aarjaneiro/planetmath huggingface dataset:
spheroid

spheroid

A

spheroid

or

ellipsoid of revolution

is an

ellipsoid

such that two of the axes are equal in length. The name ellipsoid of revolution refers to the fact that such a shape is rotationally

symmetric about

the third axis. By way of contrast, the term

triaxial ellipsoid

is used to specify that the three axes are of different lengths. Also, the term

oblate

is used to describe spheriods in which the two axes of equal length are longer than the third (think of a doorknob) whilst the term

prolate

is used to describe spheriods for which the two equally long axes are shorter than the third axis (think of a cigar).

Title

spheroid

Canonical name

Spheroid

Date of creation

2013-03-22 14:56:48

Last modified on

2013-03-22 14:56:48

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 51M05

Synonym

ellipsoid of revolution

Defines

triaxial

Defines

oblate

Defines

prolate

Generated on Fri Feb 9 22:05:37 2018 by

LaTeXML
sum of angles of triangle in Euclidean geometry

sum of angles of triangle in Euclidean geometry

The

parallel postulate

(in the form given in the parent entry (

http://planetmath.org/ParallelPostulate

)) allows to prove the important fact about the

triangles

in the

Euclidean geometry

:

Theorem

.

The sum of the

interior angles

of any triangle equals the

straight angle

.

Proof.

Let

A

⁢

B

⁢

C

be an arbitrary triangle with the interior angles

α

,

β

,

γ

.  In the plane of the triangle we set the lines

A

⁢

D

and

A

⁢

E

such that

∧

B

⁢

A

⁢

D

=

β

and

∧

C

⁢

A

⁢

E

=

γ

.  Then the lines do not

intersect

the line

B

⁢

C

.  In fact, if e.g.

A

⁢

D

would intersect

B

⁢

C

in a point

P

, then there would exist a triangle

A

⁢

B

⁢

P

where an

exterior angle

(

http://planetmath.org/ExteriorAnglesOfTriangle

) of an angle would equal to an interior angle of another angle which is impossible.  Thus

A

⁢

D

and

A

⁢

E

are both

parallel

to

B

⁢

C

.  By the parallel postulate, these lines have to coincide.  This means that the

addition

of the triangle angles

α

,

β

,

γ

gives a straight angle.

See also

http://www.cs.bham.ac.uk/research/

projects

/cogaff/misc/triangle-sum.html#pardoe-proof

this intuitive proof!

References

1

Karl Ariva

:

Lobatsevski geomeetria

.  Kirjastus “Valgus”, Tallinn (1992).

Title

sum of angles of triangle in Euclidean geometry

Canonical name

SumOfAnglesOfTriangleInEuclideanGeometry

Date of creation

2013-09-26 10:00:16

Last modified on

2013-09-26 10:00:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51M05

Generated on Fri Feb 9 22:05:39 2018 by

LaTeXML
volume of the n-sphere

volume of the

n

-sphere

The volume

contained

inside

S

n

, the

n

-sphere (or

hypersphere

), is
given by the

integral

V

⁢

(

n

)

=

∫

∑

i

=

1

n

+

1

x

i

2

≤

1

d

n

+

1

⁢

x

.

Going to

polar coordinates

(

r

2

=

∑

i

=

1

n

+

1

x

i

2

) this becomes

V

⁢

(

n

)

=

∫

S

n

𝑑

Ω

⁢

∫

0

1

r

n

⁢

𝑑

r

.

The first integral is the integral over all

solid angles

subtended by the
sphere and is equal to its area

A

⁢

(

n

)

=

2

⁢

π

n

+

1

2

Γ

⁢

(

n

+

1

2

)

,
where

Γ

⁢

(

x

)

is the

gamma function

.
The second integral is

elementary

and evaluates to

∫

0

1

r

n

⁢

𝑑

r

=

1

/

(

n

+

1

)

.

Finally, the volume is

V

⁢

(

n

)

=

π

n

+

1

2

n

+

1

2

⁢

Γ

⁢

(

n

+

1

2

)

=

π

n

+

1

2

Γ

⁢

(

n

+

3

2

)

.

If the sphere has radius

R

instead of

1

, then the correct volume is

V

⁢

(

n

)

⁢

R

n

+

1

.

