Inversion

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 $A$.

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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 $\hat{\mathbf{C}}$.

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