5677:
5676:
5724:
1661:
49:
4584:
4687:
4888:
268:
4576:
4205:
2817:
4629:
3888:
4614:) 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.
2536:
1602:
1792:
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
4142:
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
3891:
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
6421:
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
5631:
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
4602:
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
3878:
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.
3565:
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
2446:
4092:
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
860:
4019:
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
3555:
1768:
2812:{\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta \,d\varphi =2\pi \int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta =4\pi \int _{0}^{r}r'^{2}\,dr'\ ={\frac {4}{3}}\pi r^{3}.}
1448:
4069:
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
3983:
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
4157:
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
2243:
3716:
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).
682:
3642:
649:
2232:
3288:
2525:
1003:
5072:
4348:, 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
4419:
5714:
1928:
1985:
3072:
1453:
3128:
5657:
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
2967:
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
5232:
4076:
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.
3698:. 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
2058:
4978:
3724:
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
5147:
1181:
3198:
5723:
3861:
4504:
1436:
4664:
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 a
2140:
4777:
4287: – 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
3357:
394:). 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.
1309:
281:
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.
3405:
2943:
4138:
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
3449:
1258:
1211:
1109:
1032:
4056:* 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.
5661:. A similar problem – to construct a segment equal in volume to a given segment, and in surface to another segment – was solved later by
4722:
1641:
4746:
1062:
1377:
1285:
4815:
1682:
1597:{\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \\z&=z_{0}+r\cos \theta \,\end{aligned}}}
6609:
2971:. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius
4009:
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.
3957:
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
348:
is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. A particular line passing through its center defines an
4235:; 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
3917:
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.
2441:{\displaystyle V=\pi \left_{-r}^{r}=\pi \left(r^{3}-{\frac {r^{3}}{3}}\right)-\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right)={\frac {4}{3}}\pi r^{3}.}
4107:
is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.
855:{\displaystyle x_{0}={\frac {-b}{a}},\quad y_{0}={\frac {-c}{a}},\quad z_{0}={\frac {-d}{a}},\quad \rho ={\frac {b^{2}+c^{2}+d^{2}-ae}{a^{2}}}.}
3992:. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the
3921:
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
3908:
describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the
6577:
5555:
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
3583:
4162:). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the
6064:
6602:
4174:-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the
528:
4031:
For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:
2151:
442:
has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "
3213:
6010:
2464:
871:
6387:
4983:
3705:
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
5258:
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
294:. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius,
4365:
1882:
6595:
6521:
6494:
6459:
6407:
6331:
6157:
5425:-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
1939:
6433:
4250:
hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's
3027:
4281:
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e., the diameter) are called
3083:
150:
3728:) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.
1785:
first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the
87:
5153:
4121:
surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as
4907:
2880: ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1
6582:
4006:
are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
3709:
2000:
4915:
6887:
6882:
5080:
4166:. The plane is the only other surface with a three-parameter family of transformations (translations along the
1114:
3143:
2955:
cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the
3737:
1337:
is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a
6892:
4652:, the angle between its tangent and due North, is constant. Loxodromes project to straight lines under the
3786:
3712:, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the
6587:
4450:
6902:
6748:
5838:
4665:
1382:
6216:
2089:
6897:
6473:
The
Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities
5858:
5642:
4782:
4753:
3965:
of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
4606:
More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a
3311:
5651:
3550:{\displaystyle A=\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin \theta \,d\theta \,d\varphi =4\pi r^{2}.}
1290:
430:
includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the
17:
54:
6784:
6552:
5935:
4510:
4216:
4099:
The sphere has the smallest total mean curvature among all convex solids with a given surface area.
4081:
3368:
2906:
1794:
5654:
was the first to state that, for a given surface area, the sphere is the solid of maximum volume.
4239:
shows that the shortest path between two points lying on the sphere is the shorter segment of the
6857:
6788:
6120:
5808:
5479:
4549:
4533:
3680:
183:
31:
1216:
1190:
1067:
1011:
5823:
5744:
4895:
If a sphere is intersected by another surface, there may be more complicated spherical curves.
4318:
4259:
1439:
419:
6478:
6399:
5990:
The distance between two non-distinct points (i.e., a point and itself) on the sphere is zero.
4707:
1626:
6724:
6666:
5903:
5701:
experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10
5259:
4868:
4731:
4607:
4560:
4426:
4175:
3574:
3422:
2452:
1608:
1041:
6470:
5448:
1356:
326:, and spheres in this article have their center at the origin unless a center is mentioned.
284:
If a radius is extended through the center to the opposite side of the sphere, it creates a
6846:
6821:
6513:
6482:
5920:
5845:
self-portrait drawing illustrating reflection and the optical properties of a mirror sphere
5647:
5627:
5440:
4657:
4649:
4163:
3672:
1865:
1763:{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot d^{3}}
1263:
381:
77:
4227:. On the sphere, points are defined in the usual sense. The analogue of the "line" is the
335:
on the sphere has the same center and radius as the sphere, and divides it into two equal
8:
6816:
6810:
5930:
5873:
5444:
5074:
is not just one or two circles. It is the solution of the non-linear system of equations
4794:
4653:
4564:
4529:
4314:
4003:
3993:
3934:
3926:
3909:
3905:
2951:
first derived this formula from the fact that the projection to the lateral surface of a
2080:
1789:
1350:
503:
431:
173:
6486:
4861:
6711:
6320:
5914:
5614:
5563:
5552:
4786:
4661:
4597:
4552:
with possible self-intersections but without creating any creases, in a process called
4334:
4271:
4255:
4247:
4224:
4199:
4135:
1673:
447:
414:
387:
6058:
322:). For convenience, spheres are often taken to have their center at the origin of the
6917:
6912:
6690:
6517:
6506:
6490:
6471:
6455:
6403:
6348:
6327:
6179:
6153:
5813:
5774:
5764:
5633:
4537:
4441:
4345:
4310:
4275:
3980:
1813:
disks of infinitesimally small thickness stacked side by side and centered along the
1798:
1338:
659:
470:
458:
323:
288:. Like the radius, the length of a diameter is also called the diameter, and denoted
201:
187:
70:
30:
This article is about the concept in three-dimensional geometry. For other uses, see
6351:
6233:
5977:
It does not matter which direction is chosen, the distance is the sphere's radius ×
4857:
The intersection of a sphere with a quadratic cone whose vertex is the sphere center
3922:
1660:
6907:
6672:
6395:
6228:
5769:
5606:
5573:
4872:
4850:
4669:
4437:
4220:
4153:
The sphere is transformed into itself by a three-parameter family of rigid motions.
2828:
1802:
451:
355:
350:
236:
228:
212:
92:
3912:, which can be thought of as a sphere with infinite radius. These properties are:
6774:
5863:
5853:
5848:
5779:
5698:
5687:
5462:
The sphere is the inverse image of a one-point set under the continuous function
4879:
Many theorems relating to planar conic sections also extend to spherical conics.
4846:
4836:
4553:
4282:
4122:
4089:
4042:
4035:
3998:
3567:
1806:
655:
405:
345:
305:
220:
6736:
4536:
states that the hemisphere is the optimal (least area) isometric filling of the
4266:
are defined between great circles. Spherical trigonometry differs from ordinary
6372:
5898:
5888:
5784:
5759:
5730:
5285:
5238:
4422:
4349:
4330:
4295:
4104:
3721:
2456:
398:
165:
66:
5547:
If the center is a distinguished point that is considered to be the origin of
6876:
6837:
6793:
6779:
6677:
6315:
6068:. Vol. 25 (11th ed.). Cambridge University Press. pp. 647–648.
6053:
5908:
5868:
5803:
5754:
5456:
5452:
5437:
4842:
4518:
4514:
4050:
4025:
3962:
3958:
3901:
3725:
3671:; this is very similar to the traditional definition of a sphere as given in
256:
240:
146:
6182:
2862:
is the diameter of the sphere and also the length of a side of the cube and
48:
6684:
6617:
6422:
that the ratio of the resulting volumes would be the same as a given ratio.
5940:
5883:
5878:
5828:
5729:
Deck of playing cards illustrating engineering instruments, England, 1702.
5691:
5658:
5497:
5491:
4820:
4360:
4267:
4240:
4232:
4208:
4159:
4144:
3418:
2952:
2893:
1786:
331:
101:
6543:
Higher
Geometry / An Introduction to Advanced Methods in Analytic Geometry
4002:. At an umbilic all the sectional curvatures are equal; in particular the
1810:
211:. Spheres and nearly-spherical shapes also appear in nature and industry.
5893:
5842:
5556:
5402:
4824:
4085:
312:
216:
208:
5709:
scientists had created even more nearly perfect spheres, accurate to 0.3
3866:
is also the equation of a sphere for arbitrary values of the parameters
2831:
in a cube can be approximated as 52.4% of the volume of the cube, since
5925:
5683:
5637:
5602:
4882:
4725:
4637:
4623:
4294:
segment it into one minor (i.e., shorter) and one major (i.e., longer)
4236:
3684:
2956:
2948:
1782:
4548:
Remarkably, it is possible to turn an ordinary sphere inside out in a
3637:{\displaystyle \mathrm {SSA} ={\frac {A}{V\rho }}={\frac {3}{r\rho }}}
6852:
6741:
6356:
6187:
5833:
5818:
5706:
5694:
5281:
5255:
4701:
4251:
4147:
is an example of a surface with constant negative
Gaussian curvature.
4139:
4118:
4066:
3720:
The angle between two spheres at a real point of intersection is the
3699:
401:
377:
232:
224:
155:
4700:
Another kind of spherical spiral is the Clelia curve, for which the
4686:
4681:
4583:
6729:
5798:
5429:
5387:
5275:
4270:
in many respects. For example, the sum of the interior angles of a
4228:
4179:
3695:
3688:
3668:
3440:
654:
Since it can be expressed as a quadratic polynomial, a sphere is a
386:. Small circles on the sphere that are parallel to the equator are
285:
200:. The earliest known mentions of spheres appear in the work of the
161:
6434:
New
Scientist | Technology | Roundest objects in the world created
6318:; Cohn-Vossen, Stephan (1952). "Eleven properties of the sphere".
