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A sphere (from
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
—, "globe, ball") is a
geometrical Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...

geometrical
object that is a
three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
analogue to a
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
in two-dimensional space. 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
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the
ancient Greek mathematicians Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic Greece, Archaic through the Hellenistic periods, extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. ...
. Spheres and nearly-spherical shapes appear in nature and industry. Bubbles such as
soap bubble A soap bubble is an extremely thin soap film, film of soapy water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with anot ...
s take a spherical shape in equilibrium. The Earth is often approximated as a sphere in
geography Geography (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10. ...

geography
, and the
celestial sphere In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ma ...

celestial sphere
is an important concept in
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
. Manufactured items including
pressure vessels A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. Construction methods and materials may be chosen to suit the pressure application, and will depend on the size of ...
and most
curved mirror A curved mirror is a mirror Grange, East Yorkshire, UK, from World War I. The mirror magnified the sound of approaching enemy Zeppelins for a microphone placed at the Focus (geometry), focal point. A mirror is an object that Reflection ( ...
s and
lens A lens is a transmissive optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, s ...

lens
es are based on spheres. Spheres
roll Roll or Rolls may refer to: Movement about the longitudinal axis * Roll (flight), motion about the longitudinal axis of an aircraft ** Roll, an aerobatic maneuver ** Roll program, an aerodynamic maneuver performed in a rocket launch * Roll (ship), ...

roll
smoothly in any direction, so most
ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective ...

ball
s used in sports and toys are spherical, as are
ball bearings Wingquist's self-aligning ball bearing A ball bearing is a type of rolling-element bearing A rolling-element bearing, also known as a rolling bearing, is a bearing which carries a load by placing rolling elements (such as balls or rollers) b ...
.


Basic terminology

As mentioned earlier is the sphere's radius; any line from the center to a point on the sphere is also called a radius. If a radius is extended through the center to the opposite side of the sphere, it creates a
diameter In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

diameter
. Like the radius, the length of a diameter is also called the diameter, and denoted . Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, =. Two points on the sphere connected by a diameter are
antipodal point Antipodal points on a circle are 180 degrees apart. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
s of each other. A
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
is a sphere with unit radius (=1). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned. A ''
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
'' on the sphere has the same center and radius as the sphere—consequently dividing it into two equal
hemisphere Hemisphere may refer to: * A half of a sphere As half of the Earth * A hemispheres of Earth, hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the ...

hemisphere
s. Although the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

Earth
is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. If a particular point on a sphere is (arbitrarily) designated as its ''north pole'', its antipodal point is called the ''south pole''. The great circle equidistant to each is then the ''
equator The Equator is a , about in circumference, that divides into the and hemispheres. It is an located at 0 degrees , halfway between the and poles. In , as applied in , the equator of a rotating (such as a ) is the parallel (circle of l ...

equator
''. Great circles through the poles are called lines of
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

longitude
or meridians. A line connecting the two poles may be called the
axis of rotation Rotation around a fixed axis is a special case of rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotati ...
. Small circles on the sphere that are parallel to the equator are lines of
latitude In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the ...

latitude
. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and ''noted'' as such, unless there is no chance of misunderstanding. Mathematicians consider a sphere to be a two-dimensional
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
embedded in three-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. They draw a distinction a ''sphere'' and a ''
ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective ...
'', which is a three-dimensional
manifold with boundary The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
that includes the volume contained by the sphere. An ''open ball'' excludes the sphere itself, while a ''closed ball'' includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of a (closed or open) ball. The distinction between ''ball'' and ''sphere'' has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
" and "
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
" in the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
is similar.


Equations

In
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, a sphere with center and radius is the
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * Locus (magazine), ''Locus'' (magazine), science fiction and fantasy magazine ...
of all points such that : (x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2. Since it can be expressed as a quadratic polynomial, a sphere is a
quadric surface In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims ...
, a type of
algebraic surface In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Let be real numbers with and put :x_0 = \frac, \quad y_0 = \frac, \quad z_0 = \frac, \quad \rho = \frac. Then the equation :f(x,y,z) = a(x^2 + y^2 +z^2) + 2(bx + cy + dz) + e = 0 has no real points as solutions if \rho < 0 and is called the equation of an imaginary sphere. If \rho = 0, the only solution of f(x,y,z) = 0 is the point P_0 = (x_0,y_0,z_0) and the equation is said to be the equation of a point sphere. Finally, in the case \rho > 0, f(x,y,z) = 0 is an equation of a sphere whose center is P_0 and whose radius is \sqrt \rho. If in the above equation is zero then is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a
point at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
..


