aarjaneiro/planetmath huggingface dataset:
area of the n-sphere

area of the

n

-sphere

The area of

S

n

the unit

n

-sphere (or

hypersphere

) is the same as the total

solid angle

it

subtends

at the origin. To calculate it, consider the following

integral

I

⁢

(

n

)

=

∫

ℝ

n

+

1

e

-

∑

i

=

1

n

+

1

x

i

2

⁢

d

n

+

1

⁢

x

.

Switching to

polar coordinates

we let

r

2

=

∑

i

=

1

n

+

1

x

i

2

and the
integral becomes

I

⁢

(

n

)

=

∫

S

n

𝑑

Ω

⁢

∫

0

∞

r

n

⁢

e

-

r

2

⁢

𝑑

r

.

The first integral is the integral over all solid angles and is exactly what we
want to evaluate. Let us denote it by

A

⁢

(

n

)

. With the change of variable

t

=

r

2

, the second integral can be evaluated in terms of the

gamma function

Γ

⁢

(

x

)

:

I

⁢

(

n

)

/

A

⁢

(

n

)

=

1

2

⁢

∫

0

∞

t

n

-

1

2

⁢

e

-

t

⁢

𝑑

t

=

1

2

⁢

Γ

⁢

(

n

+

1

2

)

.

We can also evaluate

I

⁢

(

n

)

directly in

Cartesian coordinates

:

I

⁢

(

n

)

=

[

∫

-

∞

∞

e

-

x

2

⁢

𝑑

x

]

n

+

1

=

π

n

+

1

2

,

where we have used the standard

Gaussian integral

∫

-

∞

∞

e

-

x

2

⁢

𝑑

x

=

π

.

Finally, we can solve for the area

A

⁢

(

n

)

=

2

⁢

π

n

+

1

2

Γ

⁢

(

n

+

1

2

)

.

If the radius of the sphere is

R

and not

1

, the correct area is

A

⁢

(

n

)

⁢

R

n

.

Note that this formula works only for

n

≥

0

. The first few special cases
are

n

=

0

Γ

⁢

(

1

/

2

)

=

π

, hence

A

⁢

(

0

)

=

2

(in this case, the
area just counts the number of points in

S

0

=

{

+

1

,

-

1

}

);

n

=

1

Γ

⁢

(

1

)

=

1

, hence

A

⁢

(

1

)

=

2

⁢

π

(this is the familiar result
for the

circumference

of the unit circle);

n

=

2

Γ

⁢

(

3

/

2

)

=

π

/

2

, hence

A

⁢

(

2

)

=

4

⁢

π

(this is the
familiar result for the area of the

unit sphere

);

n

=

3

Γ

⁢

(

2

)

=

1

, hence

A

⁢

(

3

)

=

2

⁢

π

2

;

n

=

4

Γ

⁢

(

5

/

2

)

=

3

⁢

π

/

4

, hence

A

⁢

(

4

)

=

8

⁢

π

2

/

3

.

Title

area of the

n

-sphere

Canonical name

AreaOfTheNsphere

Date of creation

2013-03-22 13:47:06

Last modified on

2013-03-22 13:47:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 51M05

Related topic

VolumeOfTheNSphere

Related topic

AreaOfASphericalTriangle

Related topic

AreaOfSphericalZone

Generated on Fri Feb 9 22:04:43 2018 by

LaTeXML
cylindrical coordinates

cylindrical coordinates

Cylindrical coordinates

are a system of

coordinates

for

ℝ

3

. Two of the coordinates correspond to the

polar coordinates

of

ℝ

2

, and the third coordinate corresponds with the

z

axis. Thus, the coordinates are given by

(

x

y

z

)

=

(

r

⁢

cos

⁡

θ

r

⁢

sin

⁡

θ

z

)

,

where

r

is the

distance

from

(

0

,

0

,

0

)

to

(

x

,

y

,

0

)

and

θ

is the

azimuthal angle

defined for

θ

∈

[

0

,

2

⁢

π

)

.

Just as with polar coordinates, one can convert from

Cartesian coordinates

to cylindrical coordinates for any point not

lying on

the

z

axis via

r

⁢

(

x

,

y

)

=

x

2

+

y

2

,

θ

⁢

(

x

,

y

)

=

arctan

⁡

(

x

,

y

)

,

where

arctan

is defined here (

http://planetmath.org/OperatornamearcTanWithTwoArguments

).

