# Hyperboloid

Hyperboloid of one sheet |
Common conical surface |
Hyperboloid of two sheets |

In mathematics, a **hyperboloid** is a quadric in three dimensions described by the equation

- (hyperboloid of one sheet or hyperbolic hyperboloid),

or

- (hyperboloid of two sheets or elliptic hyperboloid).

Both of these surfaces asymptotically approach the same conical surface as *x* or *y* becomes large:

If and only if *a* = *b*, it is a **hyperboloid of revolution**, and is also called a **circular hyperboloid**.

## Cartesian coordinates

Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle *θ* ∈ [0, 2*π*), but changing inclination *v* into hyperbolic trigonometric functions:

One-surface hyperboloid: *v* ∈ (−∞, ∞)

Two-surface hyperboloid: *v* ∈ [0, ∞)

## Generalised equations

More generally, an arbitrarily oriented hyperboloid, centered at **v**, is defined by the equation

where *A* is a matrix and **x**, **v** are vectors.

The eigenvectors of *A* define the principal directions of the hyperboloid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: , and . The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.

## Properties

A hyperboloid of revolution of one sheet can be obtained by revolving a hyperbola around its semi-minor axis. Alternatively, a hyperboloid of two sheets of axis AB is obtained as the set of points P such that AP−BP is a constant, AP being the distance between A and P. Points A and B are then called the foci of the hyperboloid. A hyperboloid of revolution of two sheets can be obtained by revolving a hyperbola around its semi-major axis.

A hyperboloid of one sheet is a doubly ruled surface; if it is a hyperboloid of revolution, it can also be obtained by revolving a line about a skew line.

Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.

## In more than three dimensions

Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a pseudo-Euclidean space one has the use of a quadratic form:

When *c* is any constant, then the part of the space given by

is called a *hyperboloid*. The degenerate case corresponds to *c* = 0.

As an example, consider the following passage:^{[1]}

- ... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates (
*y*_{1}, ...,*y*_{4}), its equation is*y*2

1 +*y*2

2 +*y*2

3 −*y*2

4 = −1, analogous to the hyperboloid*y*2

1 +*y*2

2 −*y*2

3 = −1 of three-dimensional space.

However, the term **quasi-sphere** is also used in this context since the sphere and hyperboloid have some commonality (See § Relation to the sphere below).

## Hyperboloid structures

One-sheeted hyperboloids are used in construction, with the structures called hyperboloid structures. A hyperboloid is a doubly ruled surface; thus, it can be built with straight steel beams, producing a strong structure at a lower cost than other methods. Examples include cooling towers, especially of power stations, and many other structures.

## Relation to the sphere

In 1853 William Rowan Hamilton published his *Lectures on Quaternions* which included presentation of biquaternions. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from quaternions to produce hyperboloids from the equation of a sphere:

- ... the
*equation of the unit sphere**ρ*^{2}+ 1 = 0, and change the vector*ρ*to a*bivector form*, such as*σ*+*τ*√−1. The equation of the sphere then breaks up into the system of the two following,*σ*^{2}−*τ*^{2}+ 1 = 0,**S**.*στ*= 0;

- and suggests our considering
*σ*and*τ*as two real and rectangular vectors, such that**T***τ*= (**T***σ*^{2}− 1 )^{1/2}.

- Hence it is easy to infer that if we assume
*σ**λ*, where*λ*is a vector in a given position, the*new real vector**σ*+*τ*will terminate on the surface of a*double-sheeted and equilateral hyperboloid*; and that if, on the other hand, we assume*τ**λ*, then the locus of the extremity of the real vector*σ*+*τ*will be an*equilateral but single-sheeted hyperboloid*. The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ...

- ... the

In this passage **S** is the operator giving the scalar part of a quaternion, and **T** is the "tensor", now called norm, of a quaternion.

A modern view of the unification of the sphere and hyperboloid uses the idea of a conic section as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in four-dimensional space with points *p* = (*w*, *x*, *y*, *z*) ∈ **R**^{4} determined by quadratic forms. First consider the conical hypersurface

- and
- which is a hyperplane.

Then is the sphere with radius *r*. On the other hand, the conical hypersurface

- provides that is a hyperboloid.

In the theory of quadratic forms, a **unit quasi-sphere** is the subset of a quadratic space *X* consisting of the *x* ∈ *X* such that the quadratic norm of *x* is one.^{[2]}

## See also

- Ellipsoid
- Hyperbola
- Hyperboloid structure
- Paraboloid / Hyperbolic paraboloid
- Ruled surface
- Rotation of axes
- de Sitter space
- Translation of axes
- Vladimir Shukhov

## References

- ↑ Thomas Hawkins (2000)
*Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869—1926*, §9.3 "The Mathematization of Physics at Göttingen", see page 340, Springer ISBN 0-387-98963-3 - ↑ Ian R. Porteous (1995)
*Clifford Algebras and the Classical Groups*, pages 22, 24 & 106, Cambridge University Press ISBN 0-521-55177-3

- Wilhelm Blaschke (1948)
*Analytische Geometrie*, Kapital V: "Quadriken", Wolfenbutteler Verlagsanstalt. - David A. Brannan, M. F. Esplen, & Jeremy J Gray (1999)
*Geometry*, pp. 39–41 Cambridge University Press. - H. S. M. Coxeter (1961)
*Introduction to Geometry*, p. 130, John Wiley & Sons.

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