In
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 c ...
, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the
surface 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 defined as the
zero set 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 exampl ...
of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a
cone 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 ...
, having a
center of symmetry, and intersecting many
planes
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''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 c ...
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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
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 ( 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 ( 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
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:
:
Two-surface hyperboloid:
:
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.
Generalised equations
More generally, an arbitrarily oriented hyperboloid, centered at , is defined by the equation
:
where is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
and , are
vectors.
The
eigenvectors 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 denote ...
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
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''. 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 (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
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''.
Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see
circular section
In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle.
Any plan ...
).
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
:
.
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
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''.
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see
circular section
In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle.
Any plan ...
).
''Remark:'' A hyperboloid of two sheets is ''projectively'' equivalent to a sphere.
Other properties
Symmetries
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).
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:
:
When is any
constant, then the part of the space given by
:
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 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 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
power station
A power station, also referred to as a power plant and sometimes generating station or generating plant, is an industrial facility for the generation of electric power. Power stations are generally connected to an electrical grid.
Many ...
s, and
many other structures.
Adziogol hyperboloid Lighthouse by Vladimir Shukhov 1911.jpg, The Adziogol Lighthouse
__NOTOC__
The Adziogol Lighthouse ( uk, Аджигольський маяк), also known as Stanislav–Adzhyhol Lighthouse or Stanislav Range Rear light, is one of two vertical lattice hyperboloid structures of steel bars, serving as active li ...
, Ukraine
Ukraine ( uk, Україна, Ukraïna, ) is a country in Eastern Europe. It is the second-largest European country after Russia, which it borders to the east and northeast. Ukraine covers approximately . Prior to the ongoing Russian inva ...
, 1911.
Kobe port tower11s3200.jpg, Kobe Port Tower, Japan
Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the n ...
, 1963.
Mcdonnell planetarium slsc.jpg, Saint Louis Science Center's James S. McDonnell Planetarium, St. Louis, Missouri
Missouri is a state in the Midwestern region of the United States. Ranking 21st in land area, it is bordered by eight states (tied for the most with Tennessee): Iowa to the north, Illinois, Kentucky and Tennessee to the east, Arkansas t ...
, 1963.
Newcastle International Airport Control Tower.jpg, Newcastle International Airport control tower, Newcastle upon Tyne
Newcastle upon Tyne ( RP: , ), or simply Newcastle, is a city and metropolitan borough in Tyne and Wear, England. The city is located on the River Tyne's northern bank and forms the largest part of the Tyneside built-up area. Newcastle is ...
, England
England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe ...
, 1967.
Jested 002.JPG, Ještěd Transmission Tower, Czech Republic
The Czech Republic, or simply Czechia, is a landlocked country in Central Europe. Historically known as Bohemia, it is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to the southeast. The ...
, 1968.
Catedral1 Rodrigo Marfan.jpg, Cathedral of Brasília, Brazil
Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
, 1970.
Ciechanow_water_tower.jpg, Hyperboloid water tower with toroidal tank, Ciechanów, Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...
, 1972.
Toronto - ON - Roy Thomson Hall.jpg, Roy Thomson Hall, Toronto
Toronto ( ; or ) is the capital city of the Canadian province of Ontario. With a recorded population of 2,794,356 in 2021, it is the most populous city in Canada and the fourth most populous city in North America. The city is the anch ...
, Canada
Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by to ...
, 1982.
Thtr300 kuehlturm.jpg, The THTR-300
The THTR-300 was a thorium cycle high-temperature nuclear reactor rated at 300 MW electric (THTR-300) in Hamm-Uentrop, Germany. It started operating in 1983, synchronized with the grid in 1985, operated at full power in February 1987 and was s ...
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 ...
for the now decommissioned thorium nuclear reactor
A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat fr ...
in Hamm
Hamm (, Latin: ''Hammona'') is a city in North Rhine-Westphalia, Germany. It is located in the northeastern part of the Ruhr area. As of 2016 its population was 179,397. The city is situated between the A1 motorway and A2 motorway. Hamm railwa ...
-Uentrop, Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...
, 1983.
Bridge over Corporation Street - geograph.org.uk - 809089.jpg, The Corporation Street Bridge, Manchester
Manchester () is a city in Greater Manchester, England. It had a population of 552,000 in 2021. It is bordered by the Cheshire Plain to the south, the Pennines to the north and east, and the neighbouring city of Salford to the west. The ...
, England
England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe ...
, 1999.
Killesberg Tower.jpg, The Killesberg observation tower, Stuttgart
Stuttgart (; Swabian: ; ) is the capital and largest city of the German state of Baden-Württemberg. It is located on the Neckar river in a fertile valley known as the ''Stuttgarter Kessel'' (Stuttgart Cauldron) and lies an hour from the Sw ...
, Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...
, 2001.
BMW-Welt at night 2.JPG, BMW Welt
The BMW Welt is a combined exhibition, delivery, adventure museum, and event venue located in Munich's district Am Riesenfeld, next to the Olympic Park, in the immediate vicinity of the BMW Headquarters and factory. It was built from August 200 ...
, (BMW World), museum and event venue, Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the third-largest city in Germany, after Berlin and ...
, Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...
, 2007.
Canton tower in asian games opening ceremony.jpg, The Canton Tower, China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
, 2010.
Les Essarts-le-Roi Château d'eau.JPG, The Essarts-le-Roi water tower, France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
.
Relation to the sphere
In 1853
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
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
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 the c ...
:
::... 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 ...
as a
slice of a quadratic form. Instead of a
conical surface, one requires conical
hypersurfaces in
four-dimensional space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
with points determined by
quadratic forms. First consider the conical hypersurface
:
and
:
which is a
hyperplane.
Then
is the sphere with radius . 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 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 Pr ...
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
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.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
*
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
* Helicoi ...
*
Paraboloid /
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. Re ...
*
Rotation of axes
*
*
Translation of axes
References
*
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
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 Pr ...
.
*
H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
(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, i ...
.
External links
*
**
**
**{{MathWorld , title=Elliptic Hyperboloid , urlname=EllipticHyperboloid
Geometric shapes
Surfaces
Quadrics
Articles containing video clips