HOME

TheInfoList



OR:

In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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 In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
. A hyperboloid is a quadric surface, that is, a surface 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 eq ...
of a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a
cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
axes of symmetry, and three pairwise perpendicular planes of symmetry. Given a hyperboloid, one can choose a
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
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 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 Limit of a function#Limits at infinity, tends to infinity. In pro ...
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 Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
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 In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
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 ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer ( origin) to a point ...
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: \mathbf 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-\mathbf)^\mathrm A (\mathbf-\mathbf) = 1, where is a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
and , are vectors. The
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
s of define the principal directions of the hyperboloid and the
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s 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.


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^_: \mathbf(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 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 \ 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 In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane (geometry), plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the ...
).


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''. 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 (geometry), plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the ...
). ''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 In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
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 , . Models can be divided in ...
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 of signature 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 vect ...
one has the use of a
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
: 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 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 heat and cool the ...
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 electricity generation, generation of electric power. Power stations are generally connected to an electr ...
s, and many other structures. Adziogol hyperboloid Lighthouse by Vladimir Shukhov 1911.jpg, The Adziogol Lighthouse,
Ukraine Ukraine is a country in Eastern Europe. It is the List of European countries by area, second-largest country in Europe after Russia, which Russia–Ukraine border, borders it to the east and northeast. Ukraine also borders Belarus to the nor ...
, 1911. Staatsmijn Emma Koeltoren III - Brunssum - 20260911 - RCE.jpg, The first 1916 patented Van Iterson
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 heat and cool the ...
of DSM Emma in
Heerlen Heerlen (; ) is a city and a Municipalities of the Netherlands, municipality in the southeast of the Netherlands. It is the third largest settlement proper in the province of Limburg (Netherlands), Limburg. Measured as municipality, it is the f ...
,
The Netherlands , Terminology of the Low Countries, informally Holland, is a country in Northwestern Europe, with Caribbean Netherlands, overseas territories in the Caribbean. It is the largest of the four constituent countries of the Kingdom of the Nether ...
, 1918 Kobe port tower11s3200.jpg, Kobe Port Tower,
Japan Japan is an island country in East Asia. Located in the Pacific Ocean off the northeast coast of the Asia, Asian mainland, it is bordered on the west by the Sea of Japan and extends from the Sea of Okhotsk in the north to the East China Sea ...
, 1963. Mcdonnell planetarium slsc.jpg,
Saint Louis Science Center The Saint Louis Science Center, founded as a planetarium in 1963, is a collection of buildings including a science museum and planetarium in St. Louis, Missouri, on the southeastern corner of Forest Park (St. Louis, Missouri), Forest Park. With o ...
's James S. McDonnell Planetarium, St. Louis,
Missouri Missouri (''see #Etymology and pronunciation, pronunciation'') is a U.S. state, state in the Midwestern United States, Midwestern region of the United States. Ranking List of U.S. states and territories by area, 21st in land area, it border ...
, 1963. Newcastle International Airport Control Tower.jpg, Newcastle International Airport control tower,
Newcastle upon Tyne Newcastle upon Tyne, or simply Newcastle ( , Received Pronunciation, RP: ), is a City status in the United Kingdom, cathedral city and metropolitan borough in Tyne and Wear, England. It is England's northernmost metropolitan borough, located o ...
,
England England is a Countries of the United Kingdom, country that is part of the United Kingdom. It is located on the island of Great Britain, of which it covers about 62%, and List of islands of England, more than 100 smaller adjacent islands. It ...
, 1967. Jested 002.JPG, Ještěd Transmission Tower,
Czech Republic The Czech Republic, also known as Czechia, and historically known as Bohemia, is a landlocked country in Central Europe. The country is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to the south ...
, 1968. Catedral1 Rodrigo Marfan.jpg, Cathedral of Brasília,
Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
, 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 extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
, 1972. Toronto - ON - Roy Thomson Hall.jpg, Roy Thomson Hall,
Toronto Toronto ( , locally pronounced or ) is the List of the largest municipalities in Canada by population, most populous city in Canada. It is the capital city of the Provinces and territories of Canada, Canadian province of Ontario. With a p ...
,
Canada Canada is a country in North America. Its Provinces and territories of Canada, ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, making it the world's List of coun ...
, 1982. Thtr300 kuehlturm.jpg, The THTR-300
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 heat and cool the ...
for the now decommissioned
thorium Thorium is a chemical element; it has symbol Th and atomic number 90. Thorium is a weakly radioactive light silver metal which tarnishes olive grey when it is exposed to air, forming thorium dioxide; it is moderately soft, malleable, and ha ...
nuclear reactor A nuclear reactor is a device used to initiate and control a Nuclear fission, fission nuclear chain reaction. They are used for Nuclear power, commercial electricity, nuclear marine propulsion, marine propulsion, Weapons-grade plutonium, weapons ...
in Hamm-Uentrop,
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
, 1983. Bridge over Corporation Street - geograph.org.uk - 809089.jpg, The Corporation Street Bridge,
Manchester Manchester () is a city and the metropolitan borough of Greater Manchester, England. It had an estimated population of in . Greater Manchester is the third-most populous metropolitan area in the United Kingdom, with a population of 2.92&nbs ...
,
England England is a Countries of the United Kingdom, country that is part of the United Kingdom. It is located on the island of Great Britain, of which it covers about 62%, and List of islands of England, more than 100 smaller adjacent islands. It ...
, 1999. Killesberg Tower.jpg, The Killesberg observation tower,
Stuttgart Stuttgart (; ; Swabian German, Swabian: ; Alemannic German, Alemannic: ; Italian language, Italian: ; ) is the capital city, capital and List of cities in Baden-Württemberg by population, largest city of the States of Germany, German state of ...
,
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
, 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 Olympiapark (Munich), Olympic Park, in the immediate vicinity of the BMW Headquarters and factory. It was ...
, (BMW World), museum and event venue,
Munich Munich is the capital and most populous city of Bavaria, Germany. As of 30 November 2024, its population was 1,604,384, making it the third-largest city in Germany after Berlin and Hamburg. Munich is the largest city in Germany that is no ...
,
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
, 2007. Canton tower in asian games opening ceremony.jpg, The
Canton Tower The Canton Tower (), formally Guangzhou TV Astronomical and Sightseeing Tower (), is a -tall multipurpose observation tower in the Haizhu District of Guangzhou (Postal Map Romanization, alternatively romanized as ''Canton''). The tower was Top ...
,
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
, 2010. Les Essarts-le-Roi Château d'eau.JPG, The Essarts-le-Roi water tower,
France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan ...
.


Relation to the sphere

In 1853
William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
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
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s to produce hyperboloids from the equation of a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
:
... 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 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, tho ...
as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in
four-dimensional space Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...
with points determined by
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s. First consider the conical hypersurface *P = \left\ and *H_r = \lbrace p \ :\ w = r \rbrace , which is a
hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
. Then P \cap H_r is the sphere with radius . On the other hand, the conical hypersurface In the theory of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s, 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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...


See also

* List of surfaces *
Ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
*
Paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar p ...
/ Hyperbolic paraboloid *
Regulus Regulus is the brightest object in the constellation Leo (constellation), Leo and one of the List of brightest stars, brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinisation of names, ...
* Rotation of axes * * Translation of axes * De Sitter space *
Light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single Event (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...


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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
. * 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 corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...
.


External links

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