Circular algebraic curve
Introduction In geometry, a '''circular algebraic curve''' is a type of plane algebraic curve determined by an equation ''F''(''x'', ''y'') = 0, where ''F'' is a polynomial with real coefficients and the highest-order terms of ''F'' form a polynomial divisible by ''x''2 + ''y''2.More precisely, if''F'' = ''F''''n'' + ''F''''n''−1 + ... + ''F''1 + ''F''0, where each ''F''''i'' is homogeneous of degree ''i'', then the curve ''F''(''x'', ''y'') = 0 is circular if and only if ''F''''n'' is divisible by ''x''2 + ''y''2.Equivalently, if the curve is determined in homogeneous coordinates by ''G''(''x'', ''y'', ''z'') = 0, where ''G'' is a homogeneous polynomial, then the curve is circular if and only if ''G''(1, ''i'', 0) = ''G''(1, −''i'', 0) = 0.In other words, the curve is circular if it contains the circular points at infinity, (1, ''i'', 0) and (1, −''i'', 0), when considered as a curve in the complex projective plane.
Multicircular algebraic curves An algebraic curve is called '''''p''-circular''' if it contains the points (1, ''i'', 0) and (1, −''i'', 0) when considered as a curve in the complex projective plane, and these points are singularities of order at least ''p''. The terms ''bicircular'', ''tricircular'', etc. apply when ''p'' = 2, 3, etc. In terms of the polynomial ''F'' given above, the curve ''F''(''x'', ''y'') = 0 is ''p''-circular if ''F''''n''−''i'' is divisible by (''x''2 + ''y''2)''p''−''i'' when ''i'' < ''p''. When ''p'' = 1 this reduces to the definition of a circular curve. The set of ''p''-circular curves is invariant under Euclidean transformations. Note that a ''p''-circular curve must have degree at least 2''p''. When ''k'' is 1 this says that the set of lines (0-circular curves of degree 1) together with the set of circles (1-circular curves of degree 2) form a set which is invariant under inversion.
Examples * The circle is the only circular conic. * Conchoids of de Sluze (which include several well-known cubic curves) are circular cubics. * Cassini ovals (including the lemniscate of Bernoulli), toric sections and limaçons (including the cardioid) are bicircular quartics. * Watt's curve is a tricircular sextic.
Footnotes
References *   "Courbe Algébrique Circulaire" at Encyclopédie des Formes Mathématiques Remarquables*   "Courbe Algébrique Multicirculaire" at Encyclopédie des Formes Mathématiques Remarquables*  Definition at 2dcurves.com