6314:
4887:
267:
5662:
5595:
5263:
5262:. An ellipsoid bears the same relationship to the sphere that an
4084:. These properties define the sphere uniquely and can be seen in
3972:
3887:
3676:
3652:
372:
248:
4575:
3663:
A sphere can be constructed as the surface formed by rotating a
644:{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}
457:
Small spheres or balls are sometimes called spherules (e.g., in
5682:
An image of one of the most accurate human-made spheres, as it
5358:
4559:
The antipodal quotient of the sphere is the surface called the
4204:
3930:
3664:
2227:{\displaystyle V=\int _{-r}^{r}\pi \left(r^{2}-x^{2}\right)dx,}
1669:
443:
197:
169:
113:
3874:. The set of all spheres satisfying this equation is called a
3573:
The surface area relative to the mass of a ball is called the
3283:{\displaystyle {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}A(r)\,dr.}
1933:
The total volume is the summation of all incremental volumes:
6643:
6057:
4263:
4046:
2520:{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi }
998:{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}
4628:
4587:
Coaxial intersection of a sphere and a cylinder: two circles
3694:
A sphere is uniquely determined by four points that are not
3293:
Differentiating both sides of this equation with respect to
1676:, but classically referred to as the volume of a sphere) is
5610:
5566:, even a large sphere may be an empty set. For example, in
5314:)-dimensional Euclidean space that are at a fixed distance
5067:{\displaystyle \;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;}
3996:. Any closed surface will have at least four points called
370:). The great circle equidistant to the poles is called the
252:
244:
3943:
The contours and plane sections of the sphere are circles.
6147:
5412:-sphere of unit radius centered at the origin is denoted
4414:{\displaystyle dA=r^{2}\sin \theta \,d\theta \,d\varphi }
4143:
boundary with constant, positive
Gaussian curvature. The
3683:. Replacing the circle with an ellipse rotated about its
397:
Mathematicians consider a sphere to be a two-dimensional
6346:
5636:. The volume and area formulas were first determined in
4038:
one sheet forms a curve and the other sheet is a surface
3577:
and can be expressed from the above stated equations as
3414:
is now considered to be the fixed radius of the sphere.
3077:
The total volume is the summation of all shell volumes:
2827:
For most practical purposes, the volume inside a sphere
1923:{\displaystyle \delta V\approx \pi y^{2}\cdot \delta x.}
6618:
Compact topological surfaces and their immersions in 3D
5705:
nm) of thickness. It was announced on 1 July 2008 that
4853:
curve which can be defined in several equivalent ways.
304:. Two points on the sphere connected by a diameter are
5968:
is being considered as a variable in this computation.
5625:
The geometry of the sphere was studied by the Greeks.
5322:
is, as before, a positive real number. In particular:
5280:
Spheres can be generalized to spaces of any number of
1980:{\displaystyle V\approx \sum \pi y^{2}\cdot \delta x.}
376:. Great circles through the poles are called lines of
358:). The sphere-axis intersection defines two antipodal
5156:
5083:
4986:
4918:
4797:
4756:
4734:
4710:
4453:
4368:
3789:
3586:
3452:
3371:
3314:
3216:
3146:
3086:
3030:
2909:
2539:
2467:
2246:
2154:
2092:
2003:
1942:
1885:
1685:
1629:
1451:
1385:
1359:
1293:
1266:
1219:
1193:
1117:
1070:
1044:
1014:
874:
685:
531:
422:
that includes the volume contained by the sphere. An
207:
The sphere is a fundamental object in many fields of
4883:
Intersection of a sphere with a more general surface
4781:. Clelia curves project to straight lines under the
4509:
This equation reflects that the position vector and
4131:
The sphere has constant positive
Gaussian curvature.
3691:; rotated about the minor axis, an oblate spheroid.
2975:
is simply the product of the surface area at radius
1607:
The symbols used here are the same as those used in
434:
of a (closed or open) ball. The distinction between
219:
take a spherical shape in equilibrium. The Earth is
4819:) is a special case. Clelia curves approximate the
4188:
3067:{\displaystyle \delta V\approx A(r)\cdot \delta r.}
3002:) equals the product of the surface area at radius
2963:because the total volume inside a sphere of radius
6535:(Third American ed.), Oxford University Press
6505:
6319:
6173:
6171:
6169:
5226:
5141:
5066:
4972:
4809:
4771:
4740:
4716:
4610:whose axis contains the center of the sphere (are
4498:
4413:
4182:are the only surfaces with a one-parameter family.
3855:
3636:
3549:
3399:
3351:
3282:
3192:
3122:
3066:
2937:
2811:
2519:
2440:
2226:
2134:
2052:
1979:
1922:
1762:
1635:
1596:
1430:
1371:
1303:
1279:
1252:
1205:
1175:
1103:
1056:
1026:
997:
854:
643:
4567:with antipodal points of the equator identified.
4088:: a soap bubble will enclose a fixed volume, and
3953:The sphere has constant width and constant girth.
3123:{\displaystyle V\approx \sum A(r)\cdot \delta r.}
1183:and the equation is said to be the equation of a
6874:
3710:common solutions of the equations of two spheres
6550:
6177:
6166:
4867:The locus of points whose sum or difference of
4517:to each other. Furthermore, the outward-facing
4274:always exceeds 180 degrees. Also, any two
3780:are the equations of two distinct spheres then
5334:: a 0-sphere consists of two discrete points,
5227:{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}
4436:A sphere of any radius centered at zero is an
4062:All geodesics of the sphere are closed curves.
4015:The sphere does not have a surface of centers.
2959:of the formula for the volume with respect to
255:used in sports and toys are spherical, as are
6603:
6221:Bulletin of the American Mathematical Society
5390:is a sphere in 4-dimensional Euclidean space.
4912:The intersection of the sphere with equation
3738:Pencil (mathematics) § Pencil of spheres
3702:points determine a unique circle in a plane.
3658:
4024:, and the set of all such centers forms the
2053:{\displaystyle V=\int _{-r}^{r}\pi y^{2}dx.}
1864:) equals the product of the cross-sectional
6326:(2nd ed.). Chelsea. pp. 215–231.
4973:{\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;}
4864:whose axis passes through the sphere center
4660:which are aligned directly North–South and
3882:
1260:is an equation of a sphere whose center is
6610:
6596:
6578:Mathematica/Uniform Spherical Distribution
6079:
6077:
6075:
6025:
6023:
5318:from a central point of that space, where
5063:
5046:
4987:
4969:
4919:
4521:is equal to the position vector scaled by
3947:This property defines the sphere uniquely.
2237:which can be evaluated to give the result
1793:the circumscribing cylinder, and applying
1672:inside a sphere (that is, the volume of a
1540:
1491:
47:
6530:
6232:
6202:
6107:
5478:is also bounded, so it is compact by the
5142:{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}
4603:sphere: others are called small circles.
4483:
4470:
4457:
4404:
4397:
3518:
3511:
3439:. The total area can thus be obtained by
3270:
3180:
2765:
2719:
2707:
2637:
2630:
2618:
2510:
2503:
2496:
1797:. This formula can also be derived using
1589:
1176:{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}
6503:
6392:Oxford Research Encyclopedia of Classics
6284:
6095:
6052:
4886:
4685:
4627:
4582:
4574:
4324:
4203:
3886:
3193:{\displaystyle V=\int _{0}^{r}A(r)\,dr.}
1994:approaches zero, this equation becomes:
1659:
266:
6400:10.1093/acrefore/9780199381135.013.8161
6143:
6141:
6121:"The volume of a sphere – Math Central"
6072:
6020:
5947:
4113:The sphere has constant mean curvature.
3988:and the curvature of this curve is the
3137:approaches zero this equation becomes:
27:Set of points equidistant from a center
14:
6875:
6468:
6449:
6386:Fried, Michael N. (25 February 2019).
6302:
6278:
6266:
6254:
6029:
6013:, Henry George Liddell, Robert Scott,
5715:to find a new global standard kilogram
4563:, which can also be thought of as the
3933:. This second part also holds for the
3675:. Since a circle is a special type of
2451:An alternative formula is found using
1849:Proof of sphere volume, using calculus
6591:
6540:
6385:
6347:
6290:
6178:
6083:
5713:nm, as part of an international hunt
5690:in the background. This sphere was a
4860:The intersection of a sphere with an
4579:Plane section of a sphere: one circle
4570:
4193:
3856:{\displaystyle sf(x,y,z)+tg(x,y,z)=0}
3667:one half revolution about any of its
2987:Proof of surface area, using calculus
6148:E.J. Borowski; J.M. Borwein (1989).
6138:
6048:
6046:
6044:
6042:
6040:
6038:
4891:General intersection sphere-cylinder
4499:{\displaystyle x\,dx+y\,dy+z\,dz=0.}
4352:introduces some form of distortion.
3731:
426:excludes the sphere itself, while a
408:. They draw a distinction between a
262:
5447:. A topological sphere need not be
4830:
4301:have the minor arc's length be the
4278:spherical triangles are congruent.
2079:to the origin; hence, applying the
2067:, a right-angled triangle connects
1431:{\displaystyle (x_{0},y_{0},z_{0})}
1318:in the above equation is zero then
1008:has no real points as solutions if
251:smoothly in any direction, so most
24:
6443:
5613:is a sphere in geometry using the
5451:; if it is smooth, it need not be
5421:and is often referred to as "the"
5375:: a 2-sphere is an ordinary sphere
5244:
3594:
3591:
3588:
3362:This is generally abbreviated as:
2135:{\displaystyle y^{2}=r^{2}-x^{2}.}
1655:
176:that are all at the same distance
25:
6929:
6571:
6551:John C. Polking (15 April 1999).
6150:Collins Dictionary of Mathematics
6035:
5917:of a curve of constant precession
5745:Ball (mathematics) § Regions
5269:
4772:{\displaystyle \varphi =c\theta }
2979:and the infinitesimal thickness.
1664:Sphere and circumscribed cylinder
1034:and is called the equation of an
6508:Advanced Engineering Mathematics
6450:Albert, Abraham Adrian (2016) ,
5722:
5675:
5485:
4675:
4668:, such a loxodrome is a kind of
4309:Spherical geometry is a form of
4189:Treatment by area of mathematics
3961:. The girth of a surface is the
3679:, a sphere is a special type of
3570:locally minimizes surface area.
3352:{\displaystyle 4\pi r^{2}=A(r).}
3017:) and the thickness of a shell (
1837:, assuming the sphere of radius
271:Two orthogonal radii of a sphere
6427:
6379:
6365:
6340:
6308:
6296:
6272:
6260:
6248:
6234:10.1090/S0002-9904-1978-14553-4
6208:
6196:
5984:
5971:
5959:
4980:and the cylinder with equation
4862:elliptic or hyperbolic cylinder
3655:(the ratio of mass to volume).
2887:
1781:is the diameter of the sphere.
1304:{\displaystyle {\sqrt {\rho }}}
781:
749:
717:
418:, which is a three-dimensional
235:. Manufactured items including
6217:"The isoperimetric inequality"
6152:. Collins. pp. 141, 149.
6113:
6101:
6089:
6004:
5632:its diameter, probably due to
5177:
5157:
5008:
4988:
4748:are in a linear relationship,
3844:
3826:
3814:
3796:
3687:, the shape becomes a prolate
3343:
3337:
3267:
3261:
3177:
3171:
3105:
3099:
3049:
3043:
2145:Using this substitution gives
1425:
1386:
1241:
1223:
1170:
1131:
1092:
1074:
980:
953:
944:
905:
896:
878:
616:
596:
584:
564:
552:
532:
315:is a sphere with unit radius (
247:are based on spheres. Spheres
221:often approximated as a sphere
168:analogue to a two-dimensional
13:
1:
6541:Woods, Frederick S. (1961) ,
6214:
5997:
5249:
4421:. This can be found from the
4291:lie on a unique great circle,
4125:have constant mean curvature.
