aarjaneiro/planetmath huggingface dataset:
Steiner’s theorem

Steiner’s theorem

Let

A

⁢

B

⁢

C

be a triangle and

M

,

N

∈

(

B

⁢

C

)

be two points such that

m

⁢

(

∠

⁢

B

⁢

A

⁢

M

)

=

m

⁢

(

∠

⁢

N

⁢

A

⁢

C

)

. Then the

cevians

A

⁢

M

and

A

⁢

N

are
called isogonic cevians and the following relation holds:

B

⁢

M

M

⁢

C

⋅

B

⁢

N

N

⁢

C

=

A

⁢

B

2

A

⁢

C

2

Title

Steiner’s theorem

Canonical name

SteinersTheorem

Date of creation

2013-03-22 13:21:41

Last modified on

2013-03-22 13:21:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 51N20

Generated on Fri Feb 9 22:14:18 2018 by

LaTeXML
supplementary chords

supplementary chords

Two chords of

ellipse

, connecting a point of the perimetre to the end-points of a

diameter

(

http://planetmath.org/ConjugateDiametersOfEllipse

), are called

supplementary chords

of each other.

A pair of supplementary chords are

parallel

to a pair of

conjugate diameters

.

Title

supplementary chords

Canonical

name

SupplementaryChords

Date of creation

2013-03-22 17:31:01

Last modified on

2013-03-22 17:31:01

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N20

Generated on Fri Feb 9 22:14:20 2018 by

LaTeXML
tangent of conic section

tangent of conic section

The equation of every

conic section

(and the degenerate cases) in the rectangular

(

x

,

y

)

-

coordinate system

may be written in the form

A

⁢

x

2

+

B

⁢

y

2

+

2

⁢

C

⁢

x

⁢

y

+

2

⁢

D

⁢

x

+

2

⁢

E

⁢

y

+

F

=

0

,

where

A

,

B

,

C

,

D

,

E

and

F

are constants and

A

2

+

B

2

+

C

2

>

0

.

1

1

This is true also in any

skew-angled coordinate

system.

(The

2

⁢

C

⁢

x

⁢

y

is present only if the axes are not

parallel

to the

coordinate

axes.)

The equation of the

tangent line

of an ordinary conic section (i.e., circle,

ellipse

,

hyperbola

and

parabola

) in the point

(

x

0

,

y

0

)

of the curve is

A

⁢

x

0

⁢

x

+

B

⁢

y

0

⁢

y

+

C

⁢

(

y

0

⁢

x

+

x

0

⁢

y

)

+

D

⁢

(

x

+

x

0

)

+

E

⁢

(

y

+

y

0

)

+

F

=

0

.

Thus, the equation of the tangent line can be obtained from the equation of the curve by

polarizing

it, i.e. by replacing

x

2

with

x

0

⁢

x

,

y

2

with

y

0

⁢

y

,

2

⁢

x

⁢

y

with

y

0

⁢

x

+

x

0

⁢

y

,

2

⁢

x

with

x

+

x

0

,

2

⁢

y

with

y

+

y

0

.

Examples:

The of the ellipse

x

2

a

2

+

y

2

b

2

=

1

is

x

0

⁢

x

a

2

+

y

0

⁢

y

b

2

=

1

, the of the hyperbola

x

⁢

y

=

1

2

is

y

0

⁢

x

+

x

0

⁢

y

=

1

.