Title

cylindrical coordinates

Canonical name

CylindricalCoordinates

Date of creation

2013-03-22 17:01:54

Last modified on

2013-03-22 17:01:54

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M05

Related topic

PolarCoordinates

Related topic

SphericalCoordinates

Generated on Fri Feb 9 22:04:46 2018 by

LaTeXML
ellipsoid

ellipsoid

An

ellipsoid

is a subset of

ℝ

3

consisting of points

(

x

,

y

,

z

)

∈

ℝ

3

such that

(

x

a

)

2

+

(

y

b

)

2

+

(

z

c

)

2

=

1

for some

a

,

b

,

c

>

0

.

Properties

1.

If

a

=

b

=

c

, the ellipsoid reduces to a sphere.

2.

If we fix the value of any of

x

,

y

,

z

to some constant, say

x

=

C

,
we obtain an

ellipse

in the plane

(

C

,

y

,

z

)

.

3.

The ellipse determined by

a

,

b

,

c

is the

unit sphere

of the
norm

∥

v

∥

=

v

T

⁢

diag

⁡

(

1

a

,

1

b

,

1

c

)

⁢

v

,

v

=

(

x

,

y

,

z

)

T

.

Title

ellipsoid

Canonical

name

Ellipsoid

Date of creation

2013-03-22 14:56:45

Last modified on

2013-03-22 14:56:45

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Definition

Classification

msc 51M05

Related topic

Sphere

Related topic

QuadraticSurfaces

Related topic

Ellipse2

Related topic

VolumeOfEllipsoid

Generated on Fri Feb 9 22:04:48 2018 by

LaTeXML
Euclidean axiom by Hilbert

Euclidean axiom by Hilbert

In Hilbert’s

Grundlagen der Geometrie

(‘Foundations of Geometry’; the original edition in 1899) there is the following argumentation.

Let

α

be an arbitrary plane,

a

a line in

α

and

A

a point in

α

which lies outside

a

. If we draw in

α

a line

c

which

passes through

A

and

intersects

a

and then through

A

a line

b

such that the line

c

intersects the lines

a

,

b

with equal

alternate interior angles

(“unter gleichen Gegenwinkeln”),

then it follows easily from the

theorem

on the outer angles, that the lines

a

,

b

have no common point, i.e., in a plane

α

one can always draw otside a line

a

another line which does not intersect the line

a

.

The Parallel Axiom reads now:

IV (

).

Let

a

be an arbitrary line and

A

be a point outside

a

: then in the plane determined by

a

and

A

there exists at most one line which passes through

A

and does not intersect

a

.

Explanation

.

According the the preceding text and on grounds of the Parallel Axiom we realize, that there is one and only one line which passes through

A

and do not intersect

a

; that is called the

parallel

of

a

through

A

.

The Parallel Axiom means the same as the following requirement:

When two lines

a

,

b

in a plane do not meet a third line

c

of the same plane, then also they do not meet each other.

The theorem on the outer angles is the following:

An outer angle of a

triangle

is greater than both non-adjacent angles of the triangle.

Using this one may indirectly justify the assertion in the first cited paragraph.

Introducing the Parallel Axiom simplifies the

foundations

and facilitates the construction of

geometry

significantly.

If we , then we obtain easily the following well-known fact:

Theorem 31

.

If two parallels intersect a third line, then the corresponding angles and the alternate interior angles are

congruent

, and

conversely

: the

congruence

(

http://planetmath.org/GeometricCongruence

) of the corresponding or alternate interior angles implies that the lines are parallel.

References

1

D. Hilbert

:

Grundlagen der Geometrie

. Neunte Auflage, revidiert und ergänzt von Paul Bernays.  B. G. Teubner Verlagsgesellschaft, Stuttgart (1962).

Title

Euclidean axiom by Hilbert

Canonical name

EuclideanAxiomByHilbert

Date of creation

2013-03-22 17:19:16

Last modified on

2013-03-22 17:19:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 51M05

Classification

msc 51-01

Related topic

CorrespondingAnglesInTransversalCutting

Related topic

AxiomaticGeometry

Generated on Fri Feb 9 22:04:50 2018 by

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exterior angles of triangle

exterior angles of triangle

The

exterior angle

of an angle of

triangle

is greater than both other
angles of the triangle.

Proof.

