Set of points equidistant from a center
"Globose" redirects here. For the neuroanatomic structure, see
Globose nucleus
.
A
sphere
(from
Greek
σφα?ρα
,
sphaira
)
[1]
is a
geometrical
object that is a
three-dimensional
analogue to a two-dimensional
circle
. Formally, a sphere is the
set of points
that are all at the same distance
r
from a given point in
three-dimensional space
.
[2]
That given point is the
center
of the sphere, and
r
is the sphere's
radius
. The earliest known mentions of spheres appear in the work of the
ancient Greek mathematicians
.
The sphere is a fundamental object in many fields of
mathematics
. Spheres and nearly-spherical shapes also appear in nature and industry.
Bubbles
such as
soap bubbles
take a spherical shape in equilibrium.
The Earth is often approximated as a sphere
in
geography
, and the
celestial sphere
is an important concept in
astronomy
. Manufactured items including
pressure vessels
and most
curved mirrors
and
lenses
are based on spheres. Spheres
roll
smoothly in any direction, so most
balls
used in sports and toys are spherical, as are
ball bearings
.
Basic terminology
[
edit
]
Two orthogonal radii of a sphere
As mentioned earlier
r
is the sphere's radius; any line from the center to a point on the sphere is also called a radius.
[3]
If a radius is extended through the center to the opposite side of the sphere, it creates a
diameter
. Like the radius, the length of a diameter is also called the diameter, and denoted
d
. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius,
d
= 2
r
. Two points on the sphere connected by a diameter are
antipodal points
of each other.
[3]
A
unit sphere
is a sphere with unit radius (
r
= 1
). For convenience, spheres are often taken to have their center at the origin of the
coordinate system
, and spheres in this article have their center at the origin unless a center is mentioned.
A
great circle
on the sphere has the same center and radius as the sphere, and divides it into two equal
hemispheres
.
Although the
figure 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 an
axis
(as in Earth's
axis of rotation
).
The sphere-axis intersection defines two antipodal
poles
(
north pole
and
south pole
). The great circle equidistant to the poles is called the
equator
. Great circles through the poles are called lines of
longitude
or
meridians
. Small circles on the sphere that are parallel to the equator are
circles of latitude
(or
parallels
). 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 a two-dimensional
closed surface
embedded
in three-dimensional
Euclidean space
. They draw a distinction between a
sphere
and a
ball
, which is a three-dimensional
manifold with boundary
that includes the volume contained by the sphere. An
open ball
excludes the sphere itself, while a
closed ball
includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the
boundary
of a (closed or open) ball. The distinction between
ball
and
sphere
has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "
circle
" and "
disk
" in the
plane
is similar.
Small spheres or balls are sometimes called spherules, e.g. in
Martian spherules
.
Equations
[
edit
]
In
analytic geometry
, a sphere with center
(
x
0
,
y
0
,
z
0
)
and radius
r
is the
locus
of all points
(
x
,
y
,
z
)
such that
![{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a476c6003522e221e5620b363ad446c25c0044b3)
Since it can be expressed as a quadratic polynomial, a sphere is a
quadric surface
, a type of
algebraic surface
.
[3]
Let
a, b, c, d, e
be real numbers with
a
≠ 0
and put
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/836ec8f22b47f49bcc6dd7dfdf439aa1229fa65b)
Then the equation
![{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a133703df70ba0dab184e3fcf4be2cd38c74eb)
has no real points as solutions if
and is called the equation of an
imaginary sphere
. If
, the only solution of
is the point
and the equation is said to be the equation of a
point sphere
. Finally, in the case
,
is an equation of a sphere whose center is
and whose radius is
.
[2]
If
a
in the above equation is zero then
f
(
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 a
point at infinity
.
[4]
Parametric
[
edit
]
A
parametric equation
for the sphere with radius
and center
can be parameterized using
trigonometric functions
.
[5]
The symbols used here are the same as those used in
spherical coordinates
.
r
is constant, while
θ
varies from 0 to
π
and
varies from 0 to 2
π
.
