Elliptic geometry and antipodal identification

Try

On the sphere, p and −p are antipodal points. In the real elliptic plane they are identified as one point [p]. A great circle and its antipodal identification become an elliptic line. Algebraically, points are one-dimensional subspaces of R³ and lines are two-dimensional subspaces: the same incidence objects as RP².

The metric adds information beyond projective incidence. For unit representatives p,q, elliptic distance is dE([p],[q])=arccos|p·q|, ranging from zero to π/2. Arbitrary projective transformations do not preserve this distance.

Predict

Predict why the absolute value is necessary. Replacing p by −p must not change its elliptic point. It reverses p·q but leaves |p·q| unchanged.

Predict whether eliminating antipodal duplication restores all neutral geometry axioms. Unique point joins return, but an elliptic line remains a closed curve, not a globally ordered real line. Local segment behavior must not be mistaken for global betweenness.

Experiment

Compare spherical and elliptic distances using unit representatives.

p·q	Spherical short distance	Elliptic distance
1	0	