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

edit
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

edit
edit

## References

edit
- (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