# 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*.

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.^{[citation needed]} 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

- (in French) "Courbe Algébrique Circulaire" at Encyclopédie des Formes Mathématiques Remarquables
- (in French) "Courbe Algébrique Multicirculaire" at Encyclopédie des Formes Mathématiques Remarquables
- Definition at 2dcurves.com

Original source: https://en.wikipedia.org/wiki/Circular algebraic curve.
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