In geometry, a generalized circle, sometimes called a cline or circline,[1] is a straight line or a circle, the curves of constant curvature in the Euclidean plane.
The natural setting for generalized circles is the extended plane, a plane along with one point at infinity 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 Euclidean geometry, which has a well-defined notion of distance 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 degenerate circle without a well-defined center and with infinite radius (zero curvature). A reflection across a line is a Euclidean isometry (distance-preserving transformation) which maps lines to lines and circles to circles; but an inversion in a circle 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 inversive geometry, 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 compositions, called Möbius transformations, map generalized circles to generalized circles, and preserve the inversive relationships between objects.
The extended plane can be identified with the sphere using a stereographic projection. The point at infinity then becomes an ordinary point on the sphere, and all generalized circles become circles on the sphere.
^
Hitchman, Michael P. (2009). Geometry with an Introduction to Cosmic Topology. Jones & Bartlett. p. 43.
and 27 Related for: Generalised circle information
In geometry, a generalized circle, sometimes called a cline or circline, is a straight line or a circle, the curves of constant curvature in the Euclidean...
equation is called a "generalisedcircle." It may either be a true circle or a line. In this sense a line is a generalisedcircle of infinite radius. In...
Director circle Fermat–Apollonius circle Ford circle Fuhrmann circleGeneralisedcircle GEOS circle Great circle Great-circle distance Circle of a sphere...
formalized and a Valeriepieris circle can be defined for any spatial area, like a single country. These generalised Valeriepieris circles can be used for studying...
fluid layer with a property that varies Cline (mathematics) or generalisedcircle, a circle or straight line in inverse geometry Cline of instantiation,...
Kepler–Bouwkamp constant. In contexts where lines are taken to be a type of generalisedcircle with infinite radius, collinear points (points along a single line)...
deviates at n = 6, showing the risk of generalising from only a few observations. If there are n points on the circle and one more point is added, n lines...
plane: for example, the Poincaré disc model where generalisedcircles perpendicular to the unit circle correspond to "hyperbolic lines" (geodesics), and...
initially developed to detect analytically defined shapes (e.g., line, circle, ellipse etc.). In these cases, we have knowledge of the shape and aim to...
version on archive.org) Hough transform Generalised Hough transform Randomized Hough transform Lecture 10: Hough Circle Transform, By Harvey Rhody, Chester...
{1}{n}}(A_{1}P+A_{2}P+\cdots +A_{n}P)=c} This suggests a simple interpretation: the generalised conic is a curve such that the average distance of every point P on the...
move on a circle of radius L. The position of the mass is defined by the coordinate vector r = (x, y) measured in the plane of the circle such that y...
Chandrasekhar–Wentzel lemma For mathematicians this fact is known, therefore the circle is redundant and often omitted. However, one should keep in mind here that...
and physically simpler, the squircle has a simpler equation and can be generalised much more easily. One consequence of this is that the squircle and other...
interval exchange transformation is a kind of dynamical system that generalisescircle rotation. The phase space consists of the unit interval, and the transformation...
the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes...
transformations permute the generalisedcircles in the complex plane, f {\displaystyle f} maps the real line to the unit circle. Furthermore, since f {\displaystyle...
G-function and the MacRobert E-function. Hypergeometric series were generalised to several variables, for example by Paul Emile Appell and Joseph Kampé...
relativity, which describes gravity and cosmology, the idea is slightly generalised to the "curvature of spacetime"; in relativity theory spacetime is a...
\mathrm {Li} _{-3}(q^{n}),} which holds for all complex q not on the unit circle, would be considered a Lambert series identity. This identity follows in...
Green, & Company. p. 133. Collins, L; Osler, T (2011). "Law of Cosines Generalised for any Polygon and any Polyhedron". The Mathematical Gazette. 95 (533):...
Arithmetic group Lattice Hyperbolic group Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n)...
and circles). For more complicated shapes in the plane (i.e., shapes that cannot be represented analytically in some 2D space), the Generalised Hough...
The second form of exchange within elementary structures is called generalised exchange, meaning that a man can only marry either his MBD (matrilateral...
colour (argent, gules, or, and vert) with the ends resting on clouds. Generalised examples in coat of arms include those of the towns of Regen and Pfreimd...
triangulation for the four points. This operation is called a flip, and can be generalised to three and higher dimensions. Many algorithms for computing Delaunay...
Global Information