A point of the complex projective plane may be described in terms of homogeneous coordinates, being a triple of complex numbers (x : y : z), 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.
The two circular points at infinity are two of these, usually taken to be those with homogeneous coordinates Let A.
C be the measures of the vertex angles of the reference triangle ABC.
[1] A real circle, defined by its center point (x0,y0) and radius r (all three of which are real numbers) 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 (of the real projective plane).
In general, an algebraic curve that passes through these two points is called circular.
The concept of angle can be defined using the circular points, natural logarithm and cross-ratio:[3] Sommerville configures two lines on the origin as
Denoting the circular points as ω and ω′, he obtains the cross ratio The transformation
Thus all circles are transformed into conics which go through these two real infinitely distant points, i.e. into equilateral hyperbolas whose asymptotes make an angle of 45° with the axes.