Note that this

formula

works for

n

≥

0

. The first few cases are

n

=

0

Γ

⁢

(

3

/

2

)

=

π

/

2

, hence

V

⁢

(

0

)

=

2

(this is the
length of the

interval

[

-

1

,

1

]

in

ℝ

);

n

=

1

Γ

⁢

(

2

)

=

1

, hence

V

⁢

(

1

)

=

π

(this is the familiar
result for the area of the

unit circle

);

n

=

2

Γ

⁢

(

5

/

2

)

=

3

⁢

π

/

4

, hence

V

⁢

(

2

)

=

4

⁢

π

/

3

(this is the familiar result for the volume of the sphere);

n

=

3

Γ

⁢

(

3

)

=

2

, hence

V

⁢

(

3

)

=

π

2

/

2

.

Title

volume of the

n

-sphere

Canonical name

VolumeOfTheNsphere

Date of creation

2013-03-22 13:47:09

Last modified on

2013-03-22 13:47:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 51M05

Related topic

AreaOfTheNSphere

Generated on Fri Feb 9 22:05:42 2018 by

LaTeXML
AAA

AAA

In both

hyperbolic geometry

and

spherical geometry

, the

AAA theorem

holds:

Theorem 1

.

If two

triangles

have all three pairs of corresponding angles

congruent

, then the triangles are congruent.

Because of this theorem, we have that, in hyperbolic geometry and spherical geometry,

similar triangles

are congruent.

Title

AAA

Canonical name

AAA

Date of creation

2013-03-22 17:08:40

Last modified on

2013-03-22 17:08:40

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 51M10

Synonym

AAA theorem

Generated on Fri Feb 9 22:05:44 2018 by

LaTeXML
AAS is not valid in spherical geometry

AAS is not valid in spherical geometry

AAS (

http://planetmath.org/AAS

) is not valid in

spherical geometry

(

http://planetmath.org/SphericalGeometry

). This fact can be determined as follows:

Let

ℓ

be a line on a sphere and

P

be one of the two points that is furthest from

ℓ

on the sphere. (It may be beneficial to think of

ℓ

as the equator and

P

as the .) Let

A

,

B

,

C

∈

ℓ

such that

•

A

,

B

, and

C

are distinct;

•

the length of

A

⁢

B

¯

is strictly less than the length of

A

⁢

C

¯

;

•

A

,

B

, and

P

are not

collinear

;

•

A

,

C

, and

P

are not collinear;

•

B

,

C

, and

P

are not collinear.

Connect

P

to each of the three points

A

,

B

, and

C

with

line segments

. (It may be beneficial to think of these line segments as

longitudes

.)

Since

ℓ

is also a circle having

P

as one of its centers (

http://planetmath.org/Center8

) with radii

A

⁢

P

¯

,

B

⁢

P

¯

, and

C

⁢

P

¯

, we have that

A

⁢

P

¯

≅

B

⁢

P

¯

≅

C

⁢

P

¯

and that

ℓ

is

perpendicular

to each of these line segments. Thus, the

triangles

△

⁢

A

⁢

B

⁢

P

and

△

⁢

A

⁢

C

⁢

P

have two pairs of angles

congruent

and a pair of sides congruent that is not between the congruent angles (actually, two pairs of sides congruent, neither of which is in between the congruent angles). On the other hand,

△

⁢

A

⁢

B

⁢

P

≇

△

⁢

A

⁢

C

⁢

P

because the length of

A

⁢

B

¯

is strictly less than the length of

A

⁢

C

¯

.

Title

AAS is not valid in spherical geometry

Canonical name

AASIsNotValidInSphericalGeometry

Date of creation

2013-03-22 17:13:00

Last modified on

2013-03-22 17:13:00

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Result

Classification

msc 51M10

Synonym

SAA is not valid in spherical geometry

Generated on Fri Feb 9 22:05:48 2018 by

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Beltrami-Klein model

Beltrami-Klein model

The

Beltrami-Klein model

for

ℍ

2

is the disc

{

(

x

,

y

)

∈

ℝ

2

:

x

2

+

y

2

<

1

}

in which a point is

similar

to the

Euclidean

point and a line is defined to be a chord (excluding its

endpoints

) of the (

circular

) boundary.

The Beltrami-Klein model has the advantage that lines in the model resemble Euclidean lines; however, it has the drawback that it is not angle preserving. That is, the Euclidean of an angle within the model is not necessarily the

angle measure

in

hyperbolic geometry

.

Some points outside of the Beltrami-Klein model are important for constructions within the model. The following is an example of such:

Let

ℓ

be a line in the Beltrami-Klein model that is not a

diameter

of the circle. The

pole

of

ℓ

is the intersection of the Euclidean lines that are

tangent

(

http://planetmath.org/TangentLine

) to the circle at the endpoints of

ℓ

.

