geometrical
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 ...
object that is a
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
analogue to a two-dimensional
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
. A sphere is the
set of points
A railroad switch (), turnout, or ''set ofpoints () is a mechanical installation enabling railway trains to be guided from one track to another, such as at a railway junction or where a spur or siding branches off.
The most common ty ...
that are all at the same distance from a given point in three-dimensional space.. That given point is the
centre
Center or centre may refer to:
Mathematics
* Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentri ...
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 and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
.
The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry.
Bubble
Bubble, Bubbles or The Bubble may refer to:
Common uses
* Bubble (physics), a globule of one substance in another, usually gas in a liquid
** Soap bubble
* Economic bubble, a situation where asset prices are much higher than underlying funda ...
geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
. 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 o ...
and most
curved mirror
A curved mirror is a mirror with a curved reflecting surface. The surface may be either ''convex'' (bulging outward) or ''concave'' (recessed inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are ...
s and
lens
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
es are based on spheres. Spheres
roll
Roll or Rolls may refer to:
Movement about the longitudinal axis
* Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis
** Roll (aviation) ...
smoothly in any direction, so most balls used in sports and toys are spherical, as are
ball bearings
A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races.
The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this ...
.
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, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
. 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
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...
s of each other.
A
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
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'' on the sphere has the same center and radius as the sphere, and divides it into two equal ''hemispheres''.
Although the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
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''. Great circles through the poles are called lines of
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
or meridians. A line connecting the two poles may be called the axis of rotation. Small circles on the sphere that are parallel to the equator are lines of
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
. 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
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. They draw a distinction a ''sphere'' and a '' ball'', which is a three-dimensional
manifold with boundary
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
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), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
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 shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
is similar.
Small spheres are sometimes called spherules, e.g. in
Martian spherules
Martian spherules (also known as hematite spherules, blueberries, & Martian blueberries) are small spherules (roughly spherical pebbles) that are rich in an iron oxide (grey hematite, α-Fe2O3) and are found at Meridiani Planum (a large plain on ...
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), science fiction and fantasy magazine
** ''Locus Award' ...
of all points such that
:
Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface, a type of algebraic surface.
Let be real numbers with and put
:
Then the equation
:
has no real points as solutions if and is called the equation of an imaginary sphere. If , the only solution of is the point and the equation is said to be the equation of a point sphere. Finally, in the case , is an equation of a sphere whose center is and whose radius is .
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
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
..
Parametric
A parametric equation for the sphere with radius and center can be parameterized using
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s.
:
The symbols used here are the same as those used in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
. is constant, while varies from 0 to and varies from 0 to 2.
Properties
Enclosed volume
In three dimensions, the
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
inside a sphere (that is, the volume of a ball, but classically referred to as the volume of a sphere) is
:
where is the radius and is the diameter of the sphere. 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, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
d
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 ...
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. This formula can also be derived using
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
infinite number
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...
of
circular
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circular ...
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 at and its thickness ():
:
The total volume is the summation of all incremental volumes:
:
In the limit as approaches zero, this equation becomes:
:
At any given , a right-angled triangle connects , and to the origin; hence, applying the Pythagorean theorem yields:
:
Using this substitution gives
:
which can be evaluated to give the result
:
An alternative formula is found using
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, with
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
:
so
:
For most practical purposes, the volume inside a sphere
inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
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 of a sphere of radius is:
:
Archimedes first derived this formula from the fact that the projection to the lateral surface of a
circumscribe
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
d cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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 ():
:
The total volume is the summation of all shell volumes:
:
In the limit as approaches zero this equation becomes:
:
Substitute :
:
Differentiating both sides of this equation with respect to yields as a function of :
:
This is generally abbreviated as:
:
where is now considered to be the fixed radius of the sphere.
Alternatively, the area element on the sphere is given in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
by . In Cartesian coordinates, the area element is
:
The total area can thus be obtained by integration:
:
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 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 solids defined as the total surface area of a material per unit of mass, (with units of m2/kg or m2/g) or solid or bulk volume (units of m2/m3 or m−1).
It is a physical value that can be used to dete ...
and can be expressed from the above stated equations as
:
where is the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
(the ratio of mass to volume).
Other geometric properties
A sphere can be constructed as the surface formed by rotating a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
about any of its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
s; this is essentially the traditional definition of a sphere as given in
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
. Since a circle is a special type of ellipse, a sphere is a special type of
ellipsoid of revolution
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
. Replacing the circle with an ellipse rotated about its
major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
, the shape becomes a prolate
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
; rotated about the minor axis, an oblate spheroid.
A sphere is uniquely determined by four points that are not
coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...
. 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, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
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) 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
:
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.
''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
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, 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
result
A result (also called upshot) is the final consequence of a sequence of actions or events expressed qualitatively or quantitatively. Possible results include advantage, disadvantage, gain, injury, loss, value and victory. There may be a range ...
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
. This second part also holds for the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
.