Title

tangent of conic section

Canonical name

TangentOfConicSection

Date of creation

2013-03-22 14:28:40

Last modified on

2013-03-22 14:28:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N20

Synonym

tangent of quadratic curve

Related topic

TangentLine

Related topic

TangentOfCircle

Related topic

TangentPlaneOfQuadraticSurface

Related topic

QuadraticInequality

Related topic

ConjugateDiametersOfEllipse

Related topic

ConjugateHyperbola

Related topic

QuadraticCurves

Related topic

EquationOfTangentOfCircle

Related topic

TangentOfHyperbola

Defines

polarising

Defines

polarizing

Defines

polarize

Defines

mixed term

Generated on Fri Feb 9 22:14:22 2018 by

LaTeXML
tangent plane of quadratic surface

tangent plane of quadratic surface

The common equation of all

quadratic surfaces

in the rectangular

(

x

,

y

,

z

)

-

coordinate system

is

A

⁢

x

2

+

B

⁢

y

2

+

C

⁢

z

2

+

2

⁢

A

′

⁢

y

⁢

z

+

2

⁢

B

′

⁢

z

⁢

x

+

2

⁢

C

′

⁢

x

⁢

y

+

2

⁢

A

′′

⁢

x

+

2

⁢

B

′′

⁢

y

+

2

⁢

C

′′

⁢

z

+

D

=

0

(1)

where

A

,

B

,

C

,

A

′

,

B

′

,

C

′

,

A

′′

,

B

′′

,

C

′′

,

D

are constants and at least one of the six first is distinct from zero.  The equation of the

tangent plane

of the surface, with

(

x

0

,

y

0

,

z

0

)

as the point of tangency, is

A

⁢

x

0

⁢

x

+

B

⁢

y

0

⁢

y

+

C

⁢

z

0

⁢

z

+

A

′

⁢

(

z

0

⁢

y

+

y

0

⁢

z

)

+

B

′

⁢

(

x

0

⁢

z

+

z

0

⁢

x

)

+

C

′

⁢

(

y

0

⁢

x

+

x

0

⁢

y

)

+

A

′′

⁢

(

x

+

x

0

)

+

B

′′

⁢

(

y

+

y

0

)

+

C

′′

⁢

(

z

+

z

0

)

+

D

=

0

.

This is said to be obtained from (1) by

polarizing

it.

Example.

The tangent plane of the

elliptic paraboloid

4

⁢

x

2

+

9

⁢

y

2

=

2

⁢

z

set in the point

(

x

0

,

y

0

,

z

0

)

of the surface is

4

⁢

x

0

⁢

x

+

9

⁢

y

0

⁢

y

=

z

+

z

0

,  and especially in the point

(

1

2

,

1

3

,

1

)

it is

2

⁢

x

+

3

⁢

y

-

z

-

1

=

0

.

Title

tangent plane of quadratic surface

Canonical name

TangentPlaneOfQuadraticSurface

Date of creation

2013-03-22 14:58:48

Last modified on

2013-03-22 14:58:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Result

Classification

msc 51N20

Related topic

TangentOfConicSection

Related topic

QuadraticSurfaces

Generated on Fri Feb 9 22:14:24 2018 by

LaTeXML
transition to skew-angled coordinates

transition to skew-angled coordinates

Let the

Euclidean plane

ℝ

be equipped with the

rectangular coordinate

system with the

x

and

y

coordinate

axes.  We choose new coordinate axes through the old origin and

project

(

http://planetmath.org/

Projection

) the new coordinates

ξ

,

η

of a point orthogonally on the

x

and

y

axes getting the old coordinates expressed as

{

x

=

ξ

⁢

cos

⁡

α

+

η

⁢

cos

⁡

β

,

y

=

ξ

⁢

sin

⁡

α

+

η

⁢

sin

⁡

β

,

(1)

where

α

and

β

are the angles which the

ξ

-axis and

η

-axis, respectively, form with the

x

-axis (positive if

x

-axis may be rotated anticlocwise to

ξ

-axis, else negative; similarly for rotating the

x

-axis to the

η

-axis).

The of (1) are got by solving from it for

ξ

and

η

, getting

ξ

=

x

⁢

sin

⁡

β

-

y

⁢

cos

⁡

β

sin

⁡

(

β

-

α

)

,

η

=

-

x

⁢

sin

⁡

α

+

y

⁢

cos

⁡

α

sin

⁡

(

β

-

α

)

.

