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In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.

Coordinates

The points of the complex plane may be described in terms of homogeneous coordinates, triples of complex numbers (x: y: z), with two triples describing the same point of the plane when one is a scalar multiple of the other. In this system, the points at infinity are the ones whose z-coordinate is zero. The two circular points are the points at infinity described by the homogeneous coordinates

(1: i: 0) and (1: −i: 0).

Complexified circles

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

$$(x-x_0)^2+(y-y_0)^2=r^2.$$

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

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

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. 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:

The angle between two lines is a certain multiple of the logarithm of the cross-ratio of the pencil formed by the two lines and the lines joining their intersection to the circular points.

Sommerville configures two lines on the origin as $$u : y = x \tan \theta, \quad u' : y = x \tan \theta '$$ . Denoting the circular points as ω and ω', he obtains the cross ratio

$$(u u' , \omega \omega ') = \frac{\tan \theta - i}{\tan \theta + i} \div \frac{\tan \theta ' - i}{\tan \theta ' + i}$$ , so that

$$\phi = \theta ' - \theta = \tfrac{i}{2} \log (u u', \omega \omega ')$$ .

References

C. E. Springer (1964) Geometry and Analysis of Projective Spaces, page 141, W. H. Freeman and Company

Duncan Sommerville (1914) Elements of Non-Euclidean Geometry, page 157, link from University of Michigan Historical Math Collection

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