Properties
[
edit
]
Enclosed volume
[
edit
]
Sphere and circumscribed cylinder
In three dimensions, the
volume
inside a sphere (that is, the volume of a
ball
, but classically referred to as the volume of a sphere) is
![{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot d^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42de7dbcc7c51105e0f10e71f53c61987454ced6)
where
r
is the radius and
d
is the diameter of the sphere.
Archimedes
first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the
circumscribed
cylinder
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 applying
Cavalieri's principle
.
[7]
This formula can also be derived using
integral calculus
, i.e.
disk integration
to sum the volumes of an
infinite number
of
circular
disks of infinitesimally small thickness stacked side by side and centered along the
x
-axis from
x
= ?
r
to
x
=
r
, assuming the sphere of radius
r
is centered at the origin.
Proof of sphere volume, using calculus
|
At any given
x
, the incremental volume (
δV
) equals the product of the cross-sectional
area of the disk
at
x
and its thickness (
δx
):
![{\displaystyle \delta V\approx \pi y^{2}\cdot \delta x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f5294b86738de4a6370477df266e31adea58e6)
The total volume is the summation of all incremental volumes:
![{\displaystyle V\approx \sum \pi y^{2}\cdot \delta x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d25825c42ffeebfbf63349381f1bb020398910a)
In the limit as
δx
approaches zero,
[8]
this equation becomes:
![{\displaystyle V=\int _{-r}^{r}\pi y^{2}dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89d1ea760eaab5ec19ef3372ea4c9ee24addee22)
At any given
x
, a right-angled triangle connects
x
,
y
and
r
to the origin; hence, applying the
Pythagorean theorem
yields:
![{\displaystyle y^{2}=r^{2}-x^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61fa971d8c5410f0cbbbb2f948b29db288e26550)
Using this substitution gives
![{\displaystyle V=\int _{-r}^{r}\pi \left(r^{2}-x^{2}\right)dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2040d2da3a61d453ac509a262a4efaf566ba484)
which can be evaluated to give the result
![{\displaystyle V=\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{-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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c081de9760153a5ab7e59be1b9de1aa97d08dec)
An alternative formula is found using
spherical coordinates
, with
volume element
![{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7baab55bb4d5559e61d50df77cca1d7f6befc27)
so
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b51d4bd953c2d8ddb0b746770be5d790eb6e01)
|
For most practical purposes, the volume inside a sphere
inscribed
in a cube can be approximated as 52.4% of the volume of the cube, since
V
=
π
/
6
d
3
, where
d
is the diameter of the sphere and also the length of a side of the cube and
π
/
6
? 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1
m, or about 0.524 m
3
.
Surface area
[
edit
]
The
surface area
of a sphere of radius
r
is:
![{\displaystyle A=4\pi r^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95625828c519b36791d56af65d21b8448472650d)
Archimedes
first derived this formula
[9]
from the fact that the projection to the lateral surface of a
circumscribed
cylinder is area-preserving.
[10]
Another approach to obtaining the formula comes from the fact that it equals the
derivative
of the formula for the volume with respect to
r
because the total volume inside a sphere of radius
r
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
r
. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius
r
is simply the product of the surface area at radius
r
and the infinitesimal thickness.
Proof of surface area, using calculus
|
At any given radius
r
,
[note 1]
the incremental volume (
δV
) equals the product of the surface area at radius
r
(
A
(
r
)
) and the thickness of a shell (
δr
):
![{\displaystyle \delta V\approx A(r)\cdot \delta r.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd6b20b042b7b8356514cafe4bfe9323f73970e)
The total volume is the summation of all shell volumes:
![{\displaystyle V\approx \sum A(r)\cdot \delta r.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff662cf129a7b7ae53bb28460dadef0715b60c4)
In the limit as
δr
approaches zero
[8]
this equation becomes:
![{\displaystyle V=\int _{0}^{r}A(r)\,dr.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dde5072ef2871126b29c8eab1cc7b83ec2d365ad)
Substitute
V
:
![{\displaystyle {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}A(r)\,dr.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/521cb0af0bc79aaa0a4e3c6d6f02d440057fb894)
Differentiating both sides of this equation with respect to
r
yields
A
as a function of
r
:
![{\displaystyle 4\pi r^{2}=A(r).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f25b8b4e3cc58f767579a25d212571f323e2f740)
This is generally abbreviated as:
![{\displaystyle A=4\pi r^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1cbdbb8a22c7db3b3f01dad4ea19f8dfcd502b)
where
r
is now considered to be the fixed radius of the sphere.
