As mentioned earlierr is the sphere's radius; any line from the center to a point on the sphere is also called a radius. 'Radius' is used in two senses: as a line segment and also as its length.[3]
If a radius is extended through the center to the opposite side of the sphere, it creates adiameter. Like the radius, the length of a diameter is also called the diameter, and denotedd. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius,d = 2r. Two points on the sphere connected by a diameter areantipodal points of each other.[3]
Aunit sphere is a sphere with unit radius (r = 1). For convenience, spheres are often taken to have their center at the origin of thecoordinate system, and spheres in this article have their center at the origin unless a center is mentioned.
Agreat circle on the sphere has the same center and radius as the sphere, and divides it into two equalhemispheres.
Although thefigure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere.A particular line passing through its center defines anaxis (as in Earth'saxis of rotation).The sphere-axis intersection defines two antipodalpoles (north pole andsouth pole). The great circle equidistant to the poles is called theequator. Great circles through the poles are called lines oflongitude ormeridians. Small circles on the sphere that are parallel to the equator arecircles of latitude (orparallels). 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.[3]
Mathematicians consider a sphere to be atwo-dimensionalclosed surfaceembedded in three-dimensionalEuclidean space. They draw a distinction between asphere and aball, which is asolid figure, a three-dimensionalmanifold with boundary that includes the volume contained by the sphere. Anopen ball excludes the sphere itself, while aclosed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is theboundary of a (closed or open) ball. The distinction betweenball andsphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in theplane is similar.
Small spheres or balls are sometimes calledspherules (e.g., inMartian spherules).
Leta, b, c, d, e be real numbers witha ≠ 0 and put
Then the equation
has no real points as solutions if and is called the equation of animaginary sphere. If, the only solution of is the point and the equation is said to be the equation of apoint sphere. Finally, in the case, is an equation of a sphere whose center is and whose radius is.[2]
Ifa in the above equation is zero thenf(x,y,z) = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is apoint at infinity.[4]
In three dimensions, thevolume inside a sphere (that is, the volume of aball, but classically referred to as the volume of a sphere) is
wherer is the radius andd is the diameter of the sphere.Archimedes first derived this formula (On the Sphere and Cylinder c. 225 BCE) by showing that the volume inside a sphere is twice the volume between the sphere and thecircumscribedcylinder of that sphere (having the height and diameter equal to the diameter of the sphere).[6] 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 applyingCavalieri's principle.[7] This formula can also be derived usingintegral calculus (i.e.,disk integration) to sum the volumes of aninfinite number ofcircular disks of infinitesimally small thickness stacked side by side and centered along thex-axis fromx = −r tox =r, assuming the sphere of radiusr is centered at the origin.
Proof of sphere volume, using calculus
At any givenx, the incremental volume (δV) equals the product of the cross-sectionalarea of the disk atx and its thickness (δx):
The total volume is the summation of all incremental volumes:
In the limit asδx approaches zero,[8] this equation becomes:
At any givenx, a right-angled triangle connectsx,y andr to the origin; hence, applying thePythagorean theorem yields:
For most practical purposes, the volume inside a sphereinscribed in a cube can be approximated as 52.4% of the volume of the cube, sinceV =π/6d3, whered is the diameter of the sphere and also the length of a side of the cube andπ/6 ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1m, or about 0.524 m3.
Archimedes first derived this formula[9] from the fact that the projection to the lateral surface of acircumscribed cylinder is area-preserving.[10] Another approach to obtaining the formula comes from the fact that it equals thederivative of the formula for the volume with respect tor because the total volume inside a sphere of radiusr 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 radiusr. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radiusr is simply the product of the surface area at radiusr and the infinitesimal thickness.
