Imaginary Transformation à la Klein

Imaginary Circular Points and the Rectangular Hyperbola

Consider the equation of the circle:
\[(x-a)^{2}+(y-b)^{2}=r^{2}\]
which can be written in the homogeneous form:
\[(\xi-a \tau)^{2}+(\eta-b \tau)^{2}-r^{2} \tau^{2}=0\]
The intersection with the line at infinity \( \tau=0 \) will thus be given by the equations:
\[\xi^{2}+\eta^{2}=0, \quad \tau=0\]
The constants \( a, b, \) and \( r, \) which characterise the present circle, do not appear at all. Hence, every circle cuts the line at infinity in the same two fixed points:
\[\xi: \eta=\pm i, \quad \tau=0\]
which are called the imaginary circular points.

The converse is also true: Every curve of second degree, which passes through the imaginary circular points in its plane is a circle.

The distance from the origin to the imaginary circular points is not definitely infinite, as might perhaps at first be thought of. Instead, this distance has the form:
 \[ \sqrt{x^{2}+y^{2}}=\sqrt{\xi^{2}+\eta^{2}} / \tau=0 / 0 \] 
and is consequently indeterminate.

In analytic geometry, it is common to speak always of only two points common to two circles, since the elimination of one unknown from their equations leads to only one quadratic equation. This presentation takes no account of the fact that the two circles have in common also the two imaginary circular points on the line at infinity. Thus, we should actually consider four intersections, the requisite number for two curves of the second degree.

In this connection, we may consider an imaginary transformation. By this it is meant, according to Klein, a collineation with imaginary coefficients, which carries imaginary points in which we are interested over into real points. Thus, in the theory of the imaginary circular points, we can introduce the transformation:
\[\xi^{\prime}=\xi, \quad \eta^{\prime}=i \eta, \quad \tau^{\prime}=\tau\]
This transformation sends the equation \( \xi^{2}+\eta^{2}=0 \) into the equation \( \xi^{\prime 2}-\eta^{\prime 2}=0 \) and changes the imaginary circular points:
\[ \xi: \eta=\pm i, \tau=0 \]
into the real infinitely distant points:
\[\xi^{\prime}: \eta^{\prime}=\pm 1, \tau=0\]
which are the points at infinity in the two directions that make an angle of \( 45^{\circ} \) with the axes.



Thus, all circles are transformed into conic sections, which go through these two real infinitely distant points, i.e., into equilateral hyperbolas whose asymptotes make an angle of \( \pm 45^{\circ} \) with the axes.

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