Alternatively, the
area element
on the sphere is given in
spherical coordinates
by
dA
=
r
2
sin
θ dθ dφ
. The total area can thus be obtained by
integration
:
![{\displaystyle A=\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin \theta \,d\theta \,d\varphi =4\pi r^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/128b56cc8737351056a8e5fd4dfc2dd163f58bc5)
|
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 the
surface tension
locally minimizes surface area.
The surface area relative to the mass of a ball is called the
specific surface area
and can be expressed from the above stated equations as
![{\displaystyle \mathrm {SSA} ={\frac {A}{V\rho }}={\frac {3}{r\rho }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a992bf548d15c31b027fcb1b6a7b6dada6587acf)
where
ρ
is the
density
(the ratio of mass to volume).
Other geometric properties
[
edit
]
A sphere can be constructed as the surface formed by rotating a
circle
one half revolution about any of its
diameters
; this is very similar to the traditional definition of a sphere as given in
Euclid's Elements
. Since a circle is a special type of
ellipse
, a sphere is a special type of
ellipsoid of revolution
. Replacing the circle with an ellipse rotated about its
major axis
, the shape becomes a prolate
spheroid
; rotated about the minor axis, an oblate spheroid.
[12]
A sphere is uniquely determined by four points that are not
coplanar
. 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 three
non-collinear
points determine a unique circle in a plane.
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
By examining the
common solutions of the equations of two spheres
, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the
radical plane
of the intersecting spheres.
[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 the
dihedral angle
determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection.
[16]
They intersect at right angles (are
orthogonal
) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.
[4]
Pencil of spheres
[
edit
]
If
f
(
x
,
y
,
z
) = 0
and
g
(
x
,
y
,
z
) = 0
are the equations of two distinct spheres then
![{\displaystyle sf(x,y,z)+tg(x,y,z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/163ef7689e46b6f2d09c1e04048b990192fef39d)
is also the equation of a sphere for arbitrary values of the parameters
s
and
t
. The set of all spheres satisfying this equation is called a
pencil of spheres
determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.
[4]
Properties of the sphere
[
edit
]
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 book
Geometry and the Imagination
,
David Hilbert
and
Stephan Cohn-Vossen
describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere.
[17]
Several properties hold for the
plane
, which can be thought of as a sphere with infinite radius. These properties are:
- The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.
- The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar
result
of
Apollonius of Perga
for the
circle
. This second part also holds for the
plane
.
- The contours and plane sections of the sphere are circles.
- This property defines the sphere uniquely.
- The sphere has constant width and constant girth.
- The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the
Meissner body
. The girth of a surface is the
circumference
of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
- All points of a sphere are
umbilics
.
- At any point on a surface a
normal direction
is at right angles to the surface because on the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a
normal section,
and the curvature of this curve is the
normal curvature
. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the
principal curvatures
. Any closed surface will have at least four points called
umbilical points
. At an umbilic all the sectional curvatures are equal; in particular the
principal curvatures
are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
- For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.
- The sphere does not have a surface of centers.
- For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the
focal points
, and the set of all such centers forms the
focal surface
.
- For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:
- * For
channel surfaces
one sheet forms a curve and the other sheet is a surface
- * For
cones
, cylinders,
tori
and
cyclides
both sheets form curves.
- * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.