Parametric

A
parametric equation In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...
for the sphere with radius r > 0 and center (x_0,y_0,z_0) can be parameterized using
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

trigonometric function
s. :\begin x &= x_0 + r \sin \theta \; \cos\varphi \\ y &= y_0 + r \sin \theta \; \sin\varphi \\ z &= z_0 + r \cos \theta \,\end The symbols used here are the same as those used in
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

spherical coordinates
. is constant, while varies from 0 to and \varphi varies from 0 to 2.


Enclosed volume

In three dimensions, the
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
inside a sphere (that is, the volume of a
ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective ...
, but classically referred to as the volume of a sphere) is : V = \frac\pi r^3 = \frac\ d^3 \approx 0.5236 \cdot d^3 where is the radius and is the diameter of the sphere.
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the
circumscribe In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
d
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

cylinder
of that sphere (having the height and diameter equal to the diameter of the sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying
Cavalieri's principle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
. This formula can also be derived using
integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Deriv ...
, i.e.
disk integration Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
to sum the volumes of an infinite number of
circular Circular may refer to: * The shape of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is ...
disks of infinitesimally small thickness stacked side by side and centered along the -axis from to , assuming the sphere of radius is centered at the origin. At any given , the incremental volume () equals the product of the cross-sectional
area of the disk Area is the quantity that expresses the extent of a two-dimensional Region (mathematics), region, shape, or planar lamina, in the Plane (geometry), plane. Surface area is its analog on the two-dimensional Surface (topology), surface of a solid ...
at and its thickness (): : \delta V \approx \pi y^2 \cdot \delta x. The total volume is the summation of all incremental volumes: : V \approx \sum \pi y^2 \cdot \delta x. In the limit as approaches zero, this equation becomes: : V = \int_^ \pi y^2 dx. At any given , a right-angled triangle connects , and to the origin; hence, applying the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

Pythagorean theorem
yields: : y^2 = r^2 - x^2. Using this substitution gives : V = \int_^ \pi \left(r^2 - x^2\right)dx, which can be evaluated to give the result : V = \pi \left ^2x - \frac \right^ = \pi \left(r^3 - \frac \right) - \pi \left(-r^3 + \frac \right) = \frac43\pi r^3. An alternative formula is found using
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

spherical coordinates
, with
volume elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: dV=r^2\sin\theta\, dr\, d\theta\, d\varphi so : V=\int_0^ \int_^ \int_0^r r'^2\sin\theta\, dr'\, d\theta\, d\varphi = 2\pi \int_^ \int_0^r r'^2\sin\theta\, dr'\, d\theta = 4\pi \int_0^r r'^2\, dr'\ =\frac43\pi r^3. For most practical purposes, the volume inside a sphere
inscribed{{unreferenced, date=August 2012 Image:Inscribed circles.svg, frame, Inscribed circles of various polygons image:Circumcentre.svg, An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid ...
in a cube can be approximated as 52.4% of the volume of the cube, since , where is the diameter of the sphere and also the length of a side of the cube and  ≈ 0.5236. For example, a sphere with diameter 1m has 52.4% the volume of a cube with edge length 1m, or about 0.524 m3.