3400:{\displaystyle A=4\pi r^{2},}
2938:{\displaystyle A=4\pi r^{2}.}
1650:
1344:
6583:Surface area of sphere proof
6553:"The Geometry of the Sphere"
6477:. New York: Wiley. pp.
6322:Geometry and the Imagination
5455:to the Euclidean sphere (an
4908:Sphere–cylinder intersection
4656:. Two special cases are the
4617:
3979:At any point on a surface a
3898:Geometry and the Imagination
464:
202:ancient Greek mathematicians
172:. Formally, a sphere is the
7:
5839:Hand with Reflecting Sphere
5791:
5436:-sphere is an example of a
5307:, is the set of points in (
4666:spherical coordinate system
4543:
4305:between them on the sphere.
3971:All points of a sphere are
3892:will be an umbilical point.
1841:is centered at the origin.
1805:) to sum the volumes of an
1438:can be parameterized using
1353:for the sphere with radius
231:is an important concept in
10:
6934:
6512:(3rd ed.), New York:
5859:Homotopy groups of spheres
5742:
5738:
5668:
5643:On the Sphere and Cylinder
5620:
5489:
5273:
4905:
4834:
4783:equirectangular projection
4679:
4621:
4595:
4591:
4243:that includes the points.
4197:
3735:
3659:Other geometric properties
3421:on the sphere is given in
2998:, the incremental volume (
2884:m, or about 0.524 m.
1860:, the incremental volume (
1253:{\displaystyle f(x,y,z)=0}
1206:{\displaystyle \rho >0}
1104:{\displaystyle f(x,y,z)=0}
1027:{\displaystyle \rho <0}
186:. That given point is the
149:
29:
6830:
6802:
6767:
6758:
6704:
6659:
6630:
6623:
6215:Osserman, Robert (1978).
5588:can be written as sum of
1668:In three dimensions, the
112:
100:
86:
76:
62:
46:
41:
6504:Kreyszig, Erwin (1972),
6469:Dunham, William (1997).
5952:
5936:Volume-equivalent radius
4717:{\displaystyle \varphi }
4217:Euclidean plane geometry
4082:isoperimetric inequality
4053:both sheets form curves.
3883:Properties of the sphere
1636:{\displaystyle \varphi }
6749:Sphere with three holes
6452:Solid Analytic Geometry
6065:Encyclopædia Britannica
6015:A Greek-English Lexicon
5809:Alexander horned sphere
5511:, the sphere of center
4741:{\displaystyle \theta }
4550:three-dimensional space
4534:filling area conjecture
4337:at each point equal to
4117:The sphere is the only
3681:ellipsoid of revolution
1440:trigonometric functions
1187:. Finally, in the case
1064:, the only solution of
1057:{\displaystyle \rho =0}
473:, a sphere with center
184:three-dimensional space
32:Sphere (disambiguation)
6533:Mathematical Snapshots
6531:Steinhaus, H. (1969),
6125:mathcentral.uregina.ca
5824:Directional statistics
5228:
5143:
5068:
4974:
4892:
4869:great-circle distances
4811:
4773:
4742:
4718:
4697:
4633:
4588:
4580:
4513:at a point are always
4500:
4415:
4319:non-Euclidean geometry
4313:, which together with
4260:spherical trigonometry
4215:The basic elements of
4212:
4176:surfaces of revolution
3893:
3857:
3638:
3551:
3401:
3353:
3284:
3194:
3124:
3068:
2939:
2896:of a sphere of radius
2813:
2521:
2442:
2228:
2136:
2054:
1981:
1924:
1764:
1665:
1637:
1598:
1432:
1373:
1372:{\displaystyle r>0}
1305:
1281:
1254:
1207:
1177:
1105:
1058:
1028:
999:
856:
645:
420:manifold with boundary
272:
182:from a given point in
55:perspective projection
6888:Differential topology
6883:Differential geometry
6667:Real projective plane
6652:Pretzel (genus 3) ...
6557:www.math.csi.cuny.edu
5904:Spherical coordinates
5576:, a sphere of radius
5522:is the set of points
5496:More generally, in a
5401:are sometimes called
5260:affine transformation
5229:
5144:
5069:
4975:
4890:
4812:
4774:
4743:
4719:
4689:
4631:
4608:surface of revolution
4586:
4578:
4561:real projective plane
4501:
4427:spherical coordinates
4416:
4325:Differential geometry
4207:
3890:
3858:
3639:
3575:specific surface area
3552:
3423:spherical coordinates
3402:
3354:
3285:
3195:
3125:
3069:
2940:
2814:
2522:
2453:spherical coordinates
2443:
2229:
2137:
2055:
1982:
1925:
1795:Cavalieri's principle
1765:
1663:
1638:
1609:spherical coordinates
1599:
1433:
1374:
1306:
1282:
1280:{\displaystyle P_{0}}
1255:
1208:
1178:
1106:
1059:
1029:
1000:
857:
669:be real numbers with
646:
404:in three-dimensional
275:As mentioned earlier
270:
6822:Euler characteristic
5948:Notes and references
5921:Spherical polyhedron
5648:method of exhaustion
5582:is nonempty only if
5441:topological manifold
5154:
5081:
4984:
4916:
4795:
4754:
4732:
4708:
4451:
4366:
4164:rotation group SO(3)
4004:principal curvatures
3994:principal curvatures
3787:
3584:
3450:
3369:
3312:
3214:
3144:
3084:
3028:
2994:At any given radius
2907:
2537:
2465:
2244:
2152:
2090:
2001:
1940:
1883:
1683:
1627:
1449:
1383:
1357:
1291:
1287:and whose radius is
1264:
1217:
1191:
1115:
1068:
1042:
1012:
872:
683:
529:
6893:Elementary geometry
6487:1994muaa.book.....D
5931:Tennis ball theorem
5874:Napkin ring problem
5480:Heine–Borel theorem
5472:, so it is closed;
4845:on the sphere is a
4810:{\displaystyle c=1}
4690:Clelia spiral with
4654:Mercator projection
4571:Curves on a sphere
4565:Northern Hemisphere
4530:Riemannian geometry
4355:A sphere of radius
4315:hyperbolic geometry
4246:Many theorems from
3927:Apollonius of Perga
3906:Stephan Cohn-Vossen
3491:
3476:
3417:Alternatively, the
3257:
3167:
2749:
2682:
2667:
2593:
2578:
2563:
2313:
2178:
2081:Pythagorean theorem
2027:
1872:and its thickness (
1615:is constant, while
1351:parametric equation
388:circles of latitude
190:of the sphere, and
6903:Homogeneous spaces
6649:Number 8 (genus 2)
6349:Weisstein, Eric W.
6180:Weisstein, Eric W.
5915:tangent indicatrix
5615:Chebyshev distance
5357:: a 1-sphere is a
5266:does to a circle.
5224:
5139:
5064:
4970:
4893:
4807:
4769:
4738:
4714:
4698:
4634:
4598:Circle of a sphere
4589:
4581:
4496:
4411:
4335:Gaussian curvature
4272:spherical triangle
4256:parallel postulate
4248:classical geometry
4213:
4200:Spherical geometry
4194:Spherical geometry
4136:Gaussian curvature
3894:
3853:
3634:
3547:
3477:
3459:
3397:
3349:
3280:
3243:
3190:
3153:
3120:
3064:
2935:
2809:
2735:
2668:
2653:
2579:
2564:
2546:
2517:
2438:
2256:
2224:
2161:
2132:
2050:
2010:
1977:
1920:
1777:is the radius and
1760:
1666:
1643:varies from 0 to 2
1633:
1594:
1592:
1428:
1369:
1301:
1277:
1250:
1203:
1173:
1101:
1054:
1024:
995:
865:Then the equation
852:
641:
273:
6898:Elementary shapes
6870:
6869:
6866:
6865:
6700:
6699:
6523:978-0-471-50728-4
6496:978-0-471-17661-9
6461:978-0-486-81026-3
6409:978-0-19-938113-5
6352:"Spheric section"
6333:978-0-8284-1087-8
6159:978-0-00-434347-1
5913:Spherical helix,
5814:Celestial spheres
5775:Spherical segment
5765:Spherical polygon
5634:Eudoxus of Cnidus
5628:Euclid's Elements
5241:and the diagram)
5220:
4823:of satellites in
4728:(or polar angle)
4538:Riemannian circle
4442:differential form
4440:of the following
4346:Theorema Egregium
4344:. As per Gauss's
4311:elliptic geometry
4303:shortest distance
3876:pencil of spheres
3732:Pencil of spheres
3708:By examining the
3673:Euclid's Elements
3632:
3614:
3562:
3561:
3301:as a function of
3225:
2824:
2823:
2791:
2779:
2420:
2402:
2353:
2293:
1799:integral calculus
1730:
1726:
1700:
1619:varies from 0 to
1339:point at infinity
1299:
847:
776:
744:
712:
660:algebraic surface
471:analytic geometry
459:Martian spherules
324:coordinate system
263:Basic terminology
166:three-dimensional
164:object that is a
139:
138:
71:Algebraic surface
16:(Redirected from
6925:
6785:Triangulatedness
6765:
6764:
6628:
6627:
6624:Without boundary
6612:
6605:
6598:
6589:
6588:
6567:
6565:
6563:
6546:
6536:
6526:
6511:
6500:
6476:
6464:
6437:
6431:
6425:
6424:
6418:
6416:
6388:"conic sections"
6383:
6377:
6376:
6369:
6363:
6362:
6361:
6344:
6338:
6337:
6325:
6312:
6306:
6300:
6294:
6288:
6282:
6276:
6270:
6264:
6258:
6252:
6246:
6245:
6243:
6241:
6236:
6212:
6206:
6200:
6194:
6193:
6192:
6175:
6164:
6163:
6145:
6136:
6135:
6133:
6131:
6117:
6111:
6105:
6099:
6093:
6087:
6081:
6070:
6069:
6061:
6050:
6033:
6027:
6018:
6008:
5991:
5988:
5982:
5975:
5969:
5967:
5963:
5770:Spherical sector
5726:
5712:
5704:
5679:
5607:taxicab geometry
5593:
5587:
5581:
5574:Euclidean metric
5571:
5550:
5543:
5525:
5521:
5514:
5510:
5477:
5471:
5469:
5435:
5424:
5420:
5419:
5411:
5400:
5385:
5384:
5374:
5373:
5356:
5355:
5346:
5340:
5333:
5332:
5321:
5317:
5313:
5306:
5305:
5295:
5290:
5233:
5231:
5230:
5225:
5218:
5211:
5210:
5198:
5197:
5185:
5184:
5175:
5174:
5148:
5146:
5145:
5140:
5132:
5131:
5119:
5118:
5106:
5105:
5093:
5092:
5073:
5071:
5070:
5065:
5056:
5055:
5042:
5041:
5029:
5028:
5016:
5015:
5006:
5005:
4979:
4977:
4976:
4971:
4968:
4967:
4955:
4954:
4942:
4941:
4929:
4928:
4841:The analog of a
4831:Spherical conics
4818:
4816:
4814:
4813:
4808:
4780:
4778:
4776:
4775:
4770:
4747:
4745:
4744:
4739:
4723:
4721:
4720:
4715:
4696:
4670:spherical spiral
4648:is a path whose
4524:
4505:
4503:
4502:
4497:
4438:integral surface
4432:
4420:
4418:
4417:
4412:
4387:
4386:
4358:
4343:
4329:The sphere is a
4284:antipodal points
4254:, including the
4173:
4169:
4123:minimal surfaces
4080:It follows from
4036:channel surfaces
3999:umbilical points
3990:normal curvature
3981:normal direction
3873:
3869:
3862:
3860:
3859:
3854:
3779:
3760:
3650:
3643:
3641:
3640:
3635:
3633:
3631:
3620:
3615:
3613:
3602:
3597:
3556:
3554:
3553:
3548:
3543:
3542:
3501:
3500:
3490:
3485:
3475:
3467:
3438:
3413:
3406:
3404:
3403:
3398:
3393:
3392:
3358:
3356:
3355:
3350:
3330:
3329:
3304:
3300:
3296:
3289:
3287:
3286:
3281:
3256:
3251:
3239:
3238:
3226:
3218:
3206:
3199:
3197:
3196:
3191:
3166:
3161:
3136:
3133:In the limit as
3129:
3127:
3126:
3121:
3073:
3071:
3070:
3065:
3020:
3016:
3005:
3001:
2997:
2983:
2982:
2978:
2974:
2970:
2966:
2962:
2944:
2942:
2941:
2936:
2931:
2930:
2899:
2883:
2879:
2877:
2876:
2873:
2870:
2869:
2861:
2857:
2853:
2851:
2850:
2847:
2844:
2843:
2818:
2816:
2815:
2810:
2805:
2804:
2792:
2784:
2777:
2776:
2764:
2763:
2762:
2748:
2743:
2718:
2697:
2696:
2695:
2681:
2676:
2666:
2661:
2629:
2608:
2607:
2606:
2592:
2587:
2577:
2572:
2562:
2554:
2526:
2524:
2523:
2518:
2486:
2485:
2447:
2445:
2444:
2439:
2434:
2433:
2421:
2413:
2408:
2404:
2403:
2398:
2397:
2388:
2383:
2382:
2359:
2355:
2354:
2349:
2348:
2339:
2334:
2333:
2312:
2307:
2299:
2295:
2294:
2289:
2288:
2279:
2271:
2270:
2233:
2231:
2230:
2225:
2214:
2210:
2209:
2208:
2196:
2195:
2177:
2172:
2141:
2139:
2138:
2133:
2128:
2127:
2115:
2114:
2102:
2101:
2078:
2074:
2070:
2066:
2059:
2057:
2056:
2051:
2040:
2039:
2026:
2021:
1993:
1990:In the limit as
1986:
1984:
1983:
1978:
1964:
1963:
1929:
1927:
1926:
1921:
1907:
1906:
1875:
1871:
1866:area of the disk
1863:
1859:
1845:
1844:
1840:
1836:
1826:
1816:
1803:disk integration
1780:
1776:
1769:
1767:
1766:
1761:
1759:
1758:
1740:
1739:
1728:
1727:
1719:
1714:
1713:
1701:
1693:
1646:
1642:
1640:
1639:
1634:
1622:
1618:
1614:
1603:
1601:
1600:
1595:
1593:
1573:
1572:
1524:
1523:
1475:
1474:
1437:
1435:
1434:
1429:
1424:
1423:
1411:
1410:
1398:
1397:
1378:
1376:
1375:
1370:
1336:
1317:
1310:
1308:
1307:
1302:
1300:
1295:
1286:
1284:
1283:
1278:
1276:
1275:
1259:
1257:
1256:
1251:
1212:
1210:
1209:
1204:
1182:
1180:
1179:
1174:
1169:
1168:
1156:
1155:
1143:
1142:
1127:
1126:
1110:
1108:
1107:
1102:
1063:
1061:
1060:
1055:
1036:imaginary sphere
1033:
1031:
1030:
1025:
1004:
1002:
1001:
996:
943:
942:
930:
929:
917:
916:
861:
859:
858:
853:
848:
846:
845:
836:
826:
825:
813:
812:
800:
799:
789:
777:
772:
764:
759:
758:
745:
740:
732:
727:
726:
713:
708:
700:
695:
694:
675:
668:
650:
648:
647:
642:
637:
636:
624:
623:
614:
613:
592:
591:
582:
581:
560:
559:
550:
549:
521:
501:
497:
356:axis of rotation
321:
306:antipodal points
303:
293:
280:
237:pressure vessels
229:celestial sphere
196:is the sphere's
195:
181:
159:
153:
135:
133:
131:
130:
127:
124:
108:
95:
51:
39:
38:
21:
6933:
6932:
6928:
6927:
6926:
6924:
6923:
6922:
6873:
6872:
6871:
6862:
6826:
6803:Characteristics
6798:
6760:
6754:
6696:
6655:
6619:
6616:
6574:
6561:
6559:
6524:
6497:
6462:
6446:
6444:Further reading
6441:
6440:
6432:
6428:
6414:
6412:
6410:
6384:
6380:
6371:
6370:
6366:
6345:
6341:
6334:
6313:
6309:
6301:
6297:
6289:
6285:
6277:
6273:
6265:
6261:
6253:
6249:
6239:
6237:
6213:
6209:
6201:
6197:
6176:
6167:
6160:
6146:
6139:
6129:
6127:
6119:
6118:
6114:
6106:
6102:
6098:, p. 342).
6094:
6090:
6082:
6073:
6051:
6036:
6028:
6021:
6009:
6005:
6000:
5995:
5994:
5989:
5985:
5976:
5972:
5965:
5964:
5960:
5955:
5950:
5945:
5864:Homotopy sphere
5854:Homology sphere
5849:Hoberman sphere
5794:
5789:
5780:Spherical wedge
5747:
5741:
5734:
5727:
5718:
5710:
5702:
5699:Gravity Probe B
5680:
5671:
5623:
5605:is a sphere in
5589:
5583:
5577:
5567:
5548:
5527:
5523:
5516:
5512:
5500:
5494:
5488:
5473:
5465:
5463:
5433:
5422:
5417:
5413:
5409:
5395:
5382:
5378:
5371:
5367:
5353:
5349:
5342:
5335:
5330:
5326:
5319:
5315:
5308:
5303:
5299:
5293:
5288:
5278:
5272:
5252:
5247:
5245:Generalizations
5206:
5202:
5193:
5189:
5180:
5176:
5170:
5166:
5155:
5152:
5151:
5127:
5123:
5114:
5110:
5101:
5097:
5088:
5084:
5082:
5079:
5078:
5051:
5047:
5037:
5033:
5024:
5020:
5011:
5007:
5001:
4997:
4985:
4982:
4981:
4963:
4959:
4950:
4946:
4937:
4933:
4924:
4920:
4917:
4914:
4913:
4910:
4902:sphere–cylinder
4885:
4871:from a pair of
4847:spherical conic
4839:
4837:Spherical conic
4833:
4796:
4793:
4792:
4790:
4787:Viviani's curve
4755:
4752:
4751:
4749:
4733:
4730:
4729:
4709:
4706:
4705:
4691:
4684:
4678:
4626:
4620:
4600:
4594:
4573:
4554:sphere eversion
4546:
4522:
4452:
4449:
4448:
4433:held constant.
4430:
4382:
4378:
4367:
4364:
4363:
4356:
4338:
4327:
4202:
4196:
4191:
4171:
4167:
4090:surface tension
3986:normal section,
3885:
3871:
3867:
3788:
3785:
3784:
3762:
3743:
3740:
3734:
3661:
3648:
3624:
3619:
3606:
3601:
3587:
3585:
3582:
3581:
3568:surface tension
3563:
3538:
3534:
3496:
3492:
3486:
3481:
3468:
3463:
3451:
3448:
3447:
3426:
3411:
3388:
3384:
3370:
3367:
3366:
3325:
3321:
3313:
3310:
3309:
3302:
3298:
3294:
3252:
3247:
3234:
3230:
3217:
3215:
3212:
3211:
3204:
3162:
3157:
3145:
3142:
3141:
3134:
3085:
3082:
3081:
3029:
3026:
3025:
3018:
3007:
3003:
2999:
2995:
2988:
2976:
2972:
2968:
2964:
2960:
2926:
2922:
2908:
2905:
2904:
2897:
2890:
2881:
2874:
2871:
2867:
2866:
2865:
2863:
2859:
2848:
2845:
2841:
2840:
2839:
2837:
2832:
2825:
2800:
2796:
2783:
2769:
2758:
2754:
2750:
2744:
2739:
2711:
2691:
2687:
2683:
2677:
2672:
2662:
2657:
2622:
2602:
2598:
2594:
2588:
2583:
2573:
2568:
2555:
2550:
2538:
2535:
2534:
2481:
2477:
2466:
2463:
2462:
2429:
2425:
2412:
2393:
2389:
2387:
2378:
2374:
2370:
2366:
2344:
2340:
2338:
2329:
2325:
2324:
2320:
2308:
2300:
2284:
2280:
2278:
2266:
2262:
2261:
2257:
2245:
2242:
2241:
2204:
2200:
2191:
2187:
2186:
2182:
2173:
2165:
2153:
2150:
2149:
2123:
2119:
2110:
2106:
2097:
2093:
2091:
2088:
2087:
2076:
2072:
2068:
2064:
2035:
2031:
2022:
2014:
2002:
1999:
1998:
1991:
1959:
1955:
1941:
1938:
1937:
1902:
1898:
1884:
1881:
1880:
1873:
1869:
1861:
1857:
1850:
1838:
1828:
1818:
1814:
1807:infinite number
1778:
1774:
1754:
1750:
1735:
1731:
1718:
1709:
1705:
1692:
1684:
1681:
1680:
1658:
1656:Enclosed volume
1653:
1644:
1628:
1625:
1624:
1620:
1616:
1612:
1591:
1590:
1568:
1564:
1557:
1551:
1550:
1519:
1515:
1508:
1502:
1501:
1470:
1466:
1459:
1452:
1450:
1447:
1446:
1419:
1415:
1406:
1402:
1393:
1389:
1384:
1381:
1380:
1358:
1355:
1354:
1347:
1319:
1315:
1294:
1292:
1289:
1288:
1271:
1267:
1265:
1262:
1261:
1218:
1215:
1214:
1192:
1189:
1188:
1164:
1160:
1151:
1147:
1138:
1134:
1122:
1118:
1116:
1113:
1112:
1069:
1066:
1065:
1043:
1040:
1039:
1013:
1010:
1009:
938:
934:
925:
921:
912:
908:
873:
870:
869:
841:
837:
821:
817:
808:
804:
795:
791:
790:
788:
765:
763:
754:
750:
733:
731:
722:
718:
701:
699:
690:
686:
684:
681:
680:
670:
666:
656:quadric surface
632:
628:
619:
615:
609:
605:
587:
583:
577:
573:
555:
551:
545:
541:
530:
527:
526:
507:
499:
495:
488:
481:
474:
467:
406:Euclidean space
354:(as in Earth's
346:figure of Earth
316:
308:of each other.