- All geodesics of the sphere are closed curves.
- Geodesics
are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.
- Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.
- It follows from
isoperimetric inequality
. These properties define the sphere uniquely and can be seen in
soap bubbles
: a soap bubble will enclose a fixed volume, and
surface tension
minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.
- The sphere has the smallest total mean curvature among all convex solids with a given surface area.
- The
mean curvature
is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.
- The sphere has constant mean curvature.
- The sphere is the only
imbedded
surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as
minimal 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 is
embedded
in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The
pseudosphere
is an example of a surface with constant negative Gaussian curvature.
- The sphere is transformed into itself by a three-parameter family of rigid motions.
- Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see
Euler angles
). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the
rotation group SO(3)
. The plane is the only other surface with a three-parameter family of transformations (translations along the
x
- and
y
-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the
surfaces of revolution
and
helicoids
are the only surfaces with a one-parameter family.
Treatment by area of mathematics
[
edit
]
Spherical geometry
[
edit
]
Great circle
on a sphere
The basic elements of
Euclidean plane geometry
are
points
and
lines
. On the sphere, points are defined in the usual sense. The analogue of the "line" is the
geodesic
, which is a
great circle
; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by
arc length
shows that the shortest path between two points lying on the sphere is the shorter segment of the
great circle
that includes the points.
Many theorems from
classical geometry
hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's
postulates
, including the
parallel postulate
. In
spherical trigonometry
,
angles
are defined between great circles. Spherical trigonometry differs from ordinary
trigonometry
in many respects. For example, the sum of the interior angles of a
spherical triangle
always exceeds 180 degrees. Also, any two
similar
spherical triangles are congruent.
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are called
antipodal points
?on the sphere, the distance between them is exactly half the length of the circumference.
[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 the
shortest distance
between them on the sphere.
[note 3]
Spherical geometry is a form of
elliptic geometry
, which together with
hyperbolic geometry
makes up
non-Euclidean geometry
.
Differential geometry
[
edit
]
The sphere is a
smooth surface
with constant
Gaussian curvature
at each point equal to
1/
r
2
.
[9]
As per Gauss's
Theorema 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, any
map projection
introduces some form of distortion.
A sphere of radius
r
has
area element
. This can be found from the
volume element
in
spherical coordinates
with
r
held constant.
[9]
A sphere of any radius centered at zero is an
integral surface
of the following
differential form
:
![{\displaystyle x\,dx+y\,dy+z\,dz=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cfcc379e60de9faca3aa371dc3f6b3ea23e965)
This equation reflects that the position vector and
tangent plane
at a point are always
orthogonal
to each other. Furthermore, the outward-facing
normal vector
is equal to the position vector scaled by
1/r
.
In
Riemannian geometry
, the
filling area conjecture
states that the hemisphere is the optimal (least area) isometric filling of the
Riemannian circle
.
Topology
[
edit
]
Remarkably, it is possible to turn an ordinary sphere inside out in a
three-dimensional space
with possible self-intersections but without creating any creases, in a process called
sphere eversion
.
The antipodal quotient of the sphere is the surface called the
real projective plane
, which can also be thought of as the
Northern Hemisphere
with antipodal points of the equator identified.
Curves on a sphere
[
edit
]
Plane section of a sphere: 1 circle
Coaxial intersection of a sphere and a cylinder: 2 circles
Circles
[
edit
]
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 a
surface of revolution
whose axis contains the center of the sphere (are
coaxial
) consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.
Loxodrome
[
edit
]
Loxodrome
In
navigation
, a
loxodrome
or
rhumb line
is a path whose
bearing
, the angle between its tangent and due North, is constant. Loxodromes project to straight lines under the
Mercator projection
. Two special cases are the
meridians
which are aligned directly North?South and
parallels
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
spherical coordinate system
, such a loxodrome is a kind of
spherical spiral
.