Proof of surface area, using calculus
At any given radiusr,[note 1] the incremental volume (δV) equals the product of the surface area at radiusr (A(r)) and the thickness of a shell (δr):
The total volume is the summation of all shell volumes:
In the limit asδr approaches zero[8] this equation becomes:
SubstituteV:
Differentiating both sides of this equation with respect tor yieldsA as a function ofr:
This is generally abbreviated as:
wherer is now considered to be the fixed radius of the sphere.
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.[11] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because thesurface tension locally minimizes surface area.
The surface area relative to the mass of a ball is called thespecific surface area and can be expressed from the above stated equations as
whereρ is thedensity (the ratio of mass to volume).
A sphere can be constructed as the surface formed by rotating acircle one half revolution about any of itsdiameters; this is very similar to the traditional definition of a sphere as given inEuclid's Elements. Since a circle is a special type ofellipse, a sphere is a special type ofellipsoid of revolution. Replacing the circle with an ellipse rotated about itsmajor axis, the shape becomes a prolatespheroid; rotated about the minor axis, an oblate spheroid.[12]
A sphere is uniquely determined by four points that are notcoplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.[13] This property is analogous to the property that threenon-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 thecommon 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 theradical plane of the intersecting spheres.[14] 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).[15]
The angle between two spheres at a real point of intersection is thedihedral 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.[16] They intersect at right angles (areorthogonal) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.[4]
Iff(x,y,z) = 0 andg(x,y,z) = 0 are the equations of two distinct spheres then
is also the equation of a sphere for arbitrary values of the parameterss andt. The set of all spheres satisfying this equation is called apencil 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.[4]
A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius: the radius of the sphere. This means that every point on the sphere will be an umbilical point.
In their bookGeometry and the Imagination,David Hilbert andStephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere.[17] Several properties hold for theplane, 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 similarresult ofApollonius of Perga for thecircle. This second part also holds for theplane.
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 theMeissner body. The girth of a surface is thecircumference of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
At any point on a surface anormal 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 anormal section, and the curvature of this curve is thenormal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called theprincipal curvatures. Any closed surface will have at least four points calledumbilical points. At an umbilic all the sectional curvatures are equal; in particular theprincipal 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 thefocal points, and the set of all such centers forms thefocal surface.
For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:
* Forchannel surfaces one sheet forms a curve and the other sheet is a surface
* 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 fromisoperimetric inequality. These properties define the sphere uniquely and can be seen insoap bubbles: a soap bubble will enclose a fixed volume, andsurface 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.
Themean 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 onlyembedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces asminimal surfaces 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 isembedded 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. Thepseudosphere 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 (seeEuler angles). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is therotation group SO(3). The plane is the only other surface with a three-parameter family of transformations (translations along thex- andy-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and thesurfaces of revolution andhelicoids are the only surfaces with a one-parameter family.
The basic elements ofEuclidean plane geometry arepoints andlines. On the sphere, points are defined in the usual sense. The analogue of the "line" is thegeodesic, which is agreat 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 byarc length shows that the shortest path between two points lying on the sphere is the shorter segment of thegreat circle that includes the points.
Many theorems fromclassical geometry hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry'spostulates, including theparallel postulate. Inspherical trigonometry,angles are defined between great circles. Spherical trigonometry differs from ordinarytrigonometry in many respects. For example, the sum of the interior angles of aspherical triangle always exceeds 180 degrees. Also, any twosimilar 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 calledantipodal points – on the sphere, the distance between them is exactly half the length of the circumference.[note 2] 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 theshortest distance between them on the sphere.[note 3]
The sphere is asmooth surface with constantGaussian curvature at each point equal to1/r2.[9] As per Gauss'sTheorema Egregium, 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, anymap projection introduces some form of distortion.
This equation reflects that the position vector andtangent plane at a point are alwaysorthogonal to each other. Furthermore, the outward-facingnormal vector is equal to the position vector scaled by1/r.
Remarkably, it is possible to turn an ordinary sphere inside out in athree-dimensional space with possible self-intersections but without creating any creases, in a process calledsphere eversion.