Poles are important for the following reason: Given a line

ℓ

that is not a diameter of the Beltrami-Klein model, one constructs a line

perpendicular

to

ℓ

by considering Euclidean lines

passing through

P

⁢

(

ℓ

)

. Thus, given two

disjointly parallel lines

ℓ

and

m

that are not diameters of the Beltrami-Klein model, one constructs their common perpendicular by connecting their poles.

In the above picture,

n

is the common perpendicular of

ℓ

and

m

.

Title

Beltrami-Klein model

Canonical name

BeltramiKleinModel

Date of creation

2013-03-22 17:06:37

Last modified on

2013-03-22 17:06:37

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

18

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M10

Classification

msc 51-00

Synonym

Klein-Beltrami model

Synonym

Klein model

Related topic

ConvertingBetweenTheBeltramiKleinModelAndThePoincareDiscModel

Defines

pole

Generated on Fri Feb 9 22:05:50 2018 by

LaTeXML
biangle

biangle

In

spherical geometry

, it is possible to form a

polygon

with only two sides. Thus, we have the following definition:

A

biangle

is a two-sided polygon that is strictly contained in one

hemisphere

of the sphere that is serving as the model for spherical geometry.

Given a biangle, its vertices must be

antipodal points

, and its two angles must be

congruent

. Therefore, every biangle is

equiangular

. Since each side of a biangle is half of a

great circle

, every biangle is

equilateral

. Hence, every biangle is

regular

.

Let

θ

be the radian measure (

http://planetmath.org/AngleMeasure

) of each angle of a biangle. Then the biangle covers (

http://planetmath.org/Cover

)

θ

2

⁢

π

of the sphere. Since the area of the sphere is

4

⁢

π

, the area of the biangle is

2

⁢

θ

. Note that this equals the

angle sum

of the biangle.

Title

biangle

Canonical name

Biangle

Date of creation

2013-03-22 17:06:10

Last modified on

2013-03-22 17:06:10

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M10

Classification

msc 51-00

Generated on Fri Feb 9 22:05:53 2018 by

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comparison of common geometries

comparison of common geometries

In this entry, the most common models of the three most common two-dimensional

geometries

(Euclidean (

http://planetmath.org/EuclideanGeometry

), hyperbolic (

http://planetmath.org/HyperbolicGeometry

), and spherical (

http://planetmath.org/SphericalGeometry

)) will be considered.

The following abbreviations will be used in this entry:

•

E

2

for the

Euclidean plane

(the most common model for two-dimensional

Euclidean geometry

);

•

ℍ

2

for two-dimensional

hyperbolic geometry

;

•

B

⁢

K

for the

Beltrami-Klein model

of

ℍ

2

;

•

P

⁢

D

for the Poincaré disc model of

ℍ

2

;

•

U

⁢

H

⁢

P

for the

upper half plane model

of

ℍ

2

;

•

S

2

for the

unit sphere

(the most common model for two-dimensional

spherical geometry

).

1

Comparison of Properties of the Models

property

E

2

B

⁢

K

P

⁢

D

U

⁢

H

⁢

P

S

2

model has area when

no

yes

yes

no

yes

considered as a subset of a

Euclidean space

lines in model look like

lines

line segments

some line segments,

some vertical rays,

circles

some arcs of circles

some

semicircles

lines have length when

no

yes

yes

yes for semicircles,

yes

considered as a subset of a

no for vertical rays

Euclidean space

angles are preserved in

yes

no

yes

yes

yes

model

2

Comparison of Properties of the Geometries

property

E

2

ℍ

2

S

2

two distinct points determine a unique line

yes

yes

no

(yes if points are not

antipodal

)

parallel lines

exist

yes

yes

no

number of lines

parallel

to a given line and

1

∞

0

passing through

a point not on the given line

space has infinite area with respect

yes

yes

no

to its own geometry

lines have infinite length

yes

yes

no

number of centers (

http://planetmath.org/Center8

) of a circle

1

1

2

angle sum

Σ

of

triangles

(in radians)

Σ

=

π

0

<

Σ

<

π

π

<

Σ

<

3

⁢

π

ASA holds

yes

yes

yes

SAS holds

yes

yes

yes

SSS holds

yes

yes

yes

AAS holds

yes

yes

no (

http://planetmath.org/AASIsNotValidInSphericalGeometry

)

