Hyperboloid

Hyperboloid of one sheet	conical surface in between	Hyperboloid of two sheets

In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. An hyperboloid is a surface that may be obtained from a paraboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

A hyperboloid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry.

Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are axes of symmetry of the hyperboloid, and origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations:

{x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1,

{x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1.

Both of these surfaces are asymptotic to the cone of equation

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

One has an hyperboloid of revolution if and only if $a^{2}=b^{2}.$ Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis.

There are two kinds of hyperboloids. In the first case ( $+1$ in the right-hand side of the equation), one has a one-sheet hyperboloid, also called hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies that the tangent plane at any point intersect the hyperboloid into two lines, and thus that the one-sheet hyperboloid is a doubly ruled surface.

In the second case ( $-1$ in the right-hand side of the equation), one has a two-sheet hyperboloid, also called elliptic hyperboloid. The surface has two connected components, and a positive Gaussian curvature at every point. Thus the surface is convex in the sense that the tangent plane at every point intersects the surface only in this point.

Parametric representations

Animation of a hyperboloid of revolution

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

One-surface hyperboloid: $v \in (- \infty, \infty)$

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

Two-surface hyperboloid: $v \in [0, \infty)$

{\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 hyperboloid of one sheet

Lines on the surface

A hyperboloid of one sheet contains two pencils of lines. It is a doubly ruled surface.

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

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 $a=b$ the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines $g_{0}^{+}$ or $g_{0}^{-}$ , 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 $\ H_{1}:x^{2}+y^{2}-z^{2}=1$ are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.

A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects $H_{1}$ in an ellipse,
A plane with a slope equal to 1 containing the origin intersects $H_{1}$ in a pair of parallel lines,
A plane with a slope equal 1 not containing the origin intersects $H_{1}$ in a parabola,
A tangential plane intersects $H_{1}$ in a pair of intersecting lines,
A non-tangential plane with a slope greater than 1 intersects $H_{1}$ in a hyperbola.^[1]

Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see circular section).

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 hyperboloid of two sheets with equation

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

which can be generated by a rotating hyperbola around one of its axes (the one that cuts the hyperbola)

A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects $H_{2}$ either in an ellipse or in a point or not at all,
A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does not intersect $H_{2}$ ,
A plane with slope equal to 1 not containing the origin intersects $H_{2}$ in a parabola,
A plane with slope greater than 1 intersects $H_{2}$ in a hyperbola.^[2]

Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see circular section).

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

Common parametric representation

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the $z$ -axis as the axis of symmetry:

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

For $d>0$ one obtains a hyperboloid of one sheet,
For $d<0$ a hyperboloid of two sheets, and
For $d=0$ a double cone.

One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of symmetry by shuffling the position of the $cs$ term to the appropriate component in the equation above.

Symmetries of a hyperboloid

The hyperboloids with equations ${\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 $a=b$ (hyperboloid of revolution).

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

({\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: ${1/a^{2}}$ , ${1/b^{2}}$ and ${1/c^{2}}$ . 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:

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

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

\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

(y 1, ..., y 4)

, its equation is

y 21 + y 22 + y 23 - y 24 = -1

, analogous to the hyperboloid

y 21 + y 22 - y 23 = -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.

Water tower in france: hyperboloid of one sheet
port tower in Kobe (Japan): hyperboloid of one sheet
Canton Tower, a one-sheet hyperboloid

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 \sqrt -1

. The equation of the sphere then breaks up into the system of the two following,

σ 2 - τ 2 + 1 = 0

S . σ τ = 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

\parallel

, 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

\parallel

, 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 $p = (w, x, y, z) \in R 4$ determined by quadratic forms. First consider the conical hypersurface

P=\lbrace p\ :\ w^{2}=x^{2}+y^{2}+z^{2}\rbrace

and

H_{r}=\lbrace p\ :\ w=r\rbrace ,

which is a hyperplane.

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

Q=\lbrace p\ :\ w^{2}+z^{2}=x^{2}+y^{2}\rbrace

provides that

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 $x \in X$ such that the quadratic norm of $x$ is one.^[4]

References

↑ CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt) (PDF; 3,4 MB), S. 116
↑ CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt) (PDF; 3,4 MB), S. 122
↑ 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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External links

Weisstein, Eric Wolfgang. "Hyperboloid". MathWorld.

- Weisstein, Eric Wolfgang. "One-sheeted hyperboloid". MathWorld.

- Weisstein, Eric Wolfgang. "Two-sheeted hyperboloid". MathWorld.

- Weisstein, Eric Wolfgang. "Elliptic Hyperboloid". MathWorld.

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