[19]
Clelia curves
[
edit
]
Clelia spiral with
c
= 8
Another kind of spherical spiral is the Clelia curve, for which the
longitude
(or azimuth)
and the
colatitude
(or polar angle)
are in a linear relationship,
. Clelia curves project to straight lines under the
equirectangular projection
.
Viviani's curve
(
) is a special case. Clelia curves approximate the
ground track
of satellites in
polar orbit
.
Spherical conics
[
edit
]
The analog of a
conic section
on the sphere is a
spherical conic
, a
quartic
curve which can be defined in several equivalent ways, including:
- as the intersection of a sphere with a quadratic cone whose vertex is the sphere center;
- as the intersection of a sphere with an
elliptic or hyperbolic cylinder
whose axis passes through the sphere center;
- as the locus of points whose sum or difference of
great-circle distances
from a pair of
foci
is a constant.
Many theorems relating to planar conic sections also extend to spherical conics.
Intersection of a sphere with a more general surface
[
edit
]
General intersection sphere-cylinder
If a sphere is intersected by another surface, there may be more complicated spherical curves.
- Example
- sphere ? cylinder
The intersection of the sphere with equation
and the cylinder with equation
is not just one or two circles. It is the solution of the non-linear system of equations
![{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91803efe99c5b176a4bccd39b154fc1c398a8510)
![{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a647e39f51682c5f61442adba20602d76042669)
(see
implicit curve
and the diagram)
Generalizations
[
edit
]
Ellipsoids
[
edit
]
An
ellipsoid
is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an
affine transformation
. An ellipsoid bears the same relationship to the sphere that an
ellipse
does to a circle.
Dimensionality
[
edit
]
Spheres can be generalized to spaces of any number of
dimensions
. For any
natural number
n
, an
n
-sphere,
often denoted
S
n
, is the set of points in (
n
+ 1
)-dimensional Euclidean space that are at a fixed distance
r
from a central point of that space, where
r
is, as before, a positive real number. In particular:
- S
0
: a 0-sphere consists of two discrete points,
?
r
and
r
- S
1
: a 1-sphere is a
circle
of radius
r
- S
2
: a 2-sphere is an ordinary sphere
- S
3
: a
3-sphere
is a sphere in 4-dimensional Euclidean space.
Spheres for
n
> 2
are sometimes called
hyperspheres
.
The
n
-sphere of unit radius centered at the origin is denoted
S
n
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.
In
topology
, the
n
-sphere is an example of a
compact
topological manifold
without
boundary
. A topological sphere need not be
smooth
; if it is smooth, it need not be
diffeomorphic
to the Euclidean sphere (an
exotic sphere
).
The sphere is the inverse image of a one-point set under the continuous function
‖
x
‖
, so it is closed;
S
n
is also bounded, so it is compact by the
Heine?Borel theorem
.
Metric spaces
[
edit
]
More generally, in a
metric space
(
E
,
d
)
, the sphere of center
x
and radius
r
> 0
is the set of points
y
such that
d
(
x
,
y
) =
r
.
If the center is a distinguished point that is considered to be the origin of
E
, as in a
normed
space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a
unit sphere
.
Unlike a
ball
, even a large sphere may be an empty set. For example, in
Z
n
with
Euclidean metric
, a sphere of radius
r
is nonempty only if
r
2
can be written as sum of
n
squares of
integers
.
An
octahedron
is a sphere in
taxicab geometry
, and a
cube
is a sphere in geometry using the
Chebyshev distance
.
History
[
edit
]
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 to
Eudoxus of Cnidus
. The volume and area formulas were first determined in
Archimedes
's
On the Sphere and Cylinder
by the
method of exhaustion
.
Zenodorus
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 by
Dionysodorus
.
[20]
A similar problem ? to construct a segment equal in volume to a given segment, and in surface to another segment ? was solved later by
al-Quhi
.
[3]
Gallery
[
edit
]
-
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]
-
Deck of playing cards illustrating engineering instruments, England, 1702.
King of spades
: Spheres
Regions
[
edit
]
See also
[
edit
]
Notes and references
[
edit
]
Notes
[
edit
]
- ^
r
is being considered as a variable in this computation.
