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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.
circle inversion examples: Image title: Examples of inversion of circles A to J with respect to the red circle at O by CMG Lee. Circles A to F which pass through O map to straight lines. Circles G to J which do not map to other circles. The reference circle and line L map to themselves. Circles intersect their inverses, if any, on the reference ...
Download as PDF; Printable version; In other projects ... Pages in category "Inversive geometry" The following 13 pages are in this category, out of 13 total ...
An analogue of the Beckman–Quarles theorem holds true for the inversive distance: if a bijection of the set of circles in the inversive plane preserves the inversive distance between pairs of circles at some chosen fixed distance , then it must be a Möbius transformation that preserves all inversive distances. [3]
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence .
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