Circular algebraic curve

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.