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A transformation taking each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524201.png" /> of the plane to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524202.png" /> on the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524203.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524204.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524205.png" /> is a constant real number. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524206.png" /> is called the centre, or pole, of the inversion and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524207.png" /> the power, or coefficient, of the inversion. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524208.png" />, then points on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i0524209.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242010.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242011.png" /> are taken to themselves under the inversion; interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242012.png" /> are taken to exterior points and vice versa (an inversion is sometimes called a symmetry with respect to a circle). The centre of an inversion does not have an image. An inversion with negative power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242013.png" /> is equivalent to the inversion with the same centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242014.png" /> and positive power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242015.png" /> followed by symmetry in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242016.png" />. An inversion with positive power is sometimes called a hyperbolic inversion, while one with negative power is called an elliptic inversion or anti-inversion. A straight line passing through the centre of an inversion is taken into itself under the inversion. A straight line not passing through the centre of an inversion is taken into a circle passing through the centre of the inversion. A circle passing through the centre of an inversion is taken into a straight line not passing through the centre of the inversion. A circle not passing through the centre of an inversion is taken into a circle not passing through the centre of the inversion. In rectangular Cartesian coordinates an inversion can be given by:
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A transformation taking each point $A$ of the plane to the point $A'$ on the [[ray]] $OA$ for which $OA'.OA = k$, where $k$ is a constant real number. The point $O$ is called the centre, or pole, of the inversion and $k$ the power, or coefficient, of the inversion. If $k=a^2$ then points on the circle $C$ with centre $O$ and radius $a$ are taken to themselves under the inversion; interior points of $C$ are taken to exterior points and vice versa (an inversion is sometimes called a symmetry with respect to a circle). The centre of an inversion does not have an image. An inversion with negative power $k$ is equivalent to the inversion with the same centre $O$ and positive power $-k$ followed by symmetry in $O$. An inversion with positive power is sometimes called a hyperbolic inversion, while one with negative power is called an elliptic inversion or anti-inversion. A straight line passing through the centre of an inversion is taken into itself under the inversion. A straight line not passing through the centre of an inversion is taken into a circle passing through the centre of the inversion. A circle passing through the centre of an inversion is taken into a straight line not passing through the centre of the inversion. A circle not passing through the centre of an inversion is taken into a circle not passing through the centre of the inversion. In rectangular Cartesian coordinates an inversion can be given by:
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$$
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x' = \frac{kx}{x^2+y^2}\,,\ \ y' = \frac{ky}{x^2+y^2}
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$$
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and in the complex plane by the formula $z' = k / \bar z$. An inversion is an anti-conformal mapping, that is, it preserves angles between lines and changes their orientation. An inversion in space is defined in a similar way.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242017.png" /></td> </tr></table>
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An inversion is sometimes defined as a mapping of the plane that associates with each point $A$ distinct from the centre of a given pencil of circles the point of intersection $A'$ of the circles of the pencil passing through $A$.
 
 
and in the complex plane by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242018.png" />. An inversion is an anti-conformal mapping, that is, it preserves angles between lines and changes their orientation. An inversion in space is defined in a similar way.
 
 
 
An inversion is sometimes defined as a mapping of the plane that associates with each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242019.png" /> distinct from the centre of a given pencil of circles the point of intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242020.png" /> of the circles of the pencil passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242021.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Alexandroff [P.S. Aleksandrov] (ed.)  et al. (ed.) , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Alexandroff [P.S. Aleksandrov] (ed.)  et al. (ed.) , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
Sometimes an ideal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242022.png" /> is regarded as the image of the centre of an inversion under this inversion, especially when one considers the inversion on the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052420/i05242023.png" />.
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Sometimes an [[ideal point]] $\infty$ is regarded as the image of the centre of an inversion under this inversion, especially when one considers the inversion on the extended complex plane $\hat{\mathbf{C}}$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Schwerdtfeger,  "Geometry of complex numbers" , Dover, reprint  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Pedoe,  "Circles" , Pergamon  (1957)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Schwerdtfeger,  "Geometry of complex numbers" , Dover, reprint  (1979)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Pedoe,  "Circles" , Pergamon  (1957)</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 18:16, 17 December 2017

A transformation taking each point $A$ of the plane to the point $A'$ on the ray $OA$ for which $OA'.OA = k$, where $k$ is a constant real number. The point $O$ is called the centre, or pole, of the inversion and $k$ the power, or coefficient, of the inversion. If $k=a^2$ then points on the circle $C$ with centre $O$ and radius $a$ are taken to themselves under the inversion; interior points of $C$ are taken to exterior points and vice versa (an inversion is sometimes called a symmetry with respect to a circle). The centre of an inversion does not have an image. An inversion with negative power $k$ is equivalent to the inversion with the same centre $O$ and positive power $-k$ followed by symmetry in $O$. An inversion with positive power is sometimes called a hyperbolic inversion, while one with negative power is called an elliptic inversion or anti-inversion. A straight line passing through the centre of an inversion is taken into itself under the inversion. A straight line not passing through the centre of an inversion is taken into a circle passing through the centre of the inversion. A circle passing through the centre of an inversion is taken into a straight line not passing through the centre of the inversion. A circle not passing through the centre of an inversion is taken into a circle not passing through the centre of the inversion. In rectangular Cartesian coordinates an inversion can be given by: $$ x' = \frac{kx}{x^2+y^2}\,,\ \ y' = \frac{ky}{x^2+y^2} $$ and in the complex plane by the formula $z' = k / \bar z$. An inversion is an anti-conformal mapping, that is, it preserves angles between lines and changes their orientation. An inversion in space is defined in a similar way.

An inversion is sometimes defined as a mapping of the plane that associates with each point $A$ distinct from the centre of a given pencil of circles the point of intersection $A'$ of the circles of the pencil passing through $A$.

References

[1] P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian)
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)


Comments

Sometimes an ideal point $\infty$ is regarded as the image of the centre of an inversion under this inversion, especially when one considers the inversion on the extended complex plane $\hat{\mathbf{C}}$.

References

[a1] H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979)
[a2] D. Pedoe, "Circles" , Pergamon (1957)
How to Cite This Entry:
Inversion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inversion&oldid=42549
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article