AAA holds

no

yes

yes

Title

comparison of common geometries

Canonical

name

ComparisonOfCommonGeometries

Date of creation

2013-03-22 17:13:06

Last modified on

2013-03-22 17:13:06

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

17

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 51M10

Classification

msc 51M05

Classification

msc 51-01

Classification

msc 51-00

Related topic

EuclideanGeometry

Related topic

NonEuclideanGeometry

Related topic

Geometry

Generated on Fri Feb 9 22:06:01 2018 by

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converting between the Poincaré disc model and the upper half plane model

converting between the Poincaré disc model and the upper half plane model

If both the Poincaré disc model and the

upper half plane model

are considered as subsets of

ℂ

rather than as subsets of

ℝ

2

(that is, the Poincaré disc model is

{

z

∈

ℂ

:

|

z

|

<

1

}

and the upper half plane model is

{

z

∈

ℂ

:

Im

⁡

(

z

)

>

0

}

), then one can use Möbius

transformations

to convert between the two models. The entry

unit disk upper half plane conformal equivalence theorem

yields that

f

:

ℂ

∪

{

∞

}

→

ℂ

∪

{

∞

}

defined by

f

⁢

(

z

)

=

z

-

i

z

+

i

maps the upper half plane model to the Poincaré disc model, and thus its

inverse

,

f

-

1

:

ℂ

∪

{

∞

}

→

ℂ

∪

{

∞

}

defined by

f

-

1

⁢

(

z

)

=

-

i

⁢

z

-

i

z

-

1

, maps the Poincaré disc model to the upper half plane model.

Note that the Möbius transformation

f

-

1

gives another justification of including

∞

in the boundary of the upper half plane model (see the entry on

parallel lines in hyperbolic geometry

for more details):

1

(or the

ordered pair

(

1

,

0

)

) is on the boundary of the Poincaré disc model and

f

-

1

⁢

(

1

)

=

∞

.

Note also that lines in the Poincaré disc model

passing through

1

(or the ordered pair

(

1

,

0

)

) are in

one-to-one correspondence

with the lines that are vertical rays in the upper half plane model.

Title

converting between the Poincaré disc model and the upper half plane model

Canonical name

ConvertingBetweenThePoincareDiscModelAndTheUpperHalfPlaneModel

Date of creation

2013-03-22 17:07:43

Last modified on

2013-03-22 17:07:43

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 51M10

Classification

msc 51-00

Related topic

PoincareDiscModel

Related topic

UpperHalfPlaneModel

Related topic

UnitDiskUpperHalfPlaneConformalEquivalenceTheorem

Related topic

PoincareUpperHalfPlaneModel

Related topic

UpperHalfPlane

Generated on Fri Feb 9 22:06:04 2018 by

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defect

defect

Consider a

triangle

△

⁢

A

⁢

B

⁢

C

in either hyperbolic or

spherical geometry

(

http://planetmath.org/NonEuclideanGeometry

) in which its

angle sum

in radians is

Σ

.

In hyperbolic geometry, the

defect

of

△

⁢

A

⁢

B

⁢

C

is

δ

⁢

(

△

⁢

A

⁢

B

⁢

C

)

=

π

-

Σ

.

In spherical geometry, the

defect

of

△

⁢

A

⁢

B

⁢

C

is

δ

⁢

(

△

⁢

A

⁢

B

⁢

C

)

=

Σ

-

π

.

Note that, in both hyperbolic and spherical geometry, the area of a is equal to its defect.

Title

defect

Canonical

name

Defect

Date of creation

2013-03-22 16:05:22

Last modified on

2013-03-22 16:05:22

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M10

Classification

msc 51-00

Related topic

AreaOfASphericalTriangle

Generated on Fri Feb 9 22:06:06 2018 by

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ideal triangle

ideal triangle

In

hyperbolic geometry

, an

ideal triangle

is a set of three lines which connect three distinct points on the boundary of the model of hyperbolic geometry.

Below is an example of an ideal triangle in the

Beltrami-Klein model

:

Below is an example of an ideal triangle in the Poincaré disc model:

Below are some examples of ideal triangles in the

upper half plane model

:

speaking, none of these figures are

triangles

in hyperbolic geometry; however, ideal triangles are useful for proving that, given

r

∈

ℝ

with

0

<

r

<

π

, there is a triangle in hyperbolic geometry whose

angle sum

in radians is equal to

r

.

