Inversionsoperation

Inversionsoperation, commonly called circle inversion, is a geometric transformation that inverts a point with respect to a fixed circle. Given a circle with center O and radius R, every point P ≠ O is mapped to a point P' on the line OP such that OP · OP' = R^2. The center O has no image, and points on the circle remain fixed.

Inversion is an involution: applying it twice returns the original point. It preserves angles (it is a conformal transformation) but does not preserve distances and reverses orientation. The transformation has characteristic effects on lines and circles: a line not passing through O maps to a circle that passes through O; a circle not passing through O maps to another circle not passing through O; a line through O maps to itself. Circles that pass through O map to lines not through O, and vice versa.

In three dimensions, inversion generalizes to a sphere with center O and radius R. Points along the line OP are mapped so that OP · OP' = R^2, and spheres and planes are mapped to spheres or planes, with similar special-case behavior for lines and circles extended to 3D.

Applications and connections: Inversions simplify many geometric problems by transforming complex configurations into simpler ones, for example in tangency problems, packing problems, and the construction of Apollonian gaskets. In complex analysis, a standard inversion corresponds to the map w = R^2 / z, a basic Möbius transformation. Compositions of inversions generate the broader class of Möbius (conformal) transformations, making the inversion a foundational tool in geometry and analysis.