Example.

Let us consider the

hyperbola

(

http://planetmath.org/Hyperbola2

)

x

2

a

2

-

y

2

b

2

=

1

(2)

and take its

asymptote

y

=

-

b

a

⁢

x

for the

ξ

-axis and the asymptote

y

=

+

b

a

⁢

c

for the

η

-axis.  If

ω

is the angle formed by the latter asymptote with the

x

-axis, then

α

=

-

ω

,

β

=

ω

.  By (1) we get first

{

x

=

ξ

⁢

cos

⁡

ω

+

η

⁢

cos

⁡

ω

=

(

η

+

ξ

)

⁢

cos

⁡

ω

,

y

=

-

ξ

⁢

sin

⁡

ω

+

η

⁢

sin

⁡

ω

=

(

η

-

ξ

)

⁢

sin

⁡

ω

.

Since

tan

⁡

ω

=

b

a

,  we see that

cos

⁡

ω

=

a

c

,

sin

⁡

ω

=

b

c

,  where

c

2

=

a

2

+

c

2

,
and accordingly

x

a

=

(

η

+

ξ

)

⁢

a

c

:

a

=

η

+

ξ

c

,

y

b

=

(

η

-

ξ

)

⁢

b

c

:

b

=

η

-

ξ

c

.

Substituting these quotients in the equation of the hyperbola we obtain

(

η

+

ξ

)

2

-

(

η

-

ξ

)

2

=

c

2

,

and after simplifying,

ξ

⁢

η

=

c

2

4

.

(3)

This is the equation of the hyperbola (2) in the

coordinate system

of its asymptotes.  Here,

c

is the

distance

of the focus (

http://planetmath.org/Hyperbola2

) from the nearer apex (

http://planetmath.org/Hyperbola2

) of the hyperbola.

If we, conversely, have in the rectangular coordinate system (

x

,

y

) an equation of the form (3), e.g.

x

⁢

y

=

constant

,

(4)

we can infer that it a hyperbola with asymptotes the coordinate axes. Since these are

perpendicular

to each other, it’s clear that the hyperbola (4) is a rectangular (

http://planetmath.org/Hyperbola2

) one.

References

1

L. Lindelöf

:

Analyyttisen geometrian oppikirja

.  Kolmas painos.  Suomalaisen Kirjallisuuden Seura, Helsinki (1924).

Title

transition to skew-angled coordinates

Canonical name

TransitionToSkewangledCoordinates

Date of creation

2013-03-22 17:09:39

Last modified on

2013-03-22 17:09:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N20

Related topic

RotationMatrix

Related topic

Hyperbola2

Related topic

ConjugateDiametersOfEllipse

Defines

skew-angled coordinate

Generated on Fri Feb 9 22:14:31 2018 by

LaTeXML
unit hyperbola

unit hyperbola

The

unit hyperbola

(cf. the

unit circle

) is the special case

x

2

-

y

2

=

1

of the

hyperbola

x

2

a

2

-

y

2

b

2

=

1

where both the

a

and the

b

have equal to 1.  The unit hyperbola is

rectangular

, i.e. its

asymptotes

(

y

=

±

x

) are at

right angles

to each other.

The unit hyperbola has the well-known parametric

x

=

±

cosh

⁡

t

,

y

=

sinh

⁡

t

,

and also a trigonometric

x

=

sec

⁡

t

,

y

=

tan

⁡

t

.

The former yields the rational

x

=

u

2

+

1

2

⁢

u

,

y

=

u

2

-

1

2

⁢

u

when one substitutes

e

t

=

u

, and the latter, via the

substitution

tan

⁡

t

2

=

u

, the rational

x

=

1

+

u

2

1

-

u

2

,

y

=

2

⁢

u

1

-

u

2

(which does not give the left apex of the hyperbola).