Title

ideal triangle

Canonical name

IdealTriangle

Date of creation

2013-03-22 17:08:26

Last modified on

2013-03-22 17:08:26

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M10

Classification

msc 51-00

Related topic

LimitingTriangle

Generated on Fri Feb 9 22:06:08 2018 by

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Lambert quadrilateral

Lambert quadrilateral

In

hyperbolic geometry

, a

Lambert quadrilateral

is a

quadrilateral

with exactly three

right angles

. Since the

angle sum

of a

triangle

in hyperbolic geometry is

strictly

less than

π

radians, the angle sum of a quadrilateral in hyperbolic geometry is strictly less than

2

⁢

π

radians. Thus, in any Lambert quadrilateral, the angle that is not a right angle must be acute.

The discovery of Lambert quadrilaterals is attributed to Johann Lambert.

Both pairs of opposite sides of a Lambert quadrilateral are

disjointly parallel

since, in both cases, they have a common

perpendicular

. Therefore, Lambert quadrilaterals are

parallelograms

. Note also that Lambert quadrilaterals are

right trapezoids

.

Below are some examples of Lambert quadrilaterals in various models. In each example, the Lambert quadrilateral is labelled as

A

⁢

B

⁢

C

⁢

D

.

•

The

Beltrami-Klein model

:

In each of these examples, blue lines indicate verification of right angles by using the poles, and green lines indicate verification of

acute angles

by using the poles. (Recall that most other models of hyperbolic geometry are angle preserving. Thus, verification of angle measures is unnecessary in those models.)

•

The Poincaré disc model:

•

The

upper half plane model

:

Title

Lambert quadrilateral

Canonical

name

LambertQuadrilateral

Date of creation

2013-03-22 17:08:04

Last modified on

2013-03-22 17:08:04

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

24

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M10

Classification

msc 51-00

Synonym

Lambert’s quadrilateral

Related topic

RightTrapezoid

Generated on Fri Feb 9 22:06:12 2018 by

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limiting triangle

limiting triangle

In

spherical geometry

, a

limiting triangle

is a

great circle

of the sphere that is serving as the model for the

geometry

.

The motivation for this definition is as follows: In

Euclidean geometry

and hyperbolic geometry, if three

collinear points

are

connected

, the result is

always

a

line segment

, which does not contain any area. In spherical geometry, if the three points are close to each other, this procedure will produce a great arc (the

equivalent

to a line segment in this geometry). On the other hand, if the three points are sufficiently spaced from each other, this procedure will yield an entire great circle (the equivalent to a line in this geometry). For example, imagine that the circle shown below is a great circle of a sphere. Then connecting the three plotted points yields the entire great circle.

Thus, limiting triangles are

geodesic triangles

determined by three collinear points that are sufficiently spaced from each other.

speaking, the resulting figure is not a

triangle

in spherical geometry; however, it is useful for demonstrating the following facts in spherical geometry:

•

540

∘

is the least

upper bound

of the

angle sum

of a triangle;

•

2

⁢

π

is the least upper bound of the area of a .

Title

limiting triangle

Canonical name

LimitingTriangle

Date of creation

2013-03-22 17:06:35

Last modified on

2013-03-22 17:06:35

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M10

Related topic

AreaOfASphericalTriangle

Related topic

IdealTriangle

Generated on Fri Feb 9 22:06:15 2018 by

LaTeXML
Lobachevsky’s formula

Lobachevsky’s formula

Let

A

⁢

B

be a line. Let

M

,

T

be two points so that

M

not lies on

A

⁢

B

,

T

lies on

A

⁢

B

, and

M

⁢

T

perpendicular

to

A

⁢

B

. Let

M

⁢

D

be any other line who meets

A

⁢

T

in

D

.In a

hyperbolic geometry

, as

D

moves off to infinity
along

A

⁢

T

the line

M

⁢

D

meets the line

M

⁢

S

which is said to be

parallel

to

A

⁢

T

. The angle

S

⁢

M

⁢

T

^

is called the

angle of parallelism

for perpendicular distance

d

, and is
given by

P

⁢

(

d

)

=

2

⁢

tan

-

1

⁡

(

e

-

d

)

,

which is called

Lobachevsky’s formula.

Title

Lobachevsky’s formula

Canonical name

LobachevskysFormula

Date of creation

2013-03-22 14:05:53

Last modified on

2013-03-22 14:05:53

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

6

Author

vmoraru (1243)

Entry type

Definition

Classification

msc 51M10

Defines

angle of parallelism

Generated on Fri Feb 9 22:06:17 2018 by

LaTeXML