# Hyperboloid

Not to be confused with Hyperbolic paraboloid.
 Hyperboloid of one sheet conical surface in between Hyperboloid of two sheets

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

${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1}$   (hyperboloid of one sheet or hyperbolic hyperboloid),

or

${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}={\color {red}-}1}$   (hyperboloid of two sheets or elliptic hyperboloid).

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

${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0.}$

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

## Parametric representations

Animation of a hyperboloid of revolution

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

One-surface hyperboloid: v(−∞, ∞)

{\displaystyle {\begin{aligned}x&=a\cosh v\cos \theta \\y&=b\cosh v\sin \theta \\z&=c\sinh v\end{aligned}}}

Two-surface hyperboloid: v[0, ∞)

{\displaystyle {\begin{aligned}x&=a\sinh v\cos \theta \\y&=b\sinh v\sin \theta \\z&=\pm c\cosh v\end{aligned}}}
hyperboloid of one sheet: generation by a rotating hyperbola (top) and line (bottom: red or blue)
hyperboloid of one sheet: plane sections

## Properties of a Hypeboloid of one sheet

### Lines on the surface

If the hyperboloid has the equation ${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1}$ then the lines

${\displaystyle g_{\alpha }^{\pm }:{\vec {x}}(t)={\begin{pmatrix}a\cos \alpha \\b\sin \alpha \\0\end{pmatrix}}+t\cdot {\begin{pmatrix}-a\sin \alpha \\b\cos \alpha \\\pm c\end{pmatrix}}\ ,\quad t\in \mathbb {R} ,\ 0\leq \alpha \leq 2\pi \ }$

are contained in the surface.

In case of ${\displaystyle a=b}$ the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines ${\displaystyle g_{0}^{+}}$ or ${\displaystyle g_{0}^{-}}$, which are skew to the rotation axis (see picture). The more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture).

Remark: A hyperboloid of two sheets is projectively equivalent to a hyperbolic paraboloid.

### Plane sections

For simplicity the plane sections of the unit hyperboloid with equation ${\displaystyle \ H_{1}:x^{2}+y^{2}-z^{2}=1}$ is considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies on the general case, too.

• Planes with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects ${\displaystyle H_{1}}$ in an ellipse,
• Planes with a slope equal 1 containing the origin intersect ${\displaystyle H_{1}}$ in a pair of parallel lines,
• Planes with a slope equal 1 containing not the origin intersect ${\displaystyle H_{1}}$ in a parabola,
• Tangential planes intersect ${\displaystyle H_{1}}$ in a pair of self intersecting lines,
• Planes with a slope greater than 1, wich are no tangential planes, intersect ${\displaystyle H_{1}}$ in a hyperbola [1].
hyperboloid of two sheets: generation by rotating a hyperbola
hyperboloid of two sheets: plane sections

## Properties of a Hyperboloid of two sheets

The hyperboloid of two sheets does not contain lines. The discussion of plane sections can be performed for the unit hyperbola of two sheets with equation

${\displaystyle H_{2}:\ x^{2}+y^{2}-z^{2}=-1}$.

which can be generated by a rotating hyperbola around its semi-minor axis.

• Planes with slopes less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersect ${\displaystyle H_{2}}$ either in an ellipse or in a point or not at all,
• Planes with slopes equal 1 containing the origin (midpoint of the hyperboloid) do not intersect ${\displaystyle H_{2}}$ ,
• Planes with slopes equal 1, which do not contain the origin, intersect ${\displaystyle H_{2}}$ in a parabola,
• Planes with slopes greater than 1 intersect ${\displaystyle H_{2}}$ in a hyperbola [2].

Remark: A hyperboloid of two sheets is projectively equivalent to a sphere.

## Common parametric representation

The following parametric representation include hyperboloids of one sheet, ...two sheets and their common boundary cone:

${\displaystyle {\vec {x}}(s,t)=\left({\begin{array}{lll}a{\sqrt {s^{2}+d}}\cos t\\b{\sqrt {s^{2}+d}}\sin t\\cs\end{array}}\right)}$

• For ${\displaystyle d=1}$ one gets a hyperboloid of one sheet,
• for ${\displaystyle d=-1}$ a hyperboloid of two sheets and
• for ${\displaystyle d=0}$ a double cone.

## Symmetries of a hyperboloid

The hyperboloids with equations ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1,\quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\ }$ are

• pointsymmetric to the origin,
• symmetric to the coordinate planes and
• rotational symmetric to the z-axis and symmetric to any plane containing the z-axis, in case of ${\displaystyle a=b}$.

## On the curvature of a hyperboloid

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.

## Generalised equations

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

${\displaystyle (\mathbf {x-v} )^{\mathrm {T} }A(\mathbf {x-v} )=1,}$

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: ${\displaystyle {1/a^{2}}}$, ${\displaystyle {1/b^{2}}}$ and ${\displaystyle {1/c^{2}}}$. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.

## 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:

${\displaystyle q(x)=\left(x_{1}^{2}+\cdots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\cdots +x_{n}^{2}\right),\quad k

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

${\displaystyle \lbrace x\ :\ q(x)=c\rbrace }$

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

As an example, consider the following passage:[3]

... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates (y1, ..., y4), its equation is y2
1
+ y2
2
+ y2
3
y2
4
= −1
, analogous to the hyperboloid y2
1
+ y2
2
y2
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

Main article: Hyperboloid structure

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 ρ2 + 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,
σ2τ2 + 1 = 0, S.στ = 0;
and suggests our considering σ and τ as two real and rectangular vectors, such that
Tτ = (Tσ2 − 1 )1/2.
Hence it is easy to infer that if we assume σ ${\displaystyle \parallel }$ 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 ${\displaystyle \parallel }$ 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; ...

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) ∈ R4 determined by quadratic forms. First consider the conical hypersurface

${\displaystyle P=\lbrace p\ :\ w^{2}=x^{2}+y^{2}+z^{2}\rbrace }$ and
${\displaystyle H_{r}=\lbrace p\ :\ w=r\rbrace ,}$ which is a hyperplane.

Then ${\displaystyle P\cap H_{r}}$ is the sphere with radius r. On the other hand, the conical hypersurface

${\displaystyle Q=\lbrace p\ :\ w^{2}+z^{2}=x^{2}+y^{2}\rbrace }$ provides that ${\displaystyle Q\cap H_{r}}$ is a hyperboloid.

In the theory of quadratic forms, a unit quasi-sphere is the subset of a quadratic space X consisting of the xX such that the quadratic norm of x is one.[4]