Generalised circle

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a generalized circle, sometimes called a cline or circline, is a <a href="/facts/Straight_line/gJqsr4Gj">straight line</a> or a <a href="/facts/Circle/9fy5ggKn">circle</a>, the <a href="/facts/Curve/xBQ0pTBW">curves</a> of constant <a href="/facts/Curvature/RFCAEsj8">curvature</a> in the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a>.
The natural setting for generalized circles is the extended plane, a <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a> along with one <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a> through which every straight line is considered to pass. Given any three distinct points in the extended plane, there exists precisely one generalized circle passing through all three.
Generalized circles sometimes appear in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, which has a well-defined notion of <a href="/facts/Euclidean_distance/9qDoQKQe">distance</a> between points, and where every circle has a center and radius: the point at infinity can be considered infinitely distant from any other point, and a line can be considered as a <a href="/facts/Degeneracy_(mathematics)/BqhqlXPH">degenerate</a> circle without a well-defined center and with infinite radius (zero <a href="/facts/Curvature/RFCAEsj8">curvature</a>). A <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a> across a line is a Euclidean <a href="/facts/Isometry/K5wX8ah6">isometry</a> (distance-preserving transformation) which maps lines to lines and circles to circles; but an <a href="/facts/Inversion_in_a_circle/VqFmbfDI">inversion in a circle</a> is not, distorting distances and mapping any line to a circle passing through the reference circles's center, and vice-versa.
However, generalized circles are fundamental to <a href="/facts/Inversive_geometry/VqFmbfDI">inversive geometry</a>, in which circles and lines are considered indistinguishable, the point at infinity is not distinguished from any other point, and the notions of curvature and distance between points are ignored. In inversive geometry, reflections, inversions, and more generally their <a href="/facts/Function_composition/r5Pvnmth">compositions</a>, called <a href="/facts/M%C3%B6bius_transformation/YEkTTLfK">Möbius transformations</a>, map generalized circles to generalized circles, and preserve the inversive relationships between objects.
The extended plane can be identified with the <a href="/facts/Sphere/jywYLNM2">sphere</a> using a <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a>. The point at infinity then becomes an ordinary point on the sphere, and all generalized circles become circles on the sphere.

Generalised circle open-in-new

Generalised circle