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ONESHEETED HYPERBOLOID H_{1}
Surface studied by Christopher Wren in 1669. 
see [Struik, p 195] and www.dma.unina.it/~nicla.palladino/catalogo/Descrizioni/21%204.htm Cylindrical equation in the case of the hyperboloid of revolution: . Surface element: Total curvature: where is the distance from O to the tangent plane at the considered point. Directrix cone, which also is the asymptote cone: . 
The onesheeted hyperboloid can be defined as:
1) a ruled quadric with a center of symmetry.
2) the union of the lines meeting three lines 2 by 2 noncoplanar and nonparallel to a fixed plane (when they are, we get the hyperbolic paraboloid)
3) the union of the lines (MN), when the points M and N move at constant speed on two parallel circles.
Here, the hyperboloid is the union of the lines
and
and is also the union of the lines
and .
The sections of the hyperboloid by vertical planes tangent to the inside ellipse are the pairs of secant lines from the two families of included lines.
The onesheeted hyperboloid of revolution can be defined as the surface of revolution generated by a line noncoplanar with the axis of revolution, or as the surface of revolution generated by the rotation of a hyperbola around its nontransverse axis.

View of the curvature lines of the onesheeted hyperboloid; they are circles and hyperbolas only in the case of the hyperboloid of revolution.
Otherwise, they are biquadratics. 
View of one of the two families of circles included in any H1, even if it is not of revolution. 
See here helices of the hyperboloid, as well as the curves of constant precession, traced on a hyperboloid.
Because of its property of being the union of lines, the onesheeted hyperboloid, like the hyperbolic paraboloid is often used in architecture.
Cooling towers of nuclear power plants. 
Water tower in La Roche de Glun in Drome 




See more beautiful pictures on the mathourist's page.
Structure designed from 2 regular polygons with n sides and 2n generatrices of a hyperboloid of revolution (joining 2 middles of sides of these polygons). The 3n resulting skew diamonds are filled by generatrices of hyperbolic paraboloids. (production: Alain Esculier) 



When two "solid spaces" have uniform rotation motions with nonsecant axes, the two loci of the instant axes of rotation of their relative motion in each of the spaces (or "axoids") are two hyperboloids of revolution that roll without slipping on one another (spatial notion equivalent to that of mating gear profile in the plane).
This beautiful kinematics theorem is at the origin of the "hyperboloidal gears" an example of which is reproduced opposite: See this book, page 144. 

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© Robert FERRÉOL, Jacques MANDONNET 2017