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Inversive geometry
In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842-3) and Kelvin (1845). The concept of inversion can be generalized to higher-dimensional spaces. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P, lying on the ray from O through P such that This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P also takes P back to P, so the result of applying the same inversion twice is the identity transformation on all the points of the plane other than O (self-inversion). To make inversion an involution it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity. It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, the nearer a point to the center, the further away its transformation, and vice versa. To construct the inverse P of a point P outside a circle Ø: Draw the segment from O (center of circle Ø) to P. Let M be the midpoint of OP. (Not shown) Draw the circle c with center M going through P. (Not labeled. It's the blue circle) Let N and N be the points where Ø and c intersect. Draw segment NN.