Circular points at infinity

In projective geometry, the circular points at infinity in the complex projective plane (also called cyclic points or isotropic points) are

(1: i: 0) and (1: −i: 0).

Here the coordinates are homogeneous coordinates (x: y: z); so that the line at infinity is defined by z = 0. These points at infinity are called circular points at infinity because they lie on the complexification of every real circle. In other words, both points satisfy the homogeneous equations of the type

$Ax^2 + Ay^2 + 2B_1xz + 2B_2yz - Cz^2 = 0.$

The case where the coefficients are all real gives the equation of a general circle (of the real projective plane). In general, an algebraic curve that passes through these two points is called circular.

The circular points at infinity are the points at infinity of the isotropic lines.

The circular points are invariant under translation and rotation.

See also

Conic section

References

• Pierre Samuel, Projective Geometry, Springer 1988, section 1.6;
• Semple and Kneebone, Algebraic projective geometry, Oxford 1952, section II-8.