Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 2 / 12A (Geometry)

For the spherical circle, use a rotation to put the center at the north pole. Points at spherical distance $r$ have polar angle $r$ and admit the parametrizationIts speed is $|{\gamma }^{\prime }(\theta )|=\sin r$, so the arc length isFor small $r$, the Taylor series gives $C(r)=2\pi r-(\pi /3)r^3+O(r^5)$. Thus its circumference is smaller than the Euclidean circumference $2\pi r$ for the same intrinsic radius. Their ratio is $\sin r/r=1-r^2/6+O(r^4)$ and tends to one. The leading agreement expresses local Euclidean behavior; the cubic deficit reflects the positive Gaussian curvature of the unit sphere.

Define the Möbius transformationThe three prescribed distinct points go to $0,1,\infty$, respectively, and the specified cross-ratio is exactly $T(z_4)$.

To identify the preimage of the extended real line directly, write $T(z)=(az+b)/(cz+d)$. The condition $T(z)=\overline{T(z)}$ becomesAfter division by $2i$, this has the real form $A|z{|}^2+Bz+\overline{B}\overline{z}+C=0$, with $A,C$ real. If $A=0$ it is a line; otherwise completing the square gives a circle. It contains $z_1,z_2,z_3$, including the pole as a point mapped to infinity, so it is nondegenerate. Three distinct noncollinear points determine one circle; if the three are collinear, their unique generalized circle is their line.

Consequently $T(z_4)$ is real exactly when the fourth point is on the same generalized circle as the first three. This proves the real cross-ratio criterion for a generalized circle: