7.2 The Riemann sphere model

The Riemann sphere adds a point at infinity to the complex plane, creating a model of elliptic geometry. This sphere's points are complex numbers and infinity, while great circles represent lines. Angles are measured by dihedral angles between planes.

The spherical metric measures distances on the Riemann sphere. Elliptic geometry axioms include unique lines through points, intersecting lines, no parallels, and triangle angle sums exceeding 180°. These concepts showcase the unique properties of this non-Euclidean geometry.

The Riemann Sphere and Elliptic Geometry

Riemann sphere and elliptic geometry

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• Model of elliptic geometry obtained by adding a point at infinity to the extended complex plane creates a surface topologically equivalent to a sphere
• Complex numbers and the point at infinity form the points on the Riemann sphere (0, i, -1+2i)
• Great circles passing through the point at infinity represent lines on the Riemann sphere (equator, meridians)
• Dihedral angle between planes containing great circles measures angles on the Riemann sphere

Elements of Riemann sphere

• Points represented by complex numbers $z = a + bi$ and the point at infinity $\infty$
• Great circles formed by intersecting the sphere with planes passing through its center represent lines
  • Vertical lines in the complex plane correspond to great circles through 0 and $\infty$ (prime meridian)
  • Non-vertical lines correspond to great circles not passing through 0 and $\infty$ (any other meridian)
• Dihedral angle between planes containing great circles measures angles
  • Angle between intersecting great circles equals angle between tangent vectors at intersection point

Spherical metric for distances

• Spherical metric on Riemann sphere given by $ds = \frac{2|dz|}{1 + |z|^2}$
• Distance between points $z_1$ and $z_2$ is $d(z_1, z_2) = 2 \arctan\left|\frac{z_1 - z_2}{1 + \bar{z_1}z_2}\right|$
  • Example: $d(0, 1) = 2 \arctan(1) \approx 1.05$ radians or about 60°
• Distance between point $z$ and $\infty$ is $d(z, \infty) = 2 \arctan\left|\frac{1}{z}\right|$
  • Example: $d(1, \infty) = 2 \arctan(1) \approx 1.05$ radians or about 60°

Axioms of elliptic geometry

• Two distinct points determine a unique line
  • Any two points on Riemann sphere lie on a unique great circle (North and South Pole)
• Two distinct lines intersect in a unique point
  • Two great circles intersect at antipodal points identified as a single point (any two meridians)
• No parallel lines exist
  • All great circles intersect, so parallel lines are impossible
• Sum of angles in a triangle exceeds 180°
  • Triangle area on Riemann sphere proportional to excess of angle sum over 180°
  • Example: A triangle with three 90° angles has an area of $\frac{1}{8}$ of the sphere's surface area