Let us study in an arbitrary triangle

A

⁢

B

⁢

C

for example the exterior angle

∧

A

⁢

C

⁢

D

where

D

is point on the lengthening of the side

B

⁢

C

nearer to

C

than to

B

.  Let

E

be the

midpoint

of

A

⁢

C

.  Let

B

⁢

E

be the median of the triangle.  We find on its lengthening the point

F

such that

E

⁢

F

=

E

⁢

B

.  Then the triangles

A

⁢

B

⁢

E

and

C

⁢

E

⁢

F

are

congruent

(SAS).  Consequently, we have

∧

E

⁢

C

⁢

F

=

∧

B

⁢

A

⁢

E

and therefore

∧

A

⁢

C

⁢

D

>

∧

B

⁢

A

⁢

C

.   Analogically one shows that

∧

A

⁢

C

⁢

D

>

∧

A

⁢

B

⁢

C

.

□

References

1

Karl Ariva

:

Lobatsevski geomeetria

.  Kirjastus “Valgus”, Tallinn (1992).

Title

exterior angles of triangle

Canonical name

ExteriorAnglesOfTriangle

Date of creation

2013-05-05 8:44:43

Last modified on

2013-05-05 8:44:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

2

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51M05

Generated on Fri Feb 9 22:04:53 2018 by

LaTeXML
homology of the sphere

homology of the sphere

Every loop on the sphere

S

2

is

contractible

to a point, so its

fundamental group

,

π

1

⁢

(

S

2

)

, is trivial.

Let

H

n

⁢

(

S

2

,

ℤ

)

denote the

n

-th

homology group

of

S

2

. We can compute all of these groups using the basic results from

algebraic topology

:

•

S

2

is a compact

orientable

smooth manifold

, so

H

2

⁢

(

S

2

,

ℤ

)

=

ℤ

;

•

S

2

is connected, so

H

0

⁢

(

S

2

,

ℤ

)

=

ℤ

;

•

H

1

⁢

(

S

2

,

ℤ

)

is the

abelianization

of

π

1

⁢

(

S

2

)

, so it is also trivial;

•

S

2

is two-dimensional, so for

k

>

2

, we have

H

k

⁢

(

S

2

,

ℤ

)

=

0

In fact, this

pattern

generalizes nicely to higher-dimensional spheres:

H

k

⁢

(

S

n

,

ℤ

)

=

{

ℤ

k

=

0

,

n

0

else

This also provides the proof that the

hyperspheres

S

n

and

S

m

are non-homotopic for

n

≠

m

, for this would imply an

isomorphism

between their

homologies

.

Title

homology of the sphere

Canonical name

HomologyOfTheSphere

Date of creation

2013-03-22 13:46:49

Last modified on

2013-03-22 13:46:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Derivation

Classification

msc 51M05

Related topic

sphere

Related topic

HomologyTopologicalSpace

Related topic

Sphere

Generated on Fri Feb 9 22:04:55 2018 by

LaTeXML
intersection of sphere and plane

intersection of sphere and plane

Theorem

.

The intersection curve of a sphere and a plane is a circle.

Proof.

We prove the theorem without the

equation

of the sphere.  Let

c

be the intersection curve,

r

the radius of the sphere and

O

⁢

Q

be the

distance

of the centre

O

of the sphere and the plane.  If

P

is an arbitrary point of

c

, then

O

⁢

P

⁢

Q

is a

right triangle

.  By the

Pythagorean theorem

,

P

⁢

Q

=

ϱ

=

r

2

-

O

⁢

Q

2

=

constant

.

Thus any point of the curve

c

is in the plane at a distance

ϱ

from the point

Q

, whence

c

is a circle.

Remark.

There are two special cases of the

intersection

of a sphere and a plane:  the

empty set

of points (

O

⁢

Q

>

r

) and a single point (

O

⁢

Q

=

r

); these of course are not curves.  In the former case one usually says that the sphere does not intersect the plane, in the latter one sometimes calls the common point a

zero circle

(it can be thought a circle with radius 0).

Title

intersection of sphere and plane

Canonical

name

IntersectionOfSphereAndPlane

Date of creation

2013-03-22 18:18:39

Last modified on

2013-03-22 18:18:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

22

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51M05

Related topic

ConformalityOfStereographicProjection

Defines

zero circle

Generated on Fri Feb 9 22:05:07 2018 by

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Legendre’s theorem on angles of triangle

Legendre’s theorem on angles of triangle

Adrien-Marie Legendre

has proved some

theorems

concerning the sum of the angles of

triangle

.  Here we give one of them, being the

inverse

of the theorem in the entry “sum of angles of triangle in

Euclidean geometry

”.