Title

unit hyperbola

Canonical name

UnitHyperbola

Date of creation

2015-02-04 11:10:22

Last modified on

2015-02-04 11:10:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N20

Related topic

HyperbolicFunctions

Related topic

AreaFunctions

Related topic

ConjugateHyperbola

Generated on Fri Feb 9 22:14:33 2018 by

LaTeXML
Van Aubel theorem

Van Aubel theorem

Let

A

⁢

B

⁢

C

be a

triangle

and

A

⁢

D

,

B

⁢

E

,

C

⁢

F

concurrent

cevians at

P

.

Then

C

⁢

P

P

⁢

F

=

C

⁢

D

D

⁢

B

+

C

⁢

E

E

⁢

A

Title

Van Aubel theorem

Canonical name

VanAubelTheorem

Date of creation

2013-03-22 14:03:26

Last modified on

2013-03-22 14:03:26

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Theorem

Classification

msc 51N20

Related topic

ProofOfVanAubelsTheorem

Related topic

CevasTheorem

Related topic

VanAubelsTheorem

Related topic

ProofOfVanAubelTheorem

Related topic

TrigonometricVersionOfCevasTheorem

Generated on Fri Feb 9 22:14:35 2018 by

LaTeXML
cross ratio

cross ratio

The

cross ratio

of the points

a

,

b

,

c

, and

d

in

ℂ

∪

{

∞

}

is denoted by

[

a

,

b

,

c

,

d

]

and is defined by

[

a

,

b

,

c

,

d

]

=

a

-

c

a

-

d

⋅

b

-

d

b

-

c

.

Some authors denote the cross ratio by

(

a

,

b

,

c

,

d

)

.

Examples

Example 1

.

The cross ratio of

1

,

i

,

-

1

, and

-

i

is

1

-

(

-

1

)

1

-

(

-

i

)

⋅

i

-

(

-

i

)

i

-

(

-

1

)

=

4

⁢

i

(

1

+

i

)

2

=

2

.

Example 2

.

The cross ratio of

1

,

2

⁢

i

,

3

, and

4

⁢

i

is

1

-

3

1

-

4

⁢

i

⋅

2

⁢

i

-

4

⁢

i

2

⁢

i

-

3

=

4

⁢

i

5

+

14

⁢

i

=

56

+

20

⁢

i

221

.

Properties

1.

The cross ratio is invariant under Möbius

transformations

and

projective transformations

. This fact can be used to determine

distances

between objects in a photograph when the distance between certain reference points is known.

2.

The cross ratio

[

a

,

b

,

c

,

d

]

is real if and only if

a

,

b

,

c

, and

d

lie on a single circle on the Riemann sphere.

3.

The

function

T

:

ℂ

∪

{

∞

}

→

ℂ

∪

{

∞

}

defined by

T

⁢

(

z

)

=

[

z

,

b

,

c

,

d

]

is the unique Möbius transformation which sends

b

to

1

,

c

to

0

, and

d

to

∞

.

References

1

Ahlfors, L.,

Complex Analysis

. McGraw-Hill, 1966.

2

Beardon, A. F.,

The

Geometry

of Discrete Groups

. Springer-Verlag, 1983.

3

Henle, M.,

Modern Geometries: Non-Euclidean, Projective, and Discrete

. Prentice-Hall, 1997 [2001].

Title

cross ratio

Canonical name

CrossRatio

Date of creation

2013-03-22 15:23:31

Last modified on

2013-03-22 15:23:31

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 51N25

Classification

msc 30C20

Classification

msc 30F40

Synonym

cross-ratio

Related topic

MobiusTransformationCrossRatioPreservationTheorem

Generated on Fri Feb 9 22:14:37 2018 by

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asymptote

asymptote

If a

plane curve

γ

has a continuing infinitely far from the origin

O

, then

γ

may have an

asymptote

:  The direct line

l

is an asymptote of

γ

, if

lim

d

⁢

(

P

,

O

)

→

∞

⁡

d

⁢

(

P

,

l

)

=

0

,

where

d

⁢

(

P

,

O

)

means the of the point

P

of the from the origin and

d

⁢

(

P

,

l

)

the of

P

from the line

l

.

