Hyperboloid of two sheets
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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
generated by rotating a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, ca ...
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. A hyperboloid is a quadric surface, that is, a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
defined as the
zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...
of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
or a
cylinder A cylinder (from ) 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. A cylinder may also be defined as an infin ...
, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
axes of symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...
, 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: : + - = 1, or : + - = -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: : + - = 0 . One has a 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 ( 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
and thus the one-sheet hyperboloid is a doubly ruled surface. In the second case ( 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 In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, keeping the
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...
angle , but changing inclination into hyperbolic trigonometric functions: One-surface hyperboloid: :\begin x&=a \cosh v \cos\theta \\ y&=b \cosh v \sin\theta \\ z&=c \sinh v \end Two-surface hyperboloid: :\begin x&=a \sinh v \cos\theta \\ y&=b \sinh v \sin\theta \\ z&=\pm c \cosh v \end 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 a \sqrt \cos t\\ b \sqrt \sin t\\ c s \end \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 c s term to the appropriate component in the equation above.


Generalised equations

More generally, an arbitrarily oriented hyperboloid, centered at , is defined by the equation :(\mathbf)^\mathrm A (\mathbf) = 1, where is a matrix and , are vectors. The
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of define the principal directions of the hyperboloid and the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of A are the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
s 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


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 + - = 1 then the lines :g^_: \vec(t)=\begin a\cos\alpha\\ b\sin\alpha\\ 0\end + t\cdot \begin -a\sin\alpha\\ b\cos\alpha\\ \pm c\end\ ,\quad t\in \R,\ 0\le \alpha\le 2\pi\ are contained in the surface. In case a=b the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines g^_ or g^_, which are skew to the rotation axis (see picture). This property is called '' Wren's theorem''. The more common generation of a one-sheet hyperboloid of revolution is rotating a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, ca ...
around its
semi-minor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
(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 \ 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''. 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 :H_2: \ x^2+y^2-z^2=-1. which can be generated by a rotating
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, ca ...
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''. 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 \frac + \frac - \frac = 1 , \quad \frac + \frac - \frac = -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).


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 A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
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 In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x ...
one has the use of a quadratic form: : q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_^2+\cdots + x_n^2\right), \quad k < n . When 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 . As an example, consider the following passage: :... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates , its equation is , analogous to the hyperboloid of three-dimensional space. However, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality (See below).


Hyperboloid structures

One-sheeted hyperboloids are used in construction, with the structures called
hyperboloid structure Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the grou ...
s. 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 tower A cooling tower is a device that rejects waste heat to the atmosphere through the cooling of a coolant stream, usually a water stream to a lower temperature. Cooling towers may either use the evaporation of water to remove process heat an ...
s, especially of
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s, and many other structures. Adziogol hyperboloid Lighthouse by Vladimir Shukhov 1911.jpg, The Adziogol Lighthouse,
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control tower,
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, 1970. Ciechanow_water_tower.jpg, Hyperboloid water tower with toroidal tank,
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, 1982. Thtr300 kuehlturm.jpg, The THTR-300
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for the now decommissioned
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in Hamm-Uentrop,
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observation tower, Stuttgart,
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Relation to the sphere

In 1853
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...
published his ''Lectures on Quaternions'' which included presentation of
biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s. 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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
: ::... the ''equation of the unit sphere'' , and change the vector to a ''bivector form'', such as . The equation of the sphere then breaks up into the system of the two following, :::, ; ::and suggests our considering and as two real and rectangular vectors, such that :::. ::Hence it is easy to infer that if we assume , where is a vector in a given position, the ''new real vector'' will terminate on the surface of a ''double-sheeted and equilateral hyperboloid''; and that if, on the other hand, we assume , then the locus of the extremity of the real vector 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 is the operator giving the scalar part of a quaternion, and 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 In mathematics, a conic section, quadratic curve or 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 spe ...
as a slice of a quadratic form. Instead of a conical surface, one requires conical
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
s in four-dimensional space with points 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 . 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 In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applie ...
is the subset of a quadratic space consisting of the such that the quadratic norm of is one. Ian R. Porteous (1995) ''Clifford Algebras and the Classical Groups'', pages 22, 24 & 106,
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...


See also

*
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
* Ellipsoid *
List of surfaces This is a list of surfaces, by Wikipedia page. ''See also List of algebraic surfaces, List of curves, Riemann surface.'' Minimal surfaces * Catalan's minimal surface * Costa's minimal surface * Catenoid * Enneper surface * Gyroid * Helicoid ...
*
Paraboloid 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. Every plan ...
/ Hyperbolic paraboloid *
Regulus Regulus is the brightest object in the constellation Leo and one of the brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinized to Alpha Leonis, and abbreviated Alpha Leo or α Leo. Reg ...
*
Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
* *
Translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...


References

*
Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...
(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 Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...
. * H. S. M. Coxeter (1961) ''Introduction to Geometry'', p. 130,
John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, ...
.


External links

* ** ** **{{MathWorld , title=Elliptic Hyperboloid , urlname=EllipticHyperboloid Geometric shapes Surfaces Quadrics Articles containing video clips