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<sup>2</sup> + y<sup>2</sup>. More precisely, if F = F<sub>n</sub> + F<sub>nâÂÂ1</sub> + ... + F<sub>1</sub> + F<sub>0</sub>, where each F<sub>i</sub> is homogeneous of degree i, then the curve F(x, y) = 0 is circular if and only if F<sub>n</sub> is divisible by x<sup>2</sup> + y<sup>2</sup>.
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, âÂÂi, 0) = 0. In other words, the curve is circular if it contains the circular points at infinity, (1, i, 0) and (1, âÂÂi, 0), when considered as a curve in the complex projective plane.
An algebraic curve is called p-circular if it contains the points (1, i, 0) and (1, âÂÂi, 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<sub>nâÂÂi</sub> is divisible by (x<sup>2</sup> + y<sup>2</sup>)<sup>pâÂÂi</sup> 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.
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.