inversion of plane

inversion of plane

Let

c

be a fixed circle in the

Euclidean plane

with center

O

and radius

r

. Set for any point

P

≠

O

of the plane a corresponding point

P

′

, called the

inverse point

of

P

with respect to

c

, on the

closed ray

from

O

through

P

such that the

product

P

′

⁢

O

⋅

P

⁢

O

has the value

r

2

. This

mapping

P

↦

P

′

of the plane interchanges the inside and outside of the base circle

c

. The point

O

′

is the “infinitely distant point” of the plane.

The following is an illustration of how to obtain

P

′

for a given circle

c

and point

P

outside of

c

. The

restricted tangent

from

P

to

c

is drawn in blue, the

line segment

that determines

P

′

(

perpendicular

to

O

⁢

P

¯

, having an

endpoint

on

O

⁢

P

¯

, and having its other endpoint at the point of tangency

T

of the circle and the

tangent line

) is drawn in red, and the radius

O

⁢

T

¯

is drawn in green.

The picture justifies the correctness of

P

′

, since the

triangles

△

⁢

O

⁢

P

⁢

T

and

△

⁢

O

⁢

T

⁢

P

′

are

similar

, implying the
proportion

P

⁢

O

:

T

⁢

O

=

T

⁢

O

:

P

′

⁢

O

whence

P

′

⁢

O

⋅

P

⁢

O

=

(

T

⁢

O

)

2

=

r

2

.  Note that this same holds if

P

and

P

′

were swapped in the picture.

Inversion formulae.

If

O

is chosen as the origin of

ℝ

2

and

P

=

(

x

,

y

)

and

P

′

=

(

x

′

,

y

′

)

,
then

x

′

=

r

⁢

x

x

2

+

y

2

,

y

′

=

r

⁢

y

x

2

+

y

2

;

x

=

r

⁢

x

′

x

′

⁣

2

+

y

′

⁣

2

,

y

=

r

⁢

y

′

x

′

⁣

2

+

y

′

⁣

2

.

Note.

Determining inverse points can also be done in the

complex plane

.  Moreover, the mapping

P

↦

P

′

is always a
Möbius

transformation

.  For example, if

c

=

{

z

∈

ℤ

⋮

|

z

|

=

1

}

,  i.e. (

http://planetmath.org/Ie

)

O

=

0

and

r

=

1

, then the mapping

P

↦

P

′

is given by

f

:

ℂ

∪

{

∞

}

→

ℂ

∪

{

∞

}

defined by

f

⁢

(

z

)

=

1

z

.

Properties of

inversion

•

The inversion is

involutory

, i.e. if

P

↦

P

′

,  then

P

′

↦

P

.

•

The inversion is

inversely conformal

, i.e. the

intersection

angle of two curves is preserved (check the Cauchy–Riemann equations (

http://planetmath.org/CauchyRiemannEquations

)!).

•

A line through the center

O

is mapped onto itself.

•

Any other line is mapped onto a circle that

passes through

the center

O

.

•

Any circle through the center

O

is mapped onto a line; if the circle intersects the base circle

c

, then the line passes through both intersection points.

•

Any other circle is mapped onto its

homothetic

circle with

O

as the homothety center.

References

1

E. J. Nyström

:

Korkeamman geometrian alkeet sovellutuksineen

.  Kustannusosakeyhtiö Otava, Helsinki (1948).

Title

inversion of plane

Canonical name

InversionOfPlane

Date of creation

2015-06-14 18:40:35

Last modified on

2015-06-14 18:40:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Topic

Classification

msc 51K99

Classification

msc 53A30

Classification

msc 30E20

Synonym

mirroring in circle

Synonym

circle inversion

Related topic

MobiusTransformation

Related topic

PowerOfPoint

Defines

inverse point

Defines

inversion

Defines

inversion formulae

Defines

involutory

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