the parabolic function . == Dimensions of a paraboloidal dish ==

The dimensions of a symmetrical paraboloidal dish are related by the equation 4FD = R^2, where is the focal length, is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and 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: (or the equivalent: ) and , where , , and 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), where means the natural logarithm of , i.e. its logarithm to base . 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 \frac{\pi}{2} R^2 D, where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder (), a hemisphere (, where ), and a cone (). 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 A = \frac{\pi R\left(\sqrt{(R^2+4D^2)^3}-R^3\right)}{6D^2}. == See also ==