# Hyperboloid

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

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

## Contents

A hyperboloid is a quadric surface, that is, a surface 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 three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry.

Given a hyperboloid, one can choose a Cartesian coordinate system such that the hyperboloid is defined by one of the following equations: ${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1,}$ or ${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1.}$ The coordinate axes are axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid. In any case, the hyperboloid is asymptotic to the cone of the equations: ${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0.}$

One has a hyperboloid of revolution if and only if ${\displaystyle a^{2}=b^{2}.}$ 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): a one-sheet hyperboloid, also called a hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies near every point the intersection of the hyperboloid and its tangent plane at the point consists of two branches of curve that have distinct tangents at the point. In the case of the one-sheet hyperboloid, these branches of curves are lines and thus the one-sheet hyperboloid is a doubly ruled surface.

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

## Parametric representations

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}}}

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the ${\displaystyle z}$-axis as the axis of symmetry: ${\displaystyle \mathbf {x} (s,t)=\left({\begin{array}{lll}a{\sqrt {s^{2}+d}}\cos t\\b{\sqrt {s^{2}+d}}\sin t\\cs\end{array}}\right)}$

• For ${\displaystyle d>0}$ one obtains a hyperboloid of one sheet,
• For ${\displaystyle d<0}$ a hyperboloid of two sheets, and
• For ${\displaystyle d=0}$ 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 ${\displaystyle cs}$ term to the appropriate component in the equation above.

### Generalised equations

More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation ${\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathrm {T} }A(\mathbf {x} -\mathbf {v} )=1,}$ 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.

## Properties

### Hyperboloid 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

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

In case ${\displaystyle a=b}$ 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). This property is called Wren's theorem. [1] The more common generation of a one-sheet hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution).

A hyperboloid of one sheet is projectively equivalent to a hyperbolic paraboloid.

#### Plane sections

For simplicity the plane sections of the unit hyperboloid with equation ${\displaystyle \ H_{1}:x^{2}+y^{2}-z^{2}=1}$ 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 ${\displaystyle H_{1}}$ in an ellipse,
• A plane with a slope equal to 1 containing the origin intersects ${\displaystyle H_{1}}$ in a pair of parallel lines,
• A plane with a slope equal 1 not containing the origin intersects ${\displaystyle H_{1}}$ in a parabola,
• A tangential plane intersects ${\displaystyle H_{1}}$ in a pair of intersecting lines,
• A non-tangential plane with a slope greater than 1 intersects ${\displaystyle H_{1}}$ in a hyperbola. [2]

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

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 ${\displaystyle H_{2}:\ x^{2}+y^{2}-z^{2}=-1.}$ 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 ${\displaystyle H_{2}}$ 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${\displaystyle H_{2}}$,
• A plane with slope equal to 1 not containing the origin intersects ${\displaystyle H_{2}}$ in a parabola,
• A plane with slope greater than 1 intersects ${\displaystyle H_{2}}$ in a hyperbola. [3]

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.

### Other properties

#### Symmetries

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 ${\displaystyle a=b}$ (hyperboloid of revolution).

#### Curvature

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: ${\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: [4]

... 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. [6]

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 sphereρ2 + 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 || λ, where λ is a vector in a given position, the new real vectorσ + 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; ...

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

• ${\displaystyle P=\left\{p\;:\;w^{2}=x^{2}+y^{2}+z^{2}\right\}}$ 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. [7]

## Related Research Articles

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin. The real projective plane is then an extension of the (ordinary) plane — to every point (v1,v2) of the ordinary plane, the line spanned by (v1,v2,1) is associated (i.e., the real projective plane is the projective completion of the ordinary plane, cf. also the homogeneous coordinates below) while there are also some “points in the infinity”.

In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered c. 1802 by Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.

Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. Confusingly, the same term is also sometimes used for two-center bipolar coordinates. There is also a third system, based on two poles.

In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci and in bipolar coordinates remain points in the bispherical coordinate system.

Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular -direction. The two lines of foci and of the projected Apollonian circles are generally taken to be defined by and , respectively, in the Cartesian coordinate system.

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

In geometry, two conic sections are called confocal if they have the same foci.

In geometry, focal conics are a pair of curves consisting of either

In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement:

## References

1. K. Strubecker: Vorlesungen der Darstellenden Geometrie. Vandenhoeck & Ruprecht, Göttingen 1967, p. 218
2. 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
3. Walter, Scott A. (1999), "The non-Euclidean style of Minkowskian relativity", in J. Gray (ed.), The Symbolic Universe: Geometry and Physics 1890-1930, Oxford University Press, pp. 91–127
4. Minkowski used the term "four-dimensional hyperboloid" only once, in a posthumously-published typescript and this was non-standard usage, as Minkowski's hyperboloid is a three-dimensional submanifold of a four-dimensional Minkowski space ${\displaystyle M^{4}.}$ [5]
5. Ian R. Porteous (1995) Clifford Algebras and the Classical Groups, pages 22, 24 & 106, Cambridge University Press ISBN   0-521-55177-3