Circular points at infinity


In projective geometry, the circular points at infinity are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.

Coordinates

A point of the complex projective plane may be described in terms of homogeneous coordinates, being a triple of complex numbers, where two triples describe the same point of the plane when the coordinates of one triple are the same as those of the other aside from being multiplied by the same nonzero factor. In this system, the points at infinity may be chosen as those whose z-coordinate is zero. The two circular points at infinity are two of these, usually taken to be those with homogeneous coordinates

Complexified circles

A real circle, defined by its center point and radius r may be described as the set of real solutions to the equation
Converting this into a homogeneous equation and taking the set of all complex-number solutions gives the complexification of the circle. The two circular points have their name because they lie on the complexification of every real circle. More generally, both points satisfy the homogeneous equations of the type
The case where the coefficients are all real gives the equation of a general circle. In general, an algebraic curve that passes through these two points is called circular.

Additional properties

The circular points at infinity are the points at infinity of the isotropic lines.
They are invariant under translations and rotations of the plane.
The concept of angle can be defined using the circular points, natural logarithm and cross-ratio:
Sommerville configures two lines on the origin as Denoting the circular points as ω and ω''′, he obtains the cross ratio