Imaginary spherical circle

The imaginary spherical circle , also called infinitely distant or improper spherical circle or absolute conic section or dimensional conic section, is referred to in projective geometry as the circle on the infinitely distant plane that lies on all spheres . This circle is analogous to the two points in (the so-called circle points ), which lie on all circles. is non-Euclidean and, if one understands the intuition space as a subset of , then these and are disjoint , which is why the spherical circle eludes ordinary intuition. ${\displaystyle P^{3}(\mathbb {C} )}$ ${\displaystyle E_{\infty }}$ ${\displaystyle P^{2}(\mathbb {C} )}$ ${\displaystyle P^{3}(\mathbb {C} )}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle P^{3}(\mathbb {C} )}$ ${\displaystyle E_{\infty }}$

description

Let be a sphere in the affine space with a center and a radius . This sphere is given by the equation ${\displaystyle K}$ ${\displaystyle \mathbb {C} ^{3}}$ ${\displaystyle (a,b,c)}$ ${\displaystyle r}$

${\displaystyle K\colon \quad (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}}$

described. The homogenization of this equation yields as an equation over ${\displaystyle P^{3}(\mathbb {C} )}$

${\displaystyle K\colon \quad (x-aw)^{2}+(y-bw)^{2}+(z-cw)^{2}=r^{2}w^{2},}$

whereby the points are now represented by homogeneous coordinates . Since the following applies to identity : ${\displaystyle (x,y,z,w)}$ ${\displaystyle E_{\infty }}$ ${\displaystyle w=0}$

${\displaystyle K\cap E_{\infty }\colon \quad x^{2}+y^{2}+z^{2}=0\wedge w=0}$

${\displaystyle K\cap E_{\infty }}$is independent of the center of the sphere and of the radius , so lies on all spheres. By rearranging, for example, you get an equation for a circle with a radius ( is the imaginary unit ), which shows why Felix Klein called this curve an imaginary circle. Note that an imaginary circle with radius 0 also consists of more than one point. ${\displaystyle (a,b,c)}$ ${\displaystyle r}$ ${\displaystyle K\cap E_{\infty }}$ ${\displaystyle x^{2}+y^{2}=(\mathrm {i} z)^{2}}$ ${\displaystyle \mathrm {i} z}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle K\cap E_{\infty }}$

reversal

A reversal of the principle that the spherical circle lies on all spheres also applies: If a surface of the second order contains the spherical circle, then this surface is already a sphere, provided it is not degenerate in two planes.

source

• Felix Klein : Lectures on non-Euclidean geometry. Göttingen / Hannover 1928, reprinted by Chelsea Publishing Company, New York, pp. 135-137.

Individual evidence

1. ^ Günther Eisenreich, Ralf Sube: Technical Dictionary of Mathematics. VEB Verlag Technik, Berlin 1982, 1st edition, p. 11.