Theorem.

If the sum of the

interior angles

of every triangle equals

straight angle

, then the

parallel postulate

is true, i.e., in the plane determined by a line and a point outwards it there is exactly one line through the point which does not intersect the line.

Proof.

We consider a line

a

and a point

B

not belonging to

a

.  Let

B

⁢

A

be the

normal line

of

a

(with

A

∈

a

) and

b

be the normal line of

B

⁢

A

through the point

B

.  By the supposition of the theorem,

b

does not intersect

a

.

We will show that in the plane determined by the line

a

and the point

B

, there are through

B

no other lines than

b

not intersecting the line

a

.  For this purpose, we choose through

B

a line

b

′

which differs from

b

; let the line

b

′

form with

B

⁢

A

an

acute angle

β

.

We determine on the line

a

a point

A

1

such that

A

⁢

A

1

=

A

⁢

B

.  By the supposition of the theorem, in the

isosceles right triangle

B

⁢

A

⁢

A

1

we have

α

1

=

:

∠

A

A

1

B

=

π

4

=

π

2

2

.

Next we determine on

a

a second point

A

2

such that

A

1

⁢

A

2

=

A

1

⁢

B

.  By the supposition of the theorem, in the

isosceles triangle

B

⁢

A

1

⁢

A

2

we have

α

2

=

:

∠

A

A

2

B

=

α

1

2

=

π

2

3

.

We continue similarly by forming isosceles triangles using the points

A

3

,

A

3

,

…

,

A

n

of the line

a

such that

A

2

⁢

A

3

=

B

⁢

A

2

,

A

3

⁢

A

4

=

B

⁢

A

3

,

…

,

A

n

-

1

⁢

A

n

=

B

⁢

A

n

-

1

.

Then the acute angles being formed beside the points are

α

3

=

π

2

4

,

α

4

=

π

2

5

,

…

,

α

n

=

π

2

n

+

1

.

They form a

geometric sequence

with the

common ratio

r

=

1

2

.  When

n

is sufficiently great, the member

α

n

is less than any given

positive

angle.  As we have so much triangles

B

⁢

A

n

-

1

⁢

A

n

that

α

n

<

π

2

-

β

,  then

∠

⁢

A

⁢

B

⁢

A

n

=

π

2

-

α

n

>

β

.

Then the line

b

′

falls after penetrating

B

into the inner territory of the triangle

A

⁢

B

⁢

A

n

.  Thereafter it must leave from there and thus intersect the side

A

⁢

A

n

of this triangle.  Accordingly,

b

′

intersects the line

a

.

The above reasoning is possible for each line

b

′

≠

b

through

B

.  Consequently, the parallel axiom is in force.

References

1

Karl Ariva

:

Lobatsevski geomeetria

.  Kirjastus “Valgus”, Tallinn (1992).

Title

Legendre’s theorem on angles of triangle

Canonical

name

LegendresTheoremOnAnglesOfTriangle

Date of creation

2013-05-11 13:55:53

Last modified on

2013-05-11 13:55:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51M05

Generated on Fri Feb 9 22:05:10 2018 by

LaTeXML
lengths of angle bisectors

lengths of angle bisectors

In any

triangle

, the

w

a

,

w

b

,

w

c

of the

angle bisectors

opposing the sides

a

,

b

,

c

, respectively, are

w

a

=

b

⁢

c

⁢

[

(

b

+

c

)

2

-

a

2

]

b

+

c

,

(1)

w

b

=

c

⁢

a

⁢

[

(

c

+

a

)

2

-

b

2

]

c

+

a

,

(2)

w

c

=

a

⁢

b

⁢

[

(

a

+

b

)

2

-

c

2

]

a

+

b

.

(3)

Proof.

By the

symmetry

, it suffices to prove only (1).

According the

angle bisector theorem

, the

bisector

w

a

divides the side

a

into the portions

b

b

+

c

⋅

a

=

a

⁢

b

b

+

c

,

c

b

+

c

⋅

a

=

c

⁢

a

b

+

c

.

