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 x² + y². More precisely, if F = F_n + F_n−1 + ... + F₁ + F₀, 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 x² + y².

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 (x² + y²)^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 (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

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References

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