Circular algebraic curve


In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F = 0, where F is a polynomial with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2. More precisely, if
F = Fn + Fn−1 + ... + F1 + F0, where each Fi is homogeneous of degree i, then the curve F = 0 is circular if and only if Fn is divisible by x2 + y2.
Equivalently, if the curve is determined in homogeneous coordinates by G = 0, where G is a homogeneous polynomial, then the curve is circular if and only if G = G = 0. In other words, the curve is circular if it contains the circular points at infinity, and, 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 and 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 = 0 is p-circular if Fn−i is divisible by 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 2p.
The set of p-circular curves of degree p + k, where p may vary but k is a fixed positive integer, is invariant under inversion. When k is 1 this says that the set of lines together with the set of circles form a set which is invariant under inversion.

Examples