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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.
In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k 2. The inverse of the curve C is then the locus of P as Q runs over C.
A simple example occurs for Steiner chains of four circles (n = 4) and one wrap (m = 1). In this case, the given circles and the Steiner-chain circles are equivalent in that both types of circles are tangent to four others; more generally, Steiner-chain circles are tangent to four circles, but the two given circles are tangent to n circles.
Geometry of Complex Numbers is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non-Euclidean geometry. It was written by Hans Schwerdtfeger , and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press .
In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius R.His work initiated a large body of publications, now called inversive geometry.
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole.
In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, α and β, is a reference circle for which α and β are inverses of each other. If α and β are non-intersecting or tangent, a single circle of antisimilitude exists; if α and β intersect at two points, there are two circles of antisimilitude.
Inversive geometry#Circle inversion, a transformation of the Euclidean plane that maps generalized circles to generalized circles; Inversion in a point, or point reflection, a kind of isometric (distance-preserving) transformation in a Euclidean space; Inversion transformation, a conformal transformation (one which preserves angles of intersection)
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