Examples

:  The

hyperbola

x

2

a

2

-

y

2

b

2

=

1

has the asymptotes

y

=

±

b

a

⁢

x

;  the curve

y

=

sin

⁡

x

x

the asymptote

y

=

0

.

Title

asymptote

Canonical name

Asymptote

Date of creation

2013-03-22 14:32:59

Last modified on

2013-03-22 14:32:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N99

Related topic

SincFunction

Related topic

FamousCurvesInThePlane

Generated on Fri Feb 9 22:14:39 2018 by

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asymptote of an arctangent

asymptote of an arctangent

Given the

function

y

=

tan

-

1

⁡

(

x

)

,

we see that it has two asymptotes, namely

π

2

and

-

(

π

2

)

. This specific example of an asymptote was mentioned in an episode of the comedy TV show

The Big Bang Theory

, in which the mathematician Sheldon Eppes tries to explain his inability to reach the top of a rock wall, getting stuck at the same height each time. According to the

Mathematica

manual, the results of

ArcTan

for real

z

”are always in the range

-

π

/

2

to

π

/

2

.

References

1

Chuck Lorre,

http://chucklorre.com/index.php?p=237

Chuck Lorre

Productions

Vanity Card #237

Title

asymptote of an

arctangent

Canonical name

AsymptoteOfAnArctangent

Date of creation

2013-03-22 18:51:01

Last modified on

2013-03-22 18:51:01

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 51N99

Synonym

asymptote of an

inverse tangent

Generated on Fri Feb 9 22:14:41 2018 by

LaTeXML
asymptotes of graph of rational function

asymptotes of graph of rational function

Let

f

⁢

(

x

)

=

P

⁢

(

x

)

Q

⁢

(

x

)

be a

fractional expression

where

P

⁢

(

x

)

and

Q

⁢

(

x

)

are

polynomials

with real

coefficients

such that their

quotient

can not be reduced (

http://planetmath.org/Division

) to a polynomial.  We suppose that

P

⁢

(

x

)

and

Q

⁢

(

x

)

have no common zeros.

If the division of the polynomials is performed, then a result of the form

f

⁢

(

x

)

=

H

⁢

(

x

)

+

R

⁢

(

x

)

Q

⁢

(

x

)

is gotten, where

H

⁢

(

x

)

and

R

⁢

(

x

)

are polynomials such that

deg

⁡

R

⁢

(

x

)

<

deg

⁡

Q

⁢

(

x

)

The graph of the

rational function

f

may have

asymptotes

:

1.

Every zero

a

of the denominator

Q

⁢

(

x

)

gives a vertical asymptote

x

=

a

.

2.

If

deg

⁡

H

⁢

(

x

)

<

1

(i.e.

0

or

-

∞

) then the graph has the horizontal asymptote

y

=

H

⁢

(

x

)

.

3.

If

deg

⁡

H

⁢

(

x

)

=

1

then the graph has the skew asymptote

y

=

H

⁢

(

x

)

.

Proof of 2 and 3.

We have

f

⁢

(

x

)

-

H

⁢

(

x

)

=

R

⁢

(

x

)

Q

⁢

(

x

)

→

0

as

|

x

|

→

∞

.

Remark.

Here we use the convention that the degree of the

zero polynomial

is

-

∞

.

Title

asymptotes of graph of rational function

Canonical name

AsymptotesOfGraphOfRationalFunction

Date of creation

2013-03-22 15:09:34

Last modified on

2013-03-22 15:09:34

Owner

eshyvari (13396)

Last modified by

eshyvari (13396)

Numerical id

10

Author

eshyvari (13396)

Entry type

Result

Classification

msc 51N99

Classification

msc 26C15

Classification

msc 26A09

Related topic

Polytrope

Generated on Fri Feb 9 22:14:43 2018 by

LaTeXML
projection formula

projection formula

Theorem.

