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CIRCULAR ALGEBRAIC CURVE


An algebraic curve is said to be circular if it passes through points called "cyclic points", with homogeneous coordinates (1, i, 0) and (1, i, 0), in the complex projective completion of the plane. In other words, they contain the points at infinity of the two complex lines, called isotropic lines, of equation , the reunion of which is the circle with non-zero radius: .
The necessary and sufficient condition on the affine Cartesian equation is that the polynomial composed of the terms of highest degree be divisible by  . This notion is Euclidian (i.e. invariant by change of orthonormal basis).

The circle is the only circular conic.

See circular cubics for the case of cubics.
See multicircular curves for a generalisation.
See bispherical surface for the 3D analogue.
 
 
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© Robert FERRÉOL  2017