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
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, cal ...
around one of its
principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
s, 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 generally, ...
.
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 equi ...
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 exa ...
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 A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique centre and, more ...
, 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 can ...
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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
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 contexts, ...
to the cone of the equations:
:
One has a hyperboloid of revolution if and only if
Otherwise, the axes are uniquely defined (
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
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 surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
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
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
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
![Cylinder - hyperboloid - cone](https://upload.wikimedia.org/wikipedia/commons/5/59/Cylinder_-_hyperboloid_-_cone.gif)
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'' measu ...
, 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.
Mathematicall ...
angle , but changing inclination into
hyperbolic trigonometric functions:
One-surface hyperboloid:
:
Two-surface hyperboloid:
:
![Hyperbo-1s-cut-all](https://upload.wikimedia.org/wikipedia/commons/3/33/Hyperbo-1s-cut-all.svg)
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'' (franchis ...
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 b ...
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 b ...
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
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, th ...
.
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
Wrens are a family of brown passerine birds in the predominantly New World family Troglodytidae. The family includes 88 species divided into 19 genera. Only the Eurasian wren occurs in the Old World, where, in Anglophone regions, it is commonly ...
'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, cal ...
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 focus (geometry), foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major wikt: ...
(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
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 plane ...
.
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
![Hyperbo-2s-ca](https://upload.wikimedia.org/wikipedia/commons/a/a8/Hyperbo-2s-ca.svg)
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, cal ...
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
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
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 Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
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
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 to a ...
:
:
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 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 gro ...
s. A hyperboloid is a
doubly ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, th ...
; 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 and ...
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 p ...
s, and
many other structures.
Adziogol hyperboloid Lighthouse by Vladimir Shukhov 1911.jpg, The Adziogol Lighthouse, 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 inv ...
, 1911.
Kobe port tower11s3200.jpg, Kobe Port Tower
The is a landmark in the port city of Kobe, Japan. The sightseeing tower was completed in 1963 and was temporarily closed from late 2009 to 28 April 2010 for renovation. It is located in the Central District, Kobe, Hyogo Prefecture, Japan.
H ...
, 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 north ...
, 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. With over 750 exhibits in a complex of o ...
's James S. McDonnell Planetarium
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 ...
, St. Louis
St. Louis () is the second-largest city in Missouri, United States. It sits near the confluence of the Mississippi and the Missouri Rivers. In 2020, the city proper had a population of 301,578, while the bi-state metropolitan area, which e ...
, Missouri
Missouri 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 is bordered by eight states (tied for the most with Tennessee ...
, 1963.
Newcastle International Airport Control Tower.jpg, Newcastle International Airport
Newcastle International Airport is an international airport in Newcastle upon Tyne, England, UK. Located approximately from Newcastle City Centre, it is the primary and busiest airport in North East England, and the second busiest in Norther ...
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 b ...
, 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
The Cathedral of Brasília (Portuguese: ''Catedral Metropolitana de Brasília'', "Metropolitan Cathedral of Brasília") is the Roman Catholic cathedral serving Brasília, Brazil, and serves as the seat of the Archdiocese of Brasília. It was des ...
, 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
Ciechanów is a city in north-central Poland. From 1975 to 1998, it was the capital of the Ciechanów Voivodeship. Since 1999, it has been situated in the Masovian Voivodeship. As of December 2021, it has a population of 43,495.
History
The se ...
, 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 populous ...
, 1972.
Toronto - ON - Roy Thomson Hall.jpg, Roy Thomson Hall
Roy Thomson Hall is a concert hall in Toronto, Ontario, Canada. Located downtown in the city's entertainment district, it is home to the Toronto Symphony Orchestra, the Toronto Mendelssohn Choir, and the Toronto Defiant. Opened in 1982, its circ ...
, 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 ancho ...
, 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 tot ...
, 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 shu ...
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 and ...
for the now decommissioned thorium
Thorium is a weakly radioactive metallic chemical element with the symbol Th and atomic number 90. Thorium is silvery and tarnishes black when it is exposed to air, forming thorium dioxide; it is moderately soft and malleable and has a high me ...
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 from nu ...
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 betwe ...
, 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 t ...
, 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 b ...
, 1999.
Killesberg Tower.jpg, The Killesberg
The Killesbergpark (Höhenpark Killesberg) is an urban public park of half a square kilometre (123 acres) in Stuttgart, Germany. It is just north of the state capital, where Killesberg is a quarter of the borough of ''Stuttgart-Nord'' (North).
...
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 ...
, 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 betwe ...
, 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 b ...
, (BMW World), museum and event venue, Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the States of Germany, German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the List of cities in Germany by popu ...
, 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 betwe ...
, 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 ( alternatively romanized as ''Canton''). The tower was topped out in 2009 and it ...
, China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...
, 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
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
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. Hamilton defined a quatern ...
s to produce hyperboloids from the equation of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. 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
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
, 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 specia ...
as a
slice of a quadratic form. Instead of a
conical surface
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''dire ...
, 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 Euclidean ...
s 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 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 to a ...
s. First consider the conical hypersurface
:
and
:
which is a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
.
Then
is the sphere with radius . On the other hand, the conical hypersurface
:
provides that
is a hyperboloid.
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 to a ...
s, 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 Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
See also
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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 ...
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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 the ...
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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
* Lid ...
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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 plane ...
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Hyperbolic 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 plane ...
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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 ...
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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 ...
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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
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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 taught ...
(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 Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
.
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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, in p ...
.
External links
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**{{MathWorld , title=Elliptic Hyperboloid , urlname=EllipticHyperboloid
Geometric shapes
Surfaces
Quadrics
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