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Surface studied by Christopher Wren in 1669.

Cartesian equation: ,.
When a = b: one-sheeted hyperboloid of revolution.
When a = b = c: rectangular one-sheeted hyperboloid.
Small exercise: of what type is the quadric ?
Answer: by a change of orthonormal frame such that OZ is the line x=y=z we get 
hence H1 for , a cone for , H2 for  all of them of revolution around OZ.
Cartesian parametrizations: 
a) the coordinate lines of which are the meridian hyperbolas and the orthogonal ellipses:
, or also , or also 
b) the coordinate lines of which are one of the families of straight lines and the previous ellipses:
= 1 for the first family, -1 for the second one)
c) the coordinate lines of which are two families of straight lines (that are the asymptotic lines):
 d) the coordinate lines of which are the curvature lines (a, b, c distinct):
with (see triple orthogonal system)
Striction line with , intersection with .
see [Struik, p 195] and
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 one-sheeted 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 non-coplanar and non-parallel 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.

A portion of hyperboloid of revolution can be produced by stretching elastic bands between two circular rods (with the elastic bands hooked in a smooth way on the rods).

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 one-sheeted hyperboloid of revolution can be defined as the surface of revolution generated by a line non-coplanar with the axis of revolution, or as the surface of revolution generated by the rotation of a hyperbola around its non-transverse axis.
View of the curvature lines of the one-sheeted 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 one-sheeted 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
In Kobe, Japan


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)

Sculpture by Angel DUARTE (Lausanne, Suisse) using 6 of these structures.

When two "solid spaces" have uniform rotation motions with non-secant 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