If the angle opposite to

a

is

α

, we apply the

law of cosines

to the half-triangles by

w

a

:

{

2

⁢

w

a

⁢

b

⁢

cos

⁡

α

2

=

w

a

2

+

b

2

-

(

a

⁢

b

b

+

c

)

2

2

⁢

w

a

⁢

c

⁢

cos

⁡

α

2

=

w

a

2

+

c

2

-

(

c

⁢

a

b

+

c

)

2

(4)

For eliminating the angle

α

, the equations (4) are divided sidewise, when one gets

b

c

=

w

a

2

+

b

2

-

(

a

⁢

b

b

+

c

)

2

w

a

2

+

c

2

-

(

c

⁢

a

b

+

c

)

2

,

from which one can after some routine manipulations solve

w

a

, and this can be simplified to the form (1).

Title

lengths of angle bisectors

Canonical

name

LengthsOfAngleBisectors

Date of creation

2013-03-22 18:26:50

Last modified on

2013-03-22 18:26:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Corollary

Classification

msc 51M05

Related topic

Incenter

Related topic

AngleBisectorAsLocus

Related topic

LengthsOfMedians

Generated on Fri Feb 9 22:05:13 2018 by

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parallel postulate

parallel postulate

The

parallel postulate

is Euclid’s fifth postulate:

equivalent

to the idea that there is a unique

parallel

to any line through a point not on the line.

Title

parallel postulate

Canonical name

ParallelPostulate

Date of creation

2013-03-22 13:54:49

Last modified on

2013-03-22 13:54:49

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

5

Author

vmoraru (1243)

Entry type

Axiom

Classification

msc 51M05

Related topic

CompassAndStraightedgeConstructionOfParallelLine

Related topic

ParallellismInEuclideanPlane

Generated on Fri Feb 9 22:05:15 2018 by

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parts of a ball

parts of a ball

Let us consider in

ℝ

3

a ball of radius

r

and the sphere bounding the ball.

•

Two

parallel planes

intersecting the ball separate between them from the ball a

spherical segment

, which can also be called a

spherical frustum

(see the

frustum

).  The curved surface of the spherical segment is the

spherical zone

.

•

In the special case that one of the planes is a

tangent plane

of the sphere, the spherical segment is a

spherical cap

and the spherical zone is a

spherical calotte

.

•

The

lateral surface

of a

circular cone

with its apex in the centre (

http://planetmath.org/Sphere

) of the ball divides the ball into two

spherical sectors

.

The

distance

h

of the two planes intersecting the ball be is called the

height

.

The volume of the spherical cap is obtained from

V

=

π

⁢

h

2

⁢

(

r

-

h

3

)

and the area of the corresponding spherical calotte and also a spherical zone from

A

=

2

⁢

π

⁢

r

⁢

h

.

The volume of a spherical segment can be got as the

difference

of the volumes of two spherical caps.

The volume of a spherical sector may be calculated from

V

=

2

3

⁢

π

⁢

r

2

⁢

h

,

where

h

is the height of the spherical cap of the spherical sector.

Title

parts of a ball

Canonical

name

PartsOfABall

Date of creation

2013-03-22 18:18:36

Last modified on

2013-03-22 18:18:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 51M05

Synonym

parts of ball

Synonym

parts of sphere

Related topic

CircularSegment

Defines

spherical segment

Defines

spherical frustum

Defines

spherical cap

Defines

spherical calotte

Defines

spherical sector

Generated on Fri Feb 9 22:05:17 2018 by

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SNCF metric

SNCF metric

The following two examples of a

metric space

(one of which is a

real
tree

) obtained their name from the of the French railway
system. Especially

malicious

rumour has it that if you want to
go by train from

x

to

y

in France, the most efficient solution is
to reduce the problem to going from

x

to Paris and then from Paris
to

y

.

Since their discovery, the intrinsic laws of the French way of going
by train have made it around the world and reached the late-afternoon
tutorials of first-term mathematics courses in an effort to lighten the
moods in the guise of the following definition:

Definition 1

(SNCF metric).

Let

P

be a point in a metric space

(

F

,

d

)

. Then the

SNCF
metric

d

P

with respect to

P

is defined by

d

P

⁢

(

x

,

y

)

:=

{

0

if

⁢

x

=

y

d

⁢

(

x

,

P

)

+

d

⁢

(

P

,

y

)

otherwise.

It is easy to see that

d

P

is a metric.