The antipodal quotient of the sphere is the surface called thereal projective plane, which can also be thought of as theNorthern Hemisphere with antipodal points of the equator identified.
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.[18] 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 asurface of revolution whose axis contains the center of the sphere (arecoaxial) 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.
Innavigation, aloxodrome orrhumb line is a path whosebearing, the angle between its tangent and due North, is constant. Loxodromes project to straight lines under theMercator projection. Two special cases are themeridians which are aligned directly North–South andparallels which are aligned directly East–West. For any other bearing, a loxodrome spirals infinitely around each pole. For the Earth modeled as a sphere, or for a general sphere given aspherical coordinate system, such a loxodrome is a kind ofspherical spiral.[19]
Another kind of spherical spiral is the Clelia curve, for which thelongitude (or azimuth) and thecolatitude (or polar angle) are in a linear relationship,. Clelia curves project to straight lines under theequirectangular projection.Viviani's curve () is a special case. Clelia curves approximate theground track of satellites inpolar orbit.
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
Anellipsoid is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under anaffine transformation. An ellipsoid bears the same relationship to the sphere that anellipse does to a circle.
Spheres can be generalized to spaces of any number ofdimensions. For anynatural numbern, ann-sphere, often denotedSn, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distancer from a central point of that space, wherer is, as before, a positive real number. In particular:
S0: a 0-sphere consists of two discrete points,−r andr
S3: a3-sphere is a sphere in 4-dimensional Euclidean space.
Spheres forn > 2 are sometimes calledhyperspheres.
Then-sphere of unit radius centered at the origin is denotedSn and is often referred to as "the"n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
The sphere is the inverse image of a one-point set under the continuous function‖x‖, so it is closed;Sn is also bounded, so it is compact by theHeine–Borel theorem.
More generally, in ametric space(E,d), the sphere of centerx and radiusr > 0 is the set of pointsy such thatd(x,y) =r.
If the center is a distinguished point that is considered to be the origin ofE, as in anormed 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 aunit sphere.
Unlike aball, even a large sphere may be an empty set. For example, inZn withEuclidean metric, a sphere of radiusr is nonempty only ifr2 can be written as sum ofn squares ofintegers.
The geometry of the sphere was studied by the Greeks.Euclid's Elements 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 toEudoxus of Cnidus. The volume and area formulas were first determined inArchimedes'sOn the Sphere and Cylinder by themethod of exhaustion.Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.[3]
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 byDionysodorus.[20] A similar problem – to construct a segment equal in volume to a given segment, and in surface to another segment – was solved later byal-Quhi.[3]
An image of one of the most accurate human-made spheres, as itrefracts the image ofEinstein in the background. This sphere was afused quartzgyroscope for theGravity 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 thatAustralian scientists had created even more nearly perfect spheres, accurate to 0.3nm, as part of an international huntto find a new global standard kilogram.[21]
Deck of playing cards illustrating engineering instruments, England, 1702.King of spades: Spheres
^Fried, Michael N. (25 February 2019)."conic sections".Oxford Research Encyclopedia of Classics.doi:10.1093/acrefore/9780199381135.013.8161.ISBN978-0-19-938113-5. Retrieved4 November 2022.More significantly, Vitruvius (On Architecture, Vitr. 9.8) associated conical sundials with Dionysodorus (early 2nd century bce), and Dionysodorus, according to Eutocius of Ascalon (c. 480–540 ce), used conic sections to complete a solution for Archimedes' problem of cutting a sphere by a plane so that the ratio of the resulting volumes would be the same as a given ratio.
![An image of one of the most accurate human-made spheres, as it refracts the image of 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 10 nm) of thickness. It was announced on 1 July 2008 that Australian scientists had created even more nearly perfect spheres, accurate to 0.3 nm, as part of an international hunt to find a new global standard kilogram.[21]](/mt/?noimg=&dark=on&url=http%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aEinstein_gyro_gravity_probe_b.jpg)