295:
289:
276:
265:
191:
177:
128:
125:
122:
121:
119:
118:
106:
93:
69:
58:
35:
28:
23:
22:
15:
12:
11:
5:
6931:
6921:
6920:
6915:
6910:
6905:
6900:
6895:
6890:
6885:
6868:
6867:
6864:
6863:
6861:
6860:
6855:
6849:
6843:
6840:
6834:
6832:
6828:
6827:
6825:
6824:
6819:
6814:
6806:
6804:
6800:
6799:
6797:
6796:
6791:
6782:
6777:
6771:
6769:
6762:
6756:
6755:
6753:
6752:
6746:
6745:
6744:
6734:
6733:
6732:
6727:
6719:
6718:
6717:
6708:
6706:
6702:
6701:
6698:
6697:
6695:
6694:
6691:Dyck's surface
6688:
6682:
6681:
6680:
6675:
6663:
6661:
6660:Non-orientable
6657:
6656:
6654:
6653:
6650:
6647:
6641:
6634:
6632:
6625:
6621:
6620:
6615:
6614:
6607:
6600:
6592:
6586:
6585:
6580:
6573:
6572:External links
6570:
6569:
6568:
6548:
6538:
6528:
6522:
6501:
6495:
6466:
6460:
6445:
6442:
6439:
6438:
6426:
6408:
6378:
6364:
6339:
6332:
6316:Hilbert, David
6307:
6295:
6283:
6271:
6259:
6247:
6207:
6203:Steinhaus 1969
6195:
6165:
6158:
6137:
6112:
6108:Steinhaus 1969
6100:
6096:Kreyszig (1972
6088:
6071:
6059:"Sphere"
6056:, ed. (1911).
6054:Chisholm, Hugh
6034:
6019:
6002:
6001:
5999:
5996:
5993:
5992:
5983:
5970:
5957:
5956:
5954:
5951:
5949:
5946:
5944:
5943:
5938:
5933:
5928:
5923:
5918:
5911:
5906:
5901:
5899:Sphere packing
5896:
5891:
5889:Riemann sphere
5886:
5881:
5876:
5871:
5866:
5861:
5856:
5851:
5846:
5836:
5831:
5826:
5821:
5816:
5811:
5806:
5801:
5795:
5793:
5790:
5788:
5787:
5785:Spherical zone
5782:
5777:
5772:
5767:
5762:
5760:Spherical lune
5757:
5752:
5748:
5740:
5737:
5736:
5735:
5731:King of spades
5728:
5721:
5719:
5681:
5674:
5670:
5667:
5622:
5619:
5490:Main article:
5487:
5484:
5392:
5391:
5376:
5365:
5347:
5298:often denoted
5286:natural number
5274:Main article:
5271:
5270:Dimensionality
5268:
5251:
5248:
5246:
5243:
5239:implicit curve
5235:
5234:
5223:
5217:
5214:
5209:
5205:
5201:
5196:
5192:
5188:
5183:
5179:
5173:
5169:
5165:
5162:
5159:
5149:
5138:
5135:
5130:
5126:
5122:
5117:
5113:
5109:
5104:
5100:
5096:
5091:
5087:
5062:
5059:
5054:
5050:
5045:
5040:
5036:
5032:
5027:
5023:
5019:
5014:
5010:
5004:
5000:
4996:
4993:
4990:
4966:
4962:
4958:
4953:
4949:
4945:
4940:
4936:
4932:
4927:
4923:
4906:Main article:
4904:
4903:
4900:
4884:
4881:
4877:
4876:
4865:
4858:
4835:Main article:
4832:
4829:
4806:
4803:
4800:
4768:
4765:
4762:
4759:
4737:
4713:
4680:Main article:
4677:
4674:
4622:Main article:
4619:
4616:
4596:Main article:
4593:
4590:
4572:
4569:
4545:
4542:
4507:
4506:
4495:
4492:
4489:
4486:
4482:
4479:
4476:
4473:
4469:
4466:
4463:
4460:
4456:
4423:volume element
4410:
4407:
4403:
4400:
4396:
4393:
4390:
4385:
4381:
4377:
4374:
4371:
4350:map projection
4333:with constant
4331:smooth surface
4326:
4323:
4307:
4306:
4299:
4292:
4198:Main article:
4195:
4192:
4190:
4187:
4186:
4185:
4184:
4183:
4150:
4149:
4148:
4128:
4127:
4126:
4110:
4109:
4108:
4105:mean curvature
4096:
4095:
4094:
4073:
4072:
4071:
4059:
4058:
4057:
4054:
4039:
4032:
4029:
4012:
4011:
4010:
4007:
3968:
3967:
3966:
3950:
3949:
3948:
3940:
3939:
3938:
3896:In their book
3884:
3881:
3864:
3863:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3736:Main article:
3733:
3730:
3722:dihedral angle
3660:
3657:
3645:
3644:
3630:
3627:
3623:
3618:
3612:
3609:
3605:
3600:
3596:
3593:
3590:
3560:
3559:
3558:
3557:
3546:
3541:
3537:
3533:
3530:
3527:
3524:
3521:
3517:
3514:
3510:
3507:
3504:
3499:
3495:
3489:
3484:
3480:
3474:
3471:
3466:
3462:
3458:
3455:
3408:
3407:
3396:
3391:
3387:
3383:
3380:
3377:
3374:
3360:
3359:
3348:
3345:
3342:
3339:
3336:
3333:
3328:
3324:
3320:
3317:
3291:
3290:
3279:
3276:
3273:
3269:
3266:
3263:
3260:
3255:
3250:
3246:
3242:
3237:
3233:
3229:
3224:
3221:
3201:
3200:
3189:
3186:
3183:
3179:
3176:
3173:
3170:
3165:
3160:
3156:
3152:
3149:
3131:
3130:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3075:
3074:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
2990:
2989:
2986:
2981:
2946:
2945:
2934:
2929:
2925:
2921:
2918:
2915:
2912:
2889:
2886:
2822:
2821:
2820:
2819:
2808:
2803:
2799:
2795:
2790:
2787:
2782:
2775:
2772:
2768:
2761:
2757:
2753:
2747:
2742:
2738:
2734:
2731:
2728:
2725:
2722:
2717:
2714:
2710:
2706:
2703:
2700:
2694:
2690:
2686:
2680:
2675:
2671:
2665:
2660:
2656:
2652:
2649:
2646:
2643:
2640:
2636:
2633:
2628:
2625:
2621:
2617:
2614:
2611:
2605:
2601:
2597:
2591:
2586:
2582:
2576:
2571:
2567:
2561:
2558:
2553:
2549:
2545:
2542:
2528:
2527:
2516:
2513:
2509:
2506:
2502:
2499:
2495:
2492:
2489:
2484:
2480:
2476:
2473:
2470:
2457:volume element
2449:
2448:
2437:
2432:
2428:
2424:
2419:
2416:
2411:
2407:
2401:
2396:
2392:
2386:
2381:
2377:
2373:
2369:
2365:
2362:
2358:
2352:
2347:
2343:
2337:
2332:
2328:
2323:
2319:
2316:
2311:
2306:
2303:
2298:
2292:
2287:
2283:
2277:
2274:
2269:
2265:
2260:
2255:
2252:
2249:
2235:
2234:
2223:
2220:
2217:
2213:
2207:
2203:
2199:
2194:
2190:
2185:
2181:
2176:
2171:
2168:
2164:
2160:
2157:
2143:
2142:
2131:
2126:
2122:
2118:
2113:
2109:
2105:
2100:
2096:
2061:
2060:
2049:
2046:
2043:
2038:
2034:
2030:
2025:
2020:
2017:
2013:
2009:
2006:
1988:
1987:
1976:
1973:
1970:
1967:
1962:
1958:
1954:
1951:
1948:
1945:
1931:
1930:
1919:
1916:
1913:
1910:
1905:
1901:
1897:
1894:
1891:
1888:
1852:
1851:
1848:
1843:
1771:
1770:
1757:
1753:
1749:
1746:
1743:
1738:
1734:
1725:
1722:
1717:
1712:
1708:
1704:
1699:
1696:
1691:
1688:
1657:
1654:
1652:
1649:
1632:
1605:
1604:
1588:
1585:
1582:
1579:
1576:
1571:
1567:
1563:
1560:
1558:
1556:
1553:
1552:
1549:
1546:
1543:
1539:
1536:
1533:
1530:
1527:
1522:
1518:
1514:
1511:
1509:
1507:
1504:
1503:
1500:
1497:
1494:
1490:
1487:
1484:
1481:
1478:
1473:
1469:
1465:
1462:
1460:
1458:
1455:
1454:
1427:
1422:
1418:
1414:
1409:
1405:
1401:
1396:
1392:
1388:
1368:
1365:
1362:
1346:
1343:
1298:
1274:
1270:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1202:
1199:
1196:
1172:
1167:
1163:
1159:
1154:
1150:
1146:
1141:
1137:
1133:
1130:
1125:
1121:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1053:
1050:
1047:
1023:
1020:
1017:
1006:
1005:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
941:
937:
933:
928:
924:
920:
915:
911:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
863:
862:
851:
844:
840:
835:
832:
829:
824:
820:
816:
811:
807:
803:
798:
794:
787:
784:
780:
775:
771:
768:
762:
757:
753:
748:
743:
739:
736:
730:
725:
721:
716:
711:
707:
704:
698:
693:
689:
652:
651:
640:
635:
631:
627:
622:
618:
612:
608:
604:
601:
598:
595:
590:
586:
580:
576:
572:
569:
566:
563:
558:
554:
548:
544:
540:
537:
534:
506:of all points
493:
486:
479:
466:
463:
399:closed surface
264:
261:
241:curved mirrors
137:
136:
116:
110:
109:
104:
98:
97:
90:
88:Symmetry group
84:
83:
80:
74:
73:
67:Smooth surface
64:
60:
59:
52:
44:
43:
26:
9:
6:
4:
3:
2:
6930:
6919:
6916:
6914:
6911:
6909:
6906:
6904:
6901:
6899:
6896:
6894:
6891:
6889:
6886:
6884:
6881:
6880:
6878:
6859:
6856:
6854:
6850:
6848:
6844:
6842:Making a hole
6841:
6839:
6838:Connected sum
6836:
6835:
6833:
6829:
6823:
6820:
6818:
6815:
6812:
6808:
6807:
6805:
6801:
6795:
6794:Orientability
6792:
6790:
6786:
6783:
6781:
6778:
6776:
6775:Connectedness
6773:
6772:
6770:
6766:
6763:
6757:
6750:
6747:
6743:
6740:
6739:
6738:
6735:
6731:
6728:
6726:
6723:
6722:
6720:
6715:
6714:
6713:
6710:
6709:
6707:
6705:With boundary
6703:
6693:(genus 3) ...