Let

a

,

b

,

c

be the sides of a

triangle

and

α

,

β

the angles opposing

a

,

b

, respectively.  Then one has

c

=

a

⁢

cos

⁡

β

+

b

⁢

cos

⁡

α

,

independently whether the angles are acute, right or obtuse.

Knowing the way to determine the length of the projection (

http://planetmath.org/ProjectionOfPoint

) of a

line segment

, the truth of the theorem is apparent; the below illustrate the cases where

β

is acute and obtuse (cosine of an

obtuse angle

is negative).

Note.

Especially, if neither of

α

and

β

is

right angle

, the formula of the theorem may be written

a

cos

⁡

α

+

b

cos

⁡

β

=

c

cos

⁡

α

⁢

cos

⁡

β

.

Title

projection formula

Canonical name

ProjectionFormula

Date of creation

2013-03-22 18:27:11

Last modified on

2013-03-22 18:27:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51N99

Synonym

projection formula for triangles

Related topic

BaseAndHeightOfTriangle

Generated on Fri Feb 9 22:14:53 2018 by

LaTeXML
projection of point

projection of point

Let a line

l

be given in a

Euclidean plane

or space. The (

orthogonal

)

projection

of a

P

onto the line

l

is the point

P

′

of

l

at which the

normal line

of

l

passing through

P

intersects

l

. One says that

P

has been (

orthogonally

)

projected

onto the line

l

.

The

projection of a set

S

of points onto the line

l

is defined to be the set of projection points of all points of

S

on

l

.

Especially, the

projection of a

P

⁢

Q

onto

l

is the

line segment

P

′

⁢

Q

′

determined by the projection points

P

′

and

Q

′

of

P

and

Q

. If the length of

P

⁢

Q

is

a

and the angle between the lines (

http://planetmath.org/AngleBetweenTwoLines

)

P

⁢

Q

and

l

is

α

, then the length

p

of its projection is

p

=

a

⁢

cos

⁡

α

.

(1)

Remark.

As one speaks of the projections onto a line

l

, one can speak in the Euclidean space also of

projections onto a plane

τ

.

Title

projection of point

Canonical

name

ProjectionOfPoint

Date of creation

2013-03-22 17:09:50

Last modified on

2013-03-22 17:09:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

21

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N99

Synonym

orthogonal projection

Related topic

Projection

Related topic

CompassAndStraightedgeConstructionOfPerpendicular

Related topic

MeusniersTheorem

Defines

project

Defines

projection of line segment

Generated on Fri Feb 9 22:14:56 2018 by

LaTeXML
projection of right angle

projection of right angle

Theorem.

The projection (

http://planetmath.org/ProjectionOfPoint

) of a

right angle

in

ℝ

3

onto a plane is a right angle if and only if at least one of its sides is

parallel

to the plane.

Proof.

Consider the projection of an angle

α

with vertex (

http://planetmath.org/Angle

)

P

onto the plane

π

.  Let

P

′

be the projection of

P

onto

π

.  If neither of the sides of

α

is parallel to

π

, then the lines of the sides intersect the plane in two distinct points

A

and

B

.  In order to that the

angle of view

of the segment

A

⁢

B

seen from the point

P

would be a right angle,

P

must be on a sphere with

diameter

A

⁢

B

centered at a point

O

.  In order to that the projection angle

A

⁢

P

′

⁢

B

would be a right angle, the point

P

′

must be on a circle of the plane

π

having

A

⁢

B

as diameter.  But

O

⁢

P

′

is as the projection of the segment

O

⁢

P

shorter than

O

⁢

P

.  It follows that the angle

A

⁢

P

′

⁢

B

is obtuse and hence cannot be right.