# ''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 Reuleaux tetrahedron is the intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three verti ...
. The girth of a surface is the
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
# ''All points of a sphere are
umbilic
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
s.''
#: At any point on a surface a normal direction is at right angles to the surface because on 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 curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends b ...
s. Any closed surface will have at least four points called ''
umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...
s''. At an umbilic all the sectional curvatures are equal; in particular the
principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends b ...
s 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 a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at th ...
.
#: 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
cones
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 conn ...
, cylinders, tori and 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
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
. These properties define the sphere uniquely and can be seen in soap bubbles: a soap bubble will enclose a fixed volume, and 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 In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
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 imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s have constant mean curvature.
# ''The sphere has constant positive Gaussian curvature.''
#: 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
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface ...
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)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a ...
. 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
surfaces of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
and
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
s are the only surfaces with a one-parameter family.
Treatment by area of mathematics
Spherical geometry
The basic elements of
Euclidean plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
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 ...
. On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic, which is a 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 may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
shows that the shortest path between two points lying on the sphere is the shorter segment of the great circle that includes the points.
Many theorems from
classical geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
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 true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
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'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segmen ...
. In
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
,
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s are defined between great circles. Spherical trigonometry differs from ordinary
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
in many respects. For example, the sum of the interior angles of a
spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
always exceeds 180 degrees. Also, any two similar 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.
Spherical geometry is a form of
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
, which together with
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
makes up
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
.
Differential geometry
The sphere is a
smooth surface
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
with constant Gaussian curvature at each point equal to . As per Gauss's
Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
, 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, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longit ...
introduces some form of distortion.
A sphere of radius has area element . This can be found from the
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
with held constant.
A sphere of any radius centered at zero is an integral surface of the following differential form:
:
This equation reflects that the position vector and
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 a point are always orthogonal to each other. Furthermore, the outward-facing normal vector is equal to the position vector scaled by .
In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, the
filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.
Definition ...
states that the hemisphere is the optimal (least area) isometric filling of the
Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or ...
.
Topology
In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, an -sphere is defined as a space homeomorphic to the boundary of an -ball; thus, it is homeomorphic to the Euclidean -sphere, but perhaps lacking its
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
.
* A 0-sphere is a pair of points with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
.
* A 1-sphere is a circle ( up to homeomorphism); thus, for example, (the image of) any
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
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 surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
is a 2-sphere.
The -sphere is denoted . It is an example of a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
without
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
. A sphere need not be
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
; if it is smooth, it need not be
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two man ...
to the Euclidean sphere (an exotic sphere).
The sphere is the inverse image of a one-point set under the continuous function , so it is closed; is also bounded, so it is compact by the Heine–Borel theorem.
Remarkably, it is possible to turn an ordinary sphere inside out in a
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
with possible self-intersections but without creating any creases, in a process called sphere eversion.
The antipodal quotient of the sphere is the surface called the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
, which can also be thought of as the Northern Hemisphere with antipodal points of the equator identified.
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
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
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
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
, a rhumb line or loxodrome is an arc crossing all meridians of
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
at the same angle. Loxodromes are the same as straight lines in the Mercator projection. A rhumb line is not a
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
and the colatitude satisfy the equation
:.
Special cases are:
Viviani's curve
In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and p ...
() and
spherical spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:Seiffert's spiral
Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. If the selected diameter is the line from the north pole to the south pole, then the requir ...
. Clelia curves approximate the path of satellites in
polar orbit
A polar orbit is one in which a satellite passes above or nearly above both poles of the body being orbited (usually a planet such as the Earth, but possibly another body such as the Moon or Sun) on each revolution. It has an inclination of about ...
.
Spherical conics
The analog of a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
on the sphere is a
spherical conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...
, a 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 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
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
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 and the cylinder with equation is not just one or two circles. It is the solution of the non-linear system of equations
:
:
(see
implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every impli ...
and the diagram)
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 does to a circle.
Dimensionality
Spheres can be generalized to spaces of any number of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s. For any
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
, an ''-sphere,'' often denoted , 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 shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
of radius ''r''
* : a 2-sphere is an ordinary sphere
* : a
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
is a sphere in 4-dimensional Euclidean space.
Spheres for are sometimes called
hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
s.
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
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, 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 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. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
.
Unlike a ball, even a large sphere may be an empty set. For example, in with
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
, 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.
History
The geometry of the sphere was studied by the Greeks. ''
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
'' defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus. The volume and area formulas were first determined in Archimedes's ''On the Sphere and Cylinder'' by the method of exhaustion. Zenodorus (mathematician), Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.
Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by Dionysodorus. A similar problem — to construct a segment equal in volume to a given segment, and in surface to another segment — was solved later by al-Quhi.
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 Alternative approaches to redefining the kilogram#Alternative approaches to redefining the kilogram, 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
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
* 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
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Differential geometry
Differential topology
Elementary geometry
Elementary shapes
Homogeneous spaces
Spheres,
Surfaces
Topology