6692:
6689:
6686:
6683:
6679:
6678:Roman surface
6676:
6674:
6673:Boy's surface
6670:
6669:
6668:
6665:
6664:
6662:
6658:
6651:
6648:
6645:
6642:
6639:
6636:
6635:
6633:
6629:
6626:
6622:
6613:
6608:
6606:
6601:
6599:
6594:
6593:
6590:
6584:
6581:
6579:
6576:
6575:
6558:
6554:
6549:
6544:
6539:
6534:
6529:
6525:
6519:
6515:
6510:
6509:
6502:
6498:
6492:
6488:
6484:
6480:
6475:
6474:
6467:
6463:
6457:
6453:
6448:
6447:
6435:
6430:
6423:
6411:
6405:
6401:
6397:
6393:
6389:
6382:
6374:
6368:
6359:
6358:
6353:
6350:
6343:
6335:
6329:
6324:
6323:
6317:
6311:
6304:
6299:
6292:
6287:
6280:
6275:
6268:
6263:
6256:
6251:
6235:
6230:
6226:
6222:
6218:
6211:
6204:
6199:
6190:
6189:
6184:
6181:
6174:
6172:
6170:
6161:
6155:
6151:
6144:
6142:
6126:
6122:
6116:
6109:
6104:
6097:
6092:
6085:
6080:
6078:
6076:
6067:
6066:
6060:
6055:
6049:
6047:
6045:
6043:
6041:
6039:
6031:
6026:
6024:
6017:, on Perseus.
6016:
6012:
6007:
6003:
5987:
5980:
5974:
5962:
5958:
5942:
5939:
5937:
5934:
5932:
5929:
5927:
5924:
5922:
5919:
5916:
5912:
5910:
5909:Spherical cow
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5885:
5882:
5880:
5877:
5875:
5872:
5870:
5869:Lenart Sphere
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5850:
5847:
5844:
5840:
5837:
5835:
5832:
5830:
5827:
5825:
5822:
5820:
5817:
5815:
5812:
5810:
5807:
5805:
5804:Affine sphere
5802:
5800:
5797:
5796:
5786:
5783:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5763:
5761:
5758:
5756:
5755:Spherical cap
5753:
5750:
5749:
5746:
5732:
5725:
5720:
5716:
5708:
5700:
5696:
5693:
5689:
5686:the image of
5685:
5678:
5673:
5672:
5666:
5664:
5660:
5655:
5653:
5649:
5645:
5644:
5639:
5635:
5630:
5629:
5618:
5616:
5612:
5608:
5604:
5599:
5597:
5592:
5586:
5580:
5575:
5570:
5565:
5560:
5558:
5554:
5545:
5542:
5538:
5534:
5530:
5519:
5508:
5504:
5499:
5493:
5486:Metric spaces
5483:
5481:
5476:
5468:
5460:
5458:
5457:exotic sphere
5454:
5453:diffeomorphic
5450:
5446:
5442:
5439:
5431:
5426:
5416:
5406:
5404:
5398:
5389:
5381:
5377:
5370:
5366:
5364:
5360:
5352:
5348:
5345:
5339:
5329:
5325:
5324:
5323:
5311:
5302:
5297:
5287:
5283:
5277:
5267:
5265:
5261:
5257:
5242:
5240:
5221:
5215:
5212:
5207:
5203:
5199:
5194:
5190:
5186:
5181:
5171:
5167:
5163:
5160:
5150:
5136:
5133:
5128:
5124:
5120:
5115:
5111:
5107:
5102:
5098:
5094:
5089:
5085:
5077:
5076:
5075:
5060:
5057:
5052:
5048:
5043:
5038:
5034:
5030:
5025:
5021:
5017:
5012:
5002:
4998:
4994:
4991:
4964:
4960:
4956:
4951:
4947:
4943:
4938:
4934:
4930:
4925:
4921:
4909:
4901:
4898:
4897:
4896:
4889:
4880:
4875:is a constant
4874:
4870:
4866:
4863:
4859:
4856:
4855:
4854:
4852:
4848:
4844:
4843:conic section
4838:
4828:
4826:
4822:
4804:
4801:
4798:
4788:
4784:
4766:
4763:
4760:
4757:
4735:
4727:
4711:
4704:(or azimuth)
4703:
4694:
4688:
4683:
4676:Clelia curves
4673:
4671:
4667:
4663:
4659:
4655:
4651:
4647:
4643:
4639:
4630:
4625:
4615:
4613:
4609:
4604:
4599:
4585:
4577:
4568:
4566:
4562:
4557:
4555:
4551:
4541:
4539:
4535:
4531:
4526:
4520:
4519:normal vector
4516:
4512:
4511:tangent plane
4493:
4490:
4487:
4484:
4480:
4477:
4474:
4471:
4467:
4464:
4461:
4458:
4454:
4447:
4446:
4445:
4443:
4439:
4434:
4428:
4424:
4408:
4405:
4401:
4398:
4394:
4391:
4388:
4383:
4379:
4375:
4372:
4369:
4362:
4353:
4351:
4347:
4342:
4336:
4332:
4322:
4320:
4316:
4312:
4304:
4300:
4297:
4293:
4290:
4289:
4288:
4286:
4285:
4279:
4277:
4273:
4269:
4265:
4261:
4257:
4253:
4249:
4244:
4242:
4238:
4234:
4231:, which is a
4230:
4226:
4222:
4218:
4210:
4206:
4201:
4181:
4177:
4165:
4161:
4156:
4155:
4154:
4151:
4146:
4141:
4137:
4134:
4133:
4132:
4129:
4124:
4120:
4116:
4115:
4114:
4111:
4106:
4102:
4101:
4100:
4097:
4091:
4087:
4083:
4079:
4078:
4077:
4074:
4068:
4065:
4064:
4063:
4060:
4055:
4052:
4048:
4045:, cylinders,
4044:
4040:
4037:
4033:
4030:
4027:
4026:focal surface
4023:
4018:
4017:
4016:
4013:
4008:
4005:
4001:
4000:
3995:
3991:
3987:
3982:
3978:
3977:
3976:
3974:
3969:
3964:
3963:circumference
3960:
3959:Meissner body
3956:
3955:
3954:
3951:
3946:
3945:
3944:
3941:
3936:
3932:
3928:
3924:
3920:
3919:
3918:
3915:
3914:
3913:
3911:
3907:
3903:
3902:David Hilbert
3899:
3889:
3880:
3877:
3850:
3847:
3841:
3838:
3835:
3832:
3829:
3823:
3820:
3817:
3811:
3808:
3805:
3802:
3799:
3793:
3790:
3783:
3782:
3781:
3777:
3773:
3769:
3765:
3758:
3754:
3750:
3746:
3739:
3729:
3727:
3723:
3718:
3715:
3714:radical plane
3711:
3706:
3703:
3701:
3700:non-collinear
3697:
3692:
3690:
3686:
3682:
3678:
3674:
3670:
3666:
3656:
3654:
3628:
3625:
3621:
3616:
3610:
3607:
3603:
3598:
3580:
3579:
3578:
3576:
3571:
3569:
3544:
3539:
3535:
3531:
3528:
3525:
3522:
3519:
3515:
3512:
3508:
3505:
3502:
3497:
3493:
3487:
3482:
3478:
3472:
3469:
3464:
3460:
3456:
3453:
3446:
3445:
3444:
3442:
3437:
3433:
3429:
3424:
3420:
3415:
3394:
3389:
3385:
3381:
3378:
3375:
3372:
3365:
3364:
3363:
3346:
3340:
3334:
3331:
3326:
3322:
3318:
3315:
3308:
3307:
3306:
3277:
3274:
3271:
3264:
3258:
3253:
3248:
3244:
3240:
3235:
3231:
3227:
3222:
3219:
3210:
3209:
3208:
3187:
3184:
3181:
3174:
3168:
3163:
3158:
3154:
3150:
3147:
3140:
3139:
3138:
3117:
3114:
3111:
3108:
3102:
3096:
3093:
3090:
3087:
3080:
3079:
3078:
3061:
3058:
3055:
3052:
3046:
3040:
3037:
3034:
3031:
3024:
3023:
3022:
3014:
3010:
2992:
2991:
2985:
2984:
2980:
2958:
2954:
2953:circumscribed
2950:
2932:
2927:
2923:
2919:
2916:
2913:
2910:
2903:
2902:
2901:
2895:
2885:
2856:
2835:
2830:
2806:
2801:
2797:
2793:
2788:
2785:
2780:
2773:
2770:
2766:
2759:
2755:
2751:
2745:
2740:
2736:
2732:
2729:
2726:
2723:
2720:
2715:
2712:
2708:
2704:
2701:
2698:
2692:
2688:
2684:
2678:
2673:
2669:
2663:
2658:
2654:
2650:
2647:
2644:
2641:
2638:
2634:
2631:
2626:
2623:
2619:
2615:
2612:
2609:
2603:
2599:
2595:
2589:
2584:
2580:
2574:
2569:
2565:
2559:
2556:
2551:
2547:
2543:
2540:
2533:
2532:
2531:
2514:
2511:
2507:
2504:
2500:
2497:
2493:
2490:
2487:
2482:
2478:
2474:
2471:
2468:
2461:
2460:
2459:
2458:
2454:
2435:
2430:
2426:
2422:
2417:
2414:
2409:
2405:
2399:
2394:
2390:
2384:
2379:
2375:
2371:
2367:
2363:
2360:
2356:
2350:
2345:
2341:
2335:
2330:
2326:
2321:
2317:
2314:
2309:
2304:
2301:
2296:
2290:
2285:
2281:
2275:
2272:
2267:
2263:
2258:
2253:
2250:
2247:
2240:
2239:
2238:
2221:
2218:
2215:
2211:
2205:
2201:
2197:
2192:
2188:
2183:
2179:
2174:
2169:
2166:
2162:
2158:
2155:
2148:
2147:
2146:
2129:
2124:
2120:
2116:
2111:
2107:
2103:
2098:
2094:
2086:
2085:
2084:
2082:
2063:At any given
2047:
2044:
2041:
2036:
2032:
2028:
2023:
2018:
2015:
2011:
2007:
2004:
1997:
1996:
1995:
1974:
1971:
1968:
1965:
1960:
1956:
1952:
1949:
1946:
1943:
1936:
1935:
1934:
1917:
1914:
1911:
1908:
1903:
1899:
1895:
1892:
1889:
1886:
1879:
1878:
1877:
1867:
1856:At any given
1854:
