PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (190)
Orphanage
Unclass'd (1)
Unproven (496)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] inversion of plane (Topic)

Let $ c$ be a fixed circle in the Euclidean plane with center $ O$ and radius $ r$. Set for any point $ P \neq 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

$\displaystyle P'O \cdot PO$
has the constant value $ r^2$. This mapping $ P \mapsto 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 $ \overline{OP}$, having an endpoint on $ \overline{OP}$, 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 $ \overline{OT}$ is drawn in green.


\begin{pspicture}(-2,-2)(4,2) \psline[linecolor=blue](1,1.732)(4,0) \psline[line... ...5,1.6){$c$} \rput[b](0,-2){.} \rput[l](-2,0){.} \rput[a](0,2){.} \end{pspicture}
The picture justifies the correctness of $ P'$, since the triangles $ \triangle OPT$ and $ \triangle OTP'$ are similar, implying the proportion $ PO\!:\!TO = TO\!:\!P'O$ whence $ P'O \cdot PO = (TO)^2 = r^2$. Note that this same argument holds if $ P$ and $ P'$ were swapped in the picture.

Inversion formulae. If $ O$ is chosen as the origin of $ \mathbb{R}^2$ and $ P = (x,\,y)$ and $ P' = (x',\,y')$, then

$\displaystyle x' = \frac{rx}{x^2+y^2},\;\;y' = \frac{ry}{x^2+y^2}; \quad\; x = \frac{rx'}{x'^{\,2}+y'^{\,2}},\;\;y = \frac{ry'}{x'^{\,2}+y'^{\,2}}.$

Note. Determining inverse points can also be done in the complex plane. Moreover, the mapping $ P \mapsto P'$ is always a Möbius transformation. For example, if $ c = \{z\in\mathbb{Z}\!: \vert z\vert=1 \}$, i.e. $ O=0$ and $ r=1$, then the mapping $ P \mapsto P'$ is given by $ f\colon \mathbb{C} \cup \{ \infty \} \to \mathbb{C} \cup \{\infty\}$ defined by $ \displaystyle f(z)=\frac{1}{z}$.

Properties of inversion

  • The inversion is involutory, i.e. if $ P\mapsto P'$, then $ P'\mapsto P$.
  • The inversion is conformal, i.e. the intersection angle of two curves is preserved.
  • 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.

Bibliography

1
E. J. NYSTRÖM: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).



"inversion of plane" is owned by pahio. [ full author list (2) ]
(view preamble)

View style:

See Also: Möbius transformation, power of point

Other names:  mirroring in circle, circle inversion
Also defines:  inverse point, inversion, inversion formulae

This object's parent.

Attachments:
compass and straightedge construction of inverse point (Algorithm) by Wkbj79
Log in to rate this entry.
(view current ratings)

Cross-references: homothety center, homothetic, passes through, onto, line, curves, angle, intersection, conformal, properties, Möbius transformation, complex plane, origin, Proportion, similar, triangles, tangent line, endpoint, perpendicular, line segment, restricted tangent, mapping, product, closed ray, plane, point, radius, center, Euclidean plane, circle
There are 12 references to this entry.

This is version 17 of inversion of plane, born on 2007-06-06, modified 2007-06-20.
Object id is 9542, canonical name is InversionOfPlane.
Accessed 2072 times total.

Classification:
AMS MSC: 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)
 53A30 (Differential geometry :: Classical differential geometry :: Conformal differential geometry)
 51K99 (Geometry :: Distance geometry :: Miscellaneous)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)