Now, what if the train from

x

to Paris stops over in

y

during the
ride (or the other way round)? Sure, Paris is a beautiful city, but
you wouldn’t

always

want to go there and back again. To
implement this, the geometric notion of “

y

lies on the straight
line defined by

x

and

P

” is required, so the definition becomes
more specialised:

Definition 2

(SNCF metric, enhanced version).

Let

P

be the origin in the space

ℝ

n

with

Euclidean norm

∥

⋅

∥

2

. Then the

SNCF metric

d

P

is defined by

d

P

⁢

(

x

,

y

)

:=

{

∥

x

-

y

∥

2

if

⁢

x

⁢

and

⁢

y

⁢

lie on the same ray from the origin

∥

x

∥

2

+

∥

y

∥

2

otherwise

.

The metric space

(

ℝ

n

,

d

P

)

is, in

addition

, a real tree since
if

x

and

y

do not lie on the same

http://planetmath.org/node/6962

ray from

P

, the only arc in

(

ℝ

n

,

d

P

)

joining

x

and

y

consists of the two ray

http://planetmath.org/node/5783

segments

x

⁢

P

and

y

⁢

P

. Other

injections

which are arcs in

Euclidean

ℝ

n

do not remain

continuous

in

(

ℝ

n

,

d

P

)

.

Title

SNCF metric

Canonical name

SNCFMetric

Date of creation

2013-03-22 15:17:25

Last modified on

2013-03-22 15:17:25

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

5

Author

GrafZahl (9234)

Entry type

Example

Classification

msc 51M05

Classification

msc 97A20

Classification

msc 54E35

Related topic

RealTree

Generated on Fri Feb 9 22:05:21 2018 by

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spherical coordinates

spherical coordinates

Spherical coordinates

are a system of

coordinates

for

ℝ

3

,
or more generally

ℝ

n

.
One coordinate is the

distance

from the origin,
which can be thought of as
the radius of the sphere centred at the origin on which the point lies.
The other coordinates are angles that specify the position of the point on this sphere.

In

ℝ

3

the coordinates are given by

(

x

y

z

)

=

(

r

⁢

sin

⁡

ϕ

⁢

cos

⁡

θ

r

⁢

sin

⁡

ϕ

⁢

sin

⁡

θ

r

⁢

cos

⁡

ϕ

)

,

where

r

is the distance from the origin,

θ

is the

azimuthal angle

defined for

θ

∈

[

0

,

2

⁢

π

)

,
and

ϕ

∈

[

0

,

π

]

is the

polar angle

.
Note that

ϕ

=

0

corresponds to the top of the sphere and

ϕ

=

π

corresponds to the bottom of the sphere.
There is a clash between the mathematicians’ and the physicists’ definition of spherical coordinates, interchanging both the direction of

ϕ

and the choice of names for the two angles (physicists often use

θ

as the azimuthal angle and

ϕ

as the polar one).

Spherical coordinates are a

generalization

of

polar coordinates

,
and can be further generalized to

ℝ

n

,
with

n

-

2

polar angles

ϕ

1

,

…

,

ϕ

n

-

2

and one azimuthal angle

θ

:

(

x

1

x

2

⋮

x

k

⋮

x

n

-

1

x

n

)

=

(

r

⁢

cos

⁡

ϕ

1

r

⁢

sin

⁡

ϕ

1

⁢

cos

⁡

ϕ

2

⋮

r

⁢

(

∏

i

=

1

k

-

1

sin

⁡

ϕ

i

)

⁢

cos

⁡

ϕ

k

⋮

r

⁢

sin

⁡

ϕ

1

⁢

sin

⁡

ϕ

2

⁢

⋯

⁢

cos

⁡

θ

r

⁢

sin

⁡

ϕ

1

⁢

sin

⁡

ϕ

2

⁢

⋯

⁢

sin

⁡

ϕ

n

-

2

⁢

sin

⁡

θ

.

)

.

These are sometimes called

hyperspherical coordinates

if

n

>

3

.

Title

spherical coordinates

Canonical name

SphericalCoordinates

Date of creation

2013-05-08 10:16:57

Last modified on

2013-05-08 10:16:57

Owner

yark (2760)

Last modified by

unlord (1)

Numerical id

17

Author

yark (1)

Entry type

Definition

Classification

msc 51M05

Related topic

Sphere

Related topic

CylindricalCoordinates

Defines

hyperspherical coordinates

Defines

azimuthal angle

Defines

polar angle

Generated on Fri Feb 9 22:05:34 2018 by

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