On the other hand, it’s not hard to see that the projection of a right angle is a right angle always when one or both of its sides are parallel to the projection plane.

References

1

E. J. Nyström

:

Korkeamman geometrian alkeet sovellutuksineen

.  Kustannusosakeyhtiö Otava, Helsinki (1948).

Title

projection of right angle

Canonical name

ProjectionOfRightAngle

Date of creation

2013-03-22 19:20:51

Last modified on

2013-03-22 19:20:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51N99

Classification

msc 51N20

Related topic

AngleBetweenLineAndPlane

Related topic

AngleOfView

Related topic

AngleOfViewOfALineSegment

Generated on Fri Feb 9 22:14:58 2018 by

LaTeXML
centre of mass of polygon

centre of mass of polygon

Let

A

1

⁢

A

2

⁢

…

⁢

A

n

be an

n

-gon (

http://planetmath.org/

Polygon

) which is supposed to have a surface-density in all of its points,

M

the

centre of mass

of the polygon and

O

the origin. Then the

position vector

of

M

with respect to

O

is

O

⁢

M

→

=

1

n

⁢

∑

i

=

1

n

O

⁢

A

i

→

.

(1)

We can of course take especially

O

=

A

1

,  and thus

A

1

⁢

M

→

=

1

n

⁢

∑

i

=

1

n

A

1

⁢

A

i

→

=

1

n

⁢

∑

i

=

2

n

A

1

⁢

A

i

→

.

In the special case of the

triangle

A

⁢

B

⁢

C

we have

A

⁢

M

→

=

1

3

⁢

(

A

⁢

B

→

+

A

⁢

C

→

)

.

(2)

The centre of mass of a triangle is the common point of its medians.

Remark.

An analogical result with (2) concerns also the

tetrahedron

A

⁢

B

⁢

C

⁢

D

,

A

⁢

M

→

=

1

4

⁢

(

A

⁢

B

→

+

A

⁢

C

→

+

A

⁢

D

→

)

,

and any

n

-dimensional

simplex

(cf. the

midpoint

(

http://planetmath.org/Midpoint

) of

line segment

:

A

⁢

M

→

=

1

2

⁢

A

⁢

B

→

).

Title

centre of mass of polygon

Canonical name

CentreOfMassOfPolygon

Date of creation

2013-03-22 17:33:13

Last modified on

2013-03-22 17:33:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Result

Classification

msc 51P05

Classification

msc 51M04

Classification

msc 26B15

Classification

msc 15A72

Synonym

centroid of polygon

Related topic

ArithmeticMean

Related topic

AreaOfPolygon

Related topic

CentreOfMassOfHalfDisc

Related topic

BarycentricSubdivision

Related topic

CoordinatesOfMidpoint

Generated on Fri Feb 9 22:15:03 2018 by

LaTeXML
convex analysis

convex analysis

0.0.1

Convex sets

1.

convex set

2.

polyconvex set

3.

extreme point

4.

convex combination

5.

CarathÃÂ©odory’s theorem (

http://planetmath.org/CaratheodorysTheorem2

)

6.

Radon’s lemma

7.

Helly’s theorem

0.0.2

Topological properties of convex sets

1.

Krein-Milman theorem

0.0.3

Convex functions

1.

convex function

2.

extremal value of convex/

concave functions

3.

logarithmically convex function

4.

Jensen’s

inequality

5.

subdifferentiable mapping

0.0.4

Miscellaneous

1.

Fenchel’s duality theorem

2.

saddle

function

3.

minimax theory

Title

convex analysis

Canonical name

ConvexAnalysis

Date of creation

2013-03-22 15:11:47

Last modified on

2013-03-22 15:11:47

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Topic

Classification

msc 52-00

Generated on Fri Feb 9 22:15:25 2018 by

LaTeXML