Anahelixis

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 con...