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

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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.

- Parametric representations
- Generalised equations
- Properties
- Hyperboloid of one sheet
- Hyperboloid of two sheets
- Other properties
- In more than three dimensions
- Hyperboloid structures
- Relation to the sphere
- See also
- References
- External links

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:

or

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:

One has a hyperboloid of revolution if and only if 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.

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, ∞)

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

- For one obtains a hyperboloid of one sheet,
- For a hyperboloid of two sheets, and
- For 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 term to the appropriate component in the equation above.

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.

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

If the hyperboloid has the equation then the lines

are contained in the surface.

In case the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines or , 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.

For simplicity the plane sections of the *unit hyperboloid* with equation 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 in an
*ellipse*, - A plane with a slope equal to 1 containing the origin intersects in a
*pair of parallel lines*, - A plane with a slope equal 1 not containing the origin intersects in a
*parabola*, - A tangential plane intersects in a
*pair of intersecting lines*, - A non-tangential plane with a slope greater than 1 intersects 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).

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

- .

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 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*, - A plane with slope equal to 1 not containing the origin intersects in a
*parabola*, - A plane with slope greater than 1 intersects 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.

The hyperboloids with equations 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 (hyperboloid of revolution).

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.

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:^{ [4] }

- ... 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.^{ [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).

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.

- The Adziogol Lighthouse, Ukraine, 1911.
- Kobe Port Tower, Japan, 1963.
- Cathedral of Brasília, Brazil, 1970.
- The THTR-300 cooling tower for the now decommissioned thorium nuclear reactor in Hamm-Uentrop, Germany, 1983.
- The Canton Tower, China, 2010.
- The Essarts-le-Roi water tower, France.

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.^{ [7] }

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.

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

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 passing through the origin.

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 by Charles Dupin in his 1803 dissertation under Gaspard Monge. 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 **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.

In mathematics, a **hyperbolic angle** is a geometric parameter that defines a hyperbolic sector. The relationship of a hyperbolic angle to a hyperbola parallels the relationship of an "ordinary" angle to a circle.

**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 mathematics, a **function of a motor variable** is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term **motor** for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. *Motor variable* is used here in place of *split-complex variable* for euphony and tradition.

In geometry, the **elliptic coordinate system** is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

**Toroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.

**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.

**Prolate spheroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

**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.

In mathematics, a **conic section**, **quadratic curve** or simply **conic** is a curve obtained as the intersection of the surface of a cone with 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 historically it was sometimes called 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, the **unit hyperbola** is the set of points (*x*,*y*) in the Cartesian plane that satisfy the implicit equation In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an *alternative radial length*

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.

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:

- ↑ K. Strubecker:
*Vorlesungen der Darstellenden Geometrie.*Vandenhoeck & Ruprecht, Göttingen 1967, p. 218 - ↑ 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 - ↑ 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 - ↑ 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
^{ [5] } - ↑ 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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