Surface area

The
surface area The surface area of a Solid geometry, solid object is a measure of the total area that the Surface (mathematics), surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more ...

surface area
of a sphere of radius is: :A = 4\pi r^2.
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
first derived this formula from the fact that the projection to the lateral surface of a
circumscribe In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
d cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
of the formula for the volume with respect to because the total volume inside a sphere of radius can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness. At any given radius , the incremental volume () equals the product of the surface area at radius () and the thickness of a shell (): :\delta V \approx A(r) \cdot \delta r. The total volume is the summation of all shell volumes: :V \approx \sum A(r) \cdot \delta r. In the limit as approaches zero this equation becomes: :V = \int_0^r A(r) \, dr. Substitute : :\frac43\pi r^3 = \int_0^r A(r) \, dr. Differentiating both sides of this equation with respect to yields as a function of : :4\pi r^2 = A(r). This is generally abbreviated as: :A = 4\pi r^2, where is now considered to be the fixed radius of the sphere. Alternatively, the area element on the sphere is given in
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

spherical coordinates
by . In
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
, the area element is : dS=\frac\prod_dx_,\;\forall k. The total area can thus be obtained by : :A = \int_0^ \int_0^\pi r^2 \sin\theta \, d\theta \, d\varphi = 4\pi r^2. The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Gerridae, water strid ...

surface tension
locally minimizes surface area. The surface area relative to the mass of a ball is called the
specific surface area Specific surface area (SSA) is a property of solid Solid is one of the four fundamental states of matter (the others being liquid, gas and plasma). The molecules in a solid are closely packed together and contain the least amount of kin ...
and can be expressed from the above stated equations as :\mathrm = \frac = \frac, where is the
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

density
(the ratio of mass to volume).


Geometric properties

A sphere can be constructed as the surface formed by rotating a
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
about any of its
diameter In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

diameter
s. Since a circle is a special type of
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
, a sphere is a special type of
ellipsoid of revolution A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occu ...
. Replacing the circle with an ellipse rotated about its
major axis In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
, the shape becomes a prolate
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipse In math ...

spheroid
; rotated about the minor axis, an oblate spheroid. A sphere is uniquely determined by four points that are not
coplanarIn geometry, a set of points in space are coplanar if there exists a geometric Plane (mathematics), plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear points, non-collinear, t ...

coplanar
. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine a unique circle in a plane. Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle. By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).. The angle between two spheres at a real point of intersection is the
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, ele ...

dihedral angle
determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are
orthogonal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

orthogonal
) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.


Pencil of spheres

If and are the equations of two distinct spheres then :s f(x,y,z) + t g(x,y,z) = 0 is also the equation of a sphere for arbitrary values of the parameters and . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.


Spherical geometry

The basic elements of
Euclidean plane geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; el, Αλεξάνδρ ...
are
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
and
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...
. On the sphere, points are defined in the usual sense. The analogue of the "line" is the
geodesic In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

geodesic
, which is a
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by
arc length Arc length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...

arc length
shows that the shortest path between two points lying on the sphere is the shorter segment of the
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
that includes the points. Many theorems from classical geometry hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's
postulate An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s, including the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

parallel postulate
. In
spherical trigonometry Spherical trigonometry is the branch of spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (plan ...
,
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

angle
s are defined between great circles. Spherical trigonometry differs from ordinary
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

trigonometry
in many respects. For example, the sum of the interior angles of a
spherical triangle Spherical trigonometry is the branch of spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (plan ...

spherical triangle
always exceeds 180 degrees. Also, any two spherical triangles are congruent. Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are called ''antipodal points''—on the sphere, the distance between them is exactly half the length of the circumference. Any other (i.e. not antipodal) pair of distinct points on a sphere * lie on a unique great circle, * segment it into one minor (i.e. shorter) and one major (i.e. longer) arc, and * have the minor arc's length be the ''shortest distance'' between them on the sphere.


Differential geometry

The sphere is a
smooth surface In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensivel ...
with constant
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

Gaussian curvature
at each point equal to . As per Gauss's
Theorema Egregium without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area. Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry Differential geometry is a Mathe ...
, this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any
map projection In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surface (mathematics), surface of the globe ...
introduces some form of distortion. A sphere of radius has has area element dA = r^2 \sin \theta\, d\theta\, d\varphi. This can be found from the
volume elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
in
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

spherical coordinates
with held constant. A sphere of any radius centered at zero is an integral surface of the following
differential form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
: : x \, dx + y \, dy + z \, dz = 0. This equation reflects that position and velocity vectors of a point, and , traveling on the sphere are always
orthogonal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

orthogonal
to each other. In
Riemannian geometry#REDIRECT Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...
, the filling area conjecture states that the hemisphere is the optimal (least area) isometric filling of the
Riemannian circleImage:Sphere halve.png, A great circle divides the sphere in two equal sphere, hemispheres In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance. It is the circle equipped wi ...
.