- ^
It does not matter which direction is chosen, the distance is the sphere's radius ×
π
.
- ^
The distance between two non-distinct points (i.e. a point and itself) on the sphere is zero.
References
[
edit
]
- ^
σφα?ρα
, Henry George Liddell, Robert Scott,
A Greek-English Lexicon
, on Perseus.
- ^
a
b
Albert 2016
, p. 54.
- ^
a
b
c
d
e
f
Chisholm, Hugh
, ed. (1911).
"Sphere"
.
Encyclopædia Britannica
. Vol. 25 (11th ed.). Cambridge University Press. pp. 647?648.
- ^
a
b
c
Woods 1961
, p. 266.
- ^
Kreyszig (1972
, p. 342).
- ^
Steinhaus 1969
, p. 223.
- ^
"The volume of a sphere ? Math Central"
.
mathcentral.uregina.ca
. Retrieved
10 June
2019
.
- ^
a
b
E.J. Borowski; J.M. Borwein (1989).
Collins Dictionary of Mathematics
. Collins. pp. 141, 149.
ISBN
978-0-00-434347-1
.
- ^
a
b
c
Weisstein, Eric W.
"Sphere"
.
MathWorld
.
- ^
Steinhaus 1969
, p. 221.
- ^
Osserman, Robert (1978).
"The isoperimetric inequality"
.
Bulletin of the American Mathematical Society
.
84
(6): 1187.
doi
:
10.1090/S0002-9904-1978-14553-4
. Retrieved
14 December
2019
.
- ^
Albert 2016
, p. 60.
- ^
Albert 2016
, p. 55.
- ^
Albert 2016
, p. 57.
- ^
Woods 1961
, p. 267.
- ^
Albert 2016
, p. 58.
- ^
Hilbert, David
; Cohn-Vossen, Stephan (1952). "Eleven properties of the sphere".
Geometry and the Imagination
(2nd ed.). Chelsea. pp. 215?231.
ISBN
978-0-8284-1087-8
.
- ^
Weisstein, Eric W.
"Spheric section"
.
MathWorld
.
- ^
https://mathworld.wolfram.com/Loxodrome.html
- ^
Fried, Michael N. (25 February 2019).
"conic sections"
.
Oxford Research Encyclopedia of Classics
.
doi
:
10.1093/acrefore/9780199381135.013.8161
.
ISBN
978-0-19-938113-5
. Retrieved
4 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.
- ^
New Scientist | Technology | Roundest objects in the world created
.
Further reading
[
edit
]
- Albert, Abraham Adrian (2016) [1949],
Solid Analytic Geometry
, Dover,
ISBN
978-0-486-81026-3
.
- Dunham, William (1997).
The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities
. New York: Wiley. pp.
28
, 226.
Bibcode
:
1994muaa.book.....D
.
ISBN
978-0-471-17661-9
.
- Kreyszig, Erwin (1972),
Advanced Engineering Mathematics
(3rd ed.), New York:
Wiley
,
ISBN
978-0-471-50728-4
.
- Steinhaus, H. (1969),
Mathematical Snapshots
(Third American ed.), Oxford University Press
.
- Woods, Frederick S. (1961) [1922],
Higher Geometry / An Introduction to Advanced Methods in Analytic Geometry
, Dover
.
- John C. Polking (15 April 1999).
"The Geometry of the Sphere"
.
www.math.csi.cuny.edu
. Retrieved
21 January
2022
.
External links
[
edit
]
Compact topological surfaces and their immersions in 3D
|
---|
Without boundary
| Orientable
|
- Sphere
(genus 0)
- Torus
(genus 1)
- Number 8 (genus 2)
- Pretzel (genus 3) ...
|
---|
Non-orientable
| |
---|
|
---|
With boundary
| |
---|
Related
notions
| Properties
| |
---|
Characteristics
| |
---|
Operations
| |
---|
|
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