1853:
1847:
1846:
1842:
1835:
1831:
1825:
1821:
1812:
1808:
1804:
1800:
1796:
1791:
1788:
1787:circumscribed
1784:
1755:
1751:
1747:
1744:
1741:
1736:
1732:
1723:
1720:
1715:
1710:
1706:
1702:
1697:
1694:
1689:
1686:
1679:
1678:
1677:
1675:
1671:
1662:
1648:
1630:
1610:
1586:
1583:
1580:
1577:
1574:
1569:
1565:
1561:
1559:
1554:
1547:
1544:
1541:
1537:
1534:
1531:
1528:
1525:
1520:
1516:
1512:
1510:
1505:
1498:
1495:
1492:
1488:
1485:
1482:
1479:
1476:
1471:
1467:
1463:
1461:
1456:
1445:
1444:
1443:
1441:
1420:
1416:
1412:
1407:
1403:
1399:
1394:
1390:
1366:
1363:
1360:
1352:
1342:
1340:
1334:
1330:
1326:
1322:
1312:
1296:
1272:
1268:
1247:
1244:
1238:
1235:
1232:
1229:
1226:
1220:
1200:
1197:
1194:
1186:
1165:
1161:
1157:
1152:
1148:
1144:
1139:
1135:
1128:
1123:
1119:
1111:is the point
1098:
1095:
1089:
1086:
1083:
1080:
1077:
1071:
1051:
1048:
1045:
1037:
1021:
1018:
1015:
992:
989:
986:
983:
977:
974:
971:
968:
965:
962:
959:
956:
950:
947:
939:
935:
931:
926:
922:
918:
913:
909:
902:
899:
893:
890:
887:
884:
881:
875:
868:
867:
866:
849:
842:
838:
833:
830:
827:
822:
818:
814:
809:
805:
801:
796:
792:
785:
782:
778:
773:
769:
766:
760:
755:
751:
746:
741:
737:
734:
728:
723:
719:
714:
709:
705:
702:
696:
691:
687:
679:
678:
677:
673:
667:a, b, c, d, e
663:
661:
657:
638:
633:
629:
625:
620:
610:
606:
602:
599:
593:
588:
578:
574:
570:
567:
561:
556:
546:
542:
538:
535:
525:
524:
523:
519:
515:
511:
505:
492:
485:
478:
472:
462:
460:
455:
453:
449:
445:
441:
437:
433:
429:
425:
421:
417:
416:
411:
407:
403:
400:
395:
393:
389:
385:
384:
379:
375:
374:
369:
365:
361:
357:
353:
352:
347:
344:Although the
342:
340:
339:
334:
333:
327:
325:
319:
314:
309:
307:
302:
298:
292:
287:
282:
279:
269:
260:
258:
257:ball bearings
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
205:
203:
199:
194:
189:
185:
180:
175:
174:set of points
171:
167:
163:
158:
152:
148:
144:
117:
115:
111:
105:
103:
99:
96:
91:
89:
85:
81:
79:
75:
72:
68:
65:
61:
56:
50:
45:
40:
37:
33:
19:
6737:Möbius strip
6685:Klein bottle
6637:
6560:. Retrieved
6556:
6542:
6532:
6507:
6472:
6451:
6429:
6420:
6413:. Retrieved
6391:
6381:
6367:
6355:
6342:
6321:
6310:
6298:
6286:
6274:
6262:
6250:
6238:. Retrieved
6224:
6220:
6210:
6198:
6186:
6149:
6128:. Retrieved
6124:
6115:
6103:
6091:
6063:
6014:
6006:
5986:
5978:
5973:
5961:
5884:Pseudosphere
5879:Orb (optics)
5829:Dyson sphere
5692:fused quartz
5659:Dionysodorus
5656:
5641:
5626:
5624:
5600:
5590:
5584:
5578:
5568:
5561:
5546:
5540:
5536:
5532:
5528:
5517:
5506:
5502:
5498:metric space
5495:
5492:Metric space
5474:
5466:
5461:
5427:
5414:
5407:
5403:hyperspheres
5396:
5394:Spheres for
5393:
5379:
5368:
5362:
5350:
5343:
5337:
5327:
5309:
5300:
5292:
5279:
5253:
5236:
4911:
4894:
4878:
4840:
4821:ground track
4699:
4692:
4645:
4641:
4635:
4611:
4605:
4601:
4558:
4547:
4527:
4508:
4435:
4361:area element
4354:
4340:
4328:
4308:
4302:
4283:
4280:
4268:trigonometry
4245:
4241:great circle
4233:great circle
4214:
4209:Great circle
4160:Euler angles
4152:
4145:pseudosphere
4130:
4112:
4098:
4086:soap bubbles
4075:
4061:
4022:focal points
4021:
4014:
3997:
3989:
3985:
3970:
3952:
3942:
3916:
3897:
3895:
3875:
3865:
3775:
3771:
3767:
3763:
3756:
3752:
3748:
3744:
3741:
3719:
3713:
3707:
3704:
3693:
3662:
3646:
3572:
3564:
3435:
3431:
3427:
3419:area element
3416:
3409:
3361:
3292:
3202:
3132:
3076:
3012:
3008:
2993:
2947:
2894:surface area
2891:
2888:Surface area
2854:
2833:
2826:
2529:
2450:
2236:
2144:
2062:
1989:
1932:
1855:
1833:
1829:
1823:
1819:
1772:
1667:
1606:
1348:
1332:
1328:
1324:
1320:
1313:
1185:point sphere
1184:
1035:
1007:
864:
671:
664:
658:, a type of
653:
517:
513:
509:
490:
483:
476:
468:
456:
454:is similar.
439:
435:
427:
423:
413:
409:
396:
391:
382:
371:
367:
363:
359:
349:
343:
337:
336:
332:great circle
330:
328:
317:
310:
300:
296:
290:
283:
277:
274:
217:soap bubbles
206:
192:
178:
156:
142:
140:
102:Surface area
36:
6780:Compactness
6373:"Loxodrome"
6303:Albert 2016
6279:Albert 2016
6267:Albert 2016
6255:Albert 2016
6240:14 December
6227:(6): 1187.
6030:Albert 2016
5941:Zoll sphere
5894:Solid angle
5843:M.C. Escher
5594:squares of
5557:unit sphere
5515:and radius
4825:polar orbit
4211:on a sphere
3441:integration
3203:Substitute
1817:-axis from
1379:and center
498:and radius
428:closed ball
338:hemispheres
313:unit sphere
209:mathematics
162:geometrical
78:Euler char.
57:of a sphere
6877:Categories
6831:Operations
6813:components
6809:Number of
6789:smoothness
6768:Properties
6716:Semisphere
6631:Orientable
6562:21 January
6415:4 November
6291:Woods 1961
6084:Woods 1961
5998:References
5926:Sphericity
5751:Hemisphere
5743:See also:
5707:Australian
5638:Archimedes
5603:octahedron
5551:, as in a
5526:such that
5361:of radius
5284:. For any
5282:dimensions
5250:Ellipsoids
4726:colatitude
4646:rhumb line
4638:navigation
4624:Rhumb line
4515:orthogonal
4252:postulates
4237:arc length
3726:orthogonal
3685:major axis
2957:derivative
2949:Archimedes
1783:Archimedes
1651:Properties
1345:Parametric
522:such that
368:south pole
364:north pole
227:, and the
6858:Immersion
6853:cross-cap
6851:Gluing a
6845:Gluing a
6742:Cross-cap
6687:(genus 2)
6671:genus 1;
6646:(genus 1)
6640:(genus 0)
6454:, Dover,
6357:MathWorld
6293:, p. 267.
6205:, p. 221.
6188:MathWorld
6110:, p. 223.
6086:, p. 266.
5834:Gauss map
5819:Curvature
5733:: Spheres
5695:gyroscope
5652:Zenodorus
5562:Unlike a
5256:ellipsoid
5200:−
5164:−
5121:−
5058:≠
4995:−
4767:θ
4758:φ
4736:θ
4712:φ
4702:longitude
4662:parallels
4658:meridians
4642:loxodrome
4632:Loxodrome
4618:Loxodrome
4409:φ
4402:θ
4395:θ
4392:
4317:makes up
4180:helicoids
4070:property.
4067:Geodesics
3669:diameters
3629:ρ
3611:ρ
3532:π
3523:φ
3516:θ
3509:θ
3506:
3488:π
3479:∫
3473:π
3461:∫
3382:π
3319:π
3245:∫
3228:π
3155:∫
3112:δ
3109:⋅
3094:∑
3091:≈
3056:δ
3053:⋅
3038:≈
3032:δ
2920:π
2829:inscribed
2794:π
2737:∫
2733:π
2724:θ
2705:θ
2702:
2670:∫
2664:π
2655:∫
2651:π
2642:φ
2635:θ
2616:θ
2613:
2581:∫
2575:π
2566:∫
2560:π
2548:∫
2515:φ
2508:θ
2494:θ
2491:
2423:π
2372:−
2364:π
2361:−
2336:−
2318:π
2302:−
2276:−
2254:π
2198:−
2180:π
2167:−
2163:∫
2117:−
2029:π
2016:−
2012:∫
1969:δ
1966:⋅
1953:π
1950:∑
1947:≈
1912:δ
1909:⋅
1896:π
1893:≈
1887:δ
1748:⋅
1742:≈
1721:π
1703:π
1631:φ
1587:θ
1584:
1548:φ
1545:
1538:θ
1535:
1499:φ
1496:
1489:θ
1486:
1297:ρ
1195:ρ
1046:ρ
1016:ρ
828:−
783:ρ
767:−
735:−
703:−
603:−
571:−
539:−
465:Equations
450:" in the
424:open ball
392:parallels
383:meridians
378:longitude
239:and most
233:astronomy
225:geography
18:Spherical
6918:Topology
6913:Surfaces
6811:boundary
6730:Cylinder
6305:, p. 58.
6281:, p. 57.
6269:, p. 55.
6257:, p. 60.
6183:"Sphere"
6032:, p. 54.
5799:3-sphere
5792:See also
5697:for the
5688:Einstein
5684:refracts
5609:, and a
5596:integers
5470:‖
5464:‖
5445:boundary
5443:without
5430:topology
5388:3-sphere
5296:-sphere,
5276:n-sphere
4724:and the
4544:Topology
4229:geodesic
4140:embedded
4119:embedded
4051:cyclides
3973:umbilics
3929:for the
3696:coplanar
3689:spheroid
2858:, where
2774:′
2756:′
2716:′
2689:′
2627:′
2600:′
2083:yields:
1811:circular
1790:cylinder
676:and put
432:boundary
402:embedded
286:diameter
215:such as
6908:Spheres
6761:notions
6759:Related
6725:Annulus
6721:Ribbon
6545:, Dover
6483:Bibcode
6481:, 226.
6130:10 June
5739:Regions
5669:Gallery
5663:al-Quhi
5646:by the
5621:History
5438:compact
5418:
5383:
5372:
5354:
5331:
5304:
5264:ellipse
4899:Example
4851:quartic
4817:
4791:
4779:
4750:
4650:bearing
4612:coaxial
4592:Circles
4276:similar
4093:bodies.
3677:ellipse
3653:density
3651:is the
3436:θ dθ dφ
3297:yields
2878:
2864:
2852:
2838:
2455:, with
1801:(i.e.,
502:is the
446:" and "
373:equator
213:Bubbles
160:) is a
157:sphaîra
132:
120:
6847:handle
6638:Sphere
6520:
6493:
6458:
6406:
6330:
6156:
6011:σφαῖρα
5711:
5703:
5553:normed
5520:> 0
5449:smooth
5432:, the
5399:> 2
5359:circle
5219:
4682:Clélie
4532:, the
4264:angles
4221:points
4170:- and
4041:* For
4034:* For
3931:circle
3923:result
3665:circle
3647:where
3410:where
2882:
2778:
1773:where
1745:0.5236
1729:
1670:volume
444:circle
440:sphere
412:and a
410:sphere
245:lenses
198:radius
188:center
170:circle
151:σφαῖρα
145:(from
143:sphere
114:Volume
42:Sphere
6817:Genus
6644:Torus
6514:Wiley
5953:Notes
5572:with
5291:, an
5237:(see
4429:with
4298:, and
4258:. In
4225:lines
4043:cones
3935:plane
3910:plane
3778:) = 0
3759:) = 0
1335:) = 0
1038:. If
504:locus
452:plane
360:poles
253:balls
147:Greek
6712:Disk
6564:2022
6518:ISBN
6491:ISBN
6456:ISBN
6417:2022
6404:ISBN
6328:ISBN
6242:2019
6154:ISBN
6132:2019
5611:cube
5564:ball
5539:) =
5408:The
5386:: a
5341:and
4873:foci
4849:, a
4640:, a
4359:has
4223:and
4219:are
4178:and
4103:The
4049:and
4047:tori
3904:and
3870:and
3761:and
3434:sin
2900:is:
2892:The
2075:and
1674:ball
1623:and
1364:>
1198:>
1019:<
665:Let
448:disk
438:and
436:ball
415:ball
390:(or
366:and
351:axis
249:roll
243:and
94:O(3)
63:Type
6787:or
6751:...
6396:doi
6229:doi
5640:'s
5601:An
5459:).
5428:In
5312:+ 1
5254:An
4695:= 8
4644:or
4636:In
4528:In
4523:1/r
4425:in
4389:sin
4296:arc
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3503:sin
3425:by
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2530:so
2488:sin
1876:):
1868:at
1827:to
1822:= −
1809:of
1581:cos
1542:sin
1532:sin
1493:cos
1483:sin
1314:If
674:≠ 0
469:In
461:).
380:or
320:= 1
299:= 2
223:in
107:4πr
6879::
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6516:,
6489:.
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6419:.
6402:.
6394:.
6390:.
6354:.
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6219:.
6185:.
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6140:^
6123:.
6074:^
6062:.
6037:^
6022:^
5841:,
5665:.
5650:.
5617:.
5598:.
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5544:.
5482:.
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4444::
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3774:,
3770:,
3755:,
3751:,
3443::
3430:=
3428:dA
3305::
3207::
3135:δr
3019:δr
3000:δV
2836:=
2071:,
1992:δx
1874:δx
1862:δV
1832:=
1647:.
1611:.
1442:.
1349:A
1341:.
1331:,
1327:,
1311:.
1213:,
662:.
516:,
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482:,
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329:A
311:A
259:.
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154:,
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134:πr
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6611:e
6604:t
6597:v
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6547:.
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6465:.
6436:.
6398::
6375:.
6360:.
6336:.
6244:.
6231::
6191:.
6162:.
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5981:.
5979:π
5966:r
5717:.
5591:n
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5338:r
5336:−
5328:S
5320:r
5316:r
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5301:S
5294:n
5289:n
5222:.
5216:0
5213:=
5208:2
5204:a
5195:2
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5161:y
5158:(
5137:0
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4992:y
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4965:2
4961:r
4957:=
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4931:+
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4805:1
4802:=
4799:c
4789:(
4764:c
4761:=
4693:c
4491:=
4488:z
4485:d
4481:z
4478:+
4475:y
4472:d
4468:y
4465:+
4462:x
4459:d
4455:x
4431:r
4406:d
4399:d
4384:2
4380:r
4376:=
4373:A
4370:d
4357:r
4341:r
4172:y
4168:x
4028:.
3975:.
3937:.
3872:t
3868:s
3851:0
3848:=
3845:)
3842:z
3839:,
3836:y
3833:,
3830:x
3827:(
3824:g
3821:t
3818:+
3815:)
3812:z
3809:,
3806:y
3803:,
3800:x
3797:(
3794:f
3791:s
3776:z
3772:y
3768:x
3766:(
3764:g
3757:z
3753:y
3749:x
3747:(
3745:f
3649:ρ
3626:r
3622:3
3617:=
3608:V
3604:A
3599:=
3595:A
3592:S
3589:S
3545:.
3540:2
3536:r
3529:4
3526:=
3520:d
3513:d
3498:2
3494:r
3483:0
3470:2
3465:0
3457:=
3454:A
3432:r
3412:r
3395:,
3390:2
3386:r
3379:4
3376:=
3373:A
3347:.
3344:)
3341:r
3338:(
3335:A
3332:=
3327:2
3323:r
3316:4
3303:r
3299:A
3295:r
3278:.
3275:r
3272:d
3268:)
3265:r
3262:(
3259:A
3254:r
3249:0
3241:=
3236:3
3232:r
3223:3
3220:4
3205:V
3188:.
3185:r
3182:d
3178:)
3175:r
3172:(
3169:A
3164:r
3159:0
3151:=
3148:V
3118:.
3115:r
3106:)
3103:r
3100:(
3097:A
3088:V
3062:.
3059:r
3050:)
3047:r
3044:(
3041:A
3035:V
3015:)
3013:r
3011:(
3009:A
3006:(
3004:r
2996:r
2977:r
2973:r
2969:r
2965:r
2961:r
2933:.
2928:2
2924:r
2917:4
2914:=
2911:A
2898:r
2875:6
2872:/
2868:π
2860:d
2855:d
2849:6
2846:/
2842:π
2834:V
2807:.
2802:3
2798:r
2789:3
2786:4
2781:=
2771:r
2767:d
2760:2
2752:r
2746:r
2741:0
2730:4
2727:=
2721:d
2713:r
2709:d
2693:2
2685:r
2679:r
2674:0
2659:0
2648:2
2645:=
2639:d
2632:d
2624:r
2620:d
2604:2
2596:r
2590:r
2585:0
2570:0
2557:2
2552:0
2544:=
2541:V
2512:d
2505:d
2501:r
2498:d
2483:2
2479:r
2475:=
2472:V
2469:d
2436:.
2431:3
2427:r
2418:3
2415:4
2410:=
2406:)
2400:3
2395:3
2391:r
2385:+
2380:3
2376:r
2368:(
2357:)
2351:3
2346:3
2342:r
2331:3
2327:r
2322:(
2315:=
2310:r
2305:r
2297:]
2291:3
2286:3
2282:x
2273:x
2268:2
2264:r
2259:[
2251:=
2248:V
2222:,
2219:x
2216:d
2212:)
2206:2
2202:x
2193:2
2189:r
2184:(
2175:r
2170:r
2159:=
2156:V
2130:.
2125:2
2121:x
2112:2
2108:r
2104:=
2099:2
2095:y
2077:r
2073:y
2069:x
2065:x
2048:.
2045:x
2042:d
2037:2
2033:y
2024:r
2019:r
2008:=
2005:V
1975:.
1972:x
1961:2
1957:y
1944:V
1918:.
1915:x
1904:2
1900:y
1890:V
1870:x
1858:x
1839:r
1834:r
1830:x
1824:r
1820:x
1815:x
1779:d
1775:r
1756:3
1752:d
1737:3
1733:d
1724:6
1716:=
1711:3
1707:r
1698:3
1695:4
1690:=
1687:V
1645:π
1621:π
1617:θ
1613:r
1578:r
1575:+
1570:0
1566:z
1562:=
1555:z
1529:r
1526:+
1521:0
1517:y
1513:=
1506:y
1480:r
1477:+
1472:0
1468:x
1464:=
1457:x
1426:)
1421:0
1417:z
1413:,
1408:0
1404:y
1400:,
1395:0
1391:x
1387:(
1367:0
1361:r
1333:z
1329:y
1325:x
1323:(
1321:f
1316:a
1273:0
1269:P
1248:0
1245:=
1242:)
1239:z
1236:,
1233:y
1230:,
1227:x
1224:(
1221:f
1201:0
1171:)
1166:0
1162:z
1158:,
1153:0
1149:y
1145:,
1140:0
1136:x
1132:(
1129:=
1124:0
1120:P
1099:0
1096:=
1093:)
1090:z
1087:,
1084:y
1081:,
1078:x
1075:(
1072:f
1052:0
1049:=
1022:0
993:0
990:=
987:e
984:+
981:)
978:z
975:d
972:+
969:y
966:c
963:+
960:x
957:b
954:(
951:2
948:+
945:)
940:2
936:z
932:+
927:2
923:y
919:+
914:2
910:x
906:(
903:a
900:=
897:)
894:z
891:,
888:y
885:,
882:x
879:(
876:f
850:.
843:2
839:a
834:e
831:a
823:2
819:d
815:+
810:2
806:c
802:+
797:2
793:b
786:=
779:,
774:a
770:d
761:=
756:0
752:z
747:,
742:a
738:c
729:=
724:0
720:y
715:,
710:a
706:b
697:=
692:0
688:x
672:a
639:.
634:2
630:r
626:=
621:2
617:)
611:0
607:z
600:z
597:(
594:+
589:2
585:)
579:0
575:y
568:y
565:(
562:+
557:2
553:)
547:0
543:x
536:x
533:(
520:)
518:z
514:y
510:x
508:(
500:r
496:)
494:0
491:z
487:0
484:y
480:0
477:x
475:(
362:(
318:r
301:r
297:d
291:d
278:r
193:r
179:r
129:3
126:/
123:4
82:2
34:.
20:)
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