Projective geometry

The antipodal quotient of the sphere is the surface called the
real projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which can also be thought of as the
Northern Hemisphere The Northern Hemisphere is the half of Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remain ...

Northern Hemisphere
with antipodal points of the equator identified.


Topology

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, an -sphere is defined as a space
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to the boundary of an -ball; thus, it is homeomorphic to the Euclidean -sphere, but perhaps lacking its Metric space, metric. * A 0-sphere is a pair of points with the discrete topology. * A 1-sphere is a circle (up to homeomorphism); thus, for example, (the image of) any Knot (mathematics), knot is a 1-sphere. * A 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipse In math ...

spheroid
is a 2-sphere. The -sphere is denoted . It is an example of a compact space, compact topological manifold without
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
. A sphere need not be Manifold#Differentiable manifolds, smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere). The Heine–Borel theorem implies that a Euclidean -sphere is compact. The sphere is the inverse image of a one-point set under the continuous function . Therefore, the sphere is closed. is also bounded; therefore it is compact. Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, in a process called sphere eversion.


Curves on a sphere


Circles

Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty. Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles. More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a surface of revolution whose axis contains the center of the sphere (are ''coaxial'') consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.


Loxodrome

In navigation, a rhumb line or loxodrome is an arc crossing all meridians of
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

longitude
at the same angle. Loxodromes are the same as straight lines in the Mercator projection. A rhumb line is not a spherical spiral. Except for some simple cases, the formula of a rhumb line is complicated.


Clelia curves

A Clelia curve is a curve on a sphere for which the
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

longitude
\varphi and the colatitude \theta satisfy the equation : \varphi=c\;\theta, \quad c>0. Special cases are: Viviani's curve ( c=1) and spherical spirals (c>2) such as Seiffert's spiral. Clelia curves approximate the path of satellites in polar orbit.


Spherical conics

The analog of a conic section on the sphere is a spherical conic, a quartic function, quartic curve which can be defined in several equivalent ways, including: * as the intersection of a sphere with a quadratic cone whose vertex is the sphere center; * as the intersection of a sphere with an cylinder#cylindrical surfaces, elliptic or hyperbolic cylinder whose axis passes through the sphere center; * as the locus of points whose sum or difference of great-circle distances from a pair of focus (geometry), foci is a constant. Many theorems relating to planar conic sections also extend to spherical conics.


Intersection of a sphere with a more general surface

If a sphere is intersected by another surface, there may be more complicated spherical curves. ; Example: sphere – cylinder The intersection of the sphere with equation \; x^2+y^2+z^2=r^2\; and the cylinder with equation \;(y-y_0)^2+z^2=a^2, \; y_0\ne 0\; is not just one or two circles. It is the solution of the non-linear system of equations :x^2+y^2+z^2-r^2=0 :(y-y_0)^2+z^2-a^2=0\ . (see implicit curve and the diagram)


Eleven properties of the sphere

In their book ''Geometry and the Imagination'', David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane (mathematics), plane, which can be thought of as a sphere with infinite radius. These properties are: # ''The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.'' #: The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar Circle#Circle of Apollonius, result of Apollonius of Perga for the
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
. This second part also holds for the plane (mathematics), plane. # ''The contours and plane sections of the sphere are circles.'' #: This property defines the sphere uniquely. # ''The sphere has constant width and constant girth.'' #: The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the Meissner body. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other. # ''All points of a sphere are umbilics.'' #: At any point on a surface a Normal (geometry), normal direction is at right angles to the surface because the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a ''normal section,'' and the curvature of this curve is the ''normal curvature''. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures. Any closed surface will have at least four points called ''umbilical points''. At an umbilic all the sectional curvatures are equal; in particular the principal curvatures are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere. #: For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property. # ''The sphere does not have a surface of centers.'' #: For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the ''focal points'', and the set of all such centers forms the focal surface. #: For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special: #: * For channel surfaces one sheet forms a curve and the other sheet is a surface #: * For Cone (geometry), cones, cylinders, torus, tori and Dupin cyclide, cyclides both sheets form curves. #: * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere. # ''All geodesics of the sphere are closed curves.'' #: Geodesics are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property. # ''Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.'' #: It follows from isoperimetric inequality. These properties define the sphere uniquely and can be seen in
soap bubble A soap bubble is an extremely thin soap film, film of soapy water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with anot ...
s: a soap bubble will enclose a fixed volume, and
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Gerridae, water strid ...

surface tension
minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies. # ''The sphere has the smallest total mean curvature among all convex solids with a given surface area.'' #: The mean curvature is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere. # ''The sphere has constant mean curvature.'' #: The sphere is the only Embedding, imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature. # ''The sphere has constant positive Gaussian curvature.'' #:
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

Gaussian curvature
is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature. # ''The sphere is transformed into itself by a three-parameter family of rigid motions.'' #: Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see Euler angles). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the rotation group SO(3). The plane is the only other surface with a three-parameter family of transformations (translations along the - and -axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the Surface of revolution, surfaces of revolution and helicoids are the only surfaces with a one-parameter family.


Generalizations


Ellipsoids

An ellipsoid is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an affine transformation. An ellipsoid bears the same relationship to the sphere that an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
does to a circle.


Dimensionality

Spheres can be generalized to spaces of any number of dimensions. For any natural number , an "-sphere," often written as , is the set of points in ()-dimensional Euclidean space that are at a fixed distance from a central point of that space, where is, as before, a positive real number. In particular: * : a 0-sphere consists of two discrete points, and * : a 1-sphere is a
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
of radius ''r'' * : a 2-sphere is an ordinary sphere * : a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for are sometimes called hyperspheres. The -sphere of unit radius centered at the origin is denoted and is often referred to as "the" -sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.


Metric spaces

More generally, in a metric space , the sphere of center and radius is the set of points such that . If the center is a distinguished point that is considered to be the origin of , as in a norm (mathematics), normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
. Unlike a
ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective ...
, even a large sphere may be an empty set. For example, in with Euclidean metric, a sphere of radius is nonempty only if can be written as sum of squares of integers. An octahedron is a sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance.


Gallery

File:Einstein gyro gravity probe b.jpg, An image of one of the most accurate human-made spheres, as it refraction, refracts the image of Albert Einstein, Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10nm) of thickness. It was announced on 1 July 2008 that Australian scientists had created even more nearly perfect spheres, accurate to 0.3nm, as part of an international hunt to find a new global standard kilogram.New Scientist , Technology , Roundest objects in the world created
File:King of spades- spheres.jpg, Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres


Regions

* Spherical cap * Spherical lune * Spherical polygon * Spherical sector * Spherical segment * Spherical wedge * Spherical zone


See also

* 3-sphere * Affine sphere * Alexander horned sphere * Celestial spheres * Curvature * Directional statistics * Dyson sphere * Gauss map * Hand with Reflecting Sphere, M.C. Escher self-portrait drawing illustrating reflection and the optical properties of a mirror sphere * Hoberman sphere * Homology sphere * Homotopy groups of spheres * Homotopy sphere * Lenart Sphere * Napkin ring problem * Orb (optics) * Pseudosphere * Riemann sphere * Solid angle * Sphere packing * Spherical coordinates * Spherical cow * Spherical helix, tangent indicatrix of a curve of constant precession * Spherical polyhedron * Sphericity * Tennis ball theorem * Zoll surface, Zoll sphere


Notes and references


Notes


References


Further reading

* . * * . * . * .


External links

* b:Mathematica/Uniform Spherical Distribution, Mathematica/Uniform Spherical Distribution
Surface area of sphere proof
{{Authority control Differential geometry Differential topology Elementary geometry Elementary shapes Homogeneous spaces Spheres, Surfaces Topology