  This is how Klein's approach works for Lobachevskian geometry on the plane. Let κ be a real conic—a conic comprising only real points—on the projective plane. Let Gκ be the set of all collineations that map κ onto itself. Gκ is a subgroup of the projective group. Consider now the cross-ratio of point quadruples &lt;P1,P2,P3,P4&gt; such that P3 and P4 belong to κ, while P1 and P2 range over the interior Int(κ) of the region of the real plane bounded by κ.  (P ∈ Int(κ) if and only if P is a real point and no real tangent to κ passes through P.) As noted above, the choice of points P1 and P2 fixes P3 and P4, so the said cross-ratio may be regarded as a function of the first pair of points only, say, fκ(P1,P2). The function fκ is clearly Gκ-invariant. Put dκ(P1,P2) = c log fκ(P1,P2), where c is an arbitrary real-valued constant, different from 0, and log x denotes the principal value of the natural logarithm of x. Klein was able to show that dκ behaves precisely like a Lobachevskian distance function on Int(κ). In other words, every theorem of Lobachevskian geometry holds for suitable figures formed from points of Int(κ), if the distance between any two of these points is given by the function dκ. Consider, for instance, four points P1,P2,P3, and P4 in Int(κ), such that dκ(P1,P2) = dκ(P2,P3) = dκ(P3,P4) = dκ(P4,P1). They are the vertices of a Lobachevskian equilateral quadrilateral Q, which can have at most three right angles, in which case the fourth interior angle of Q must be acute. (Where ‘right angle’ means, as usual, an angle equal to its adjacent angle, and two angles in Int(κ) are said to be equal if one is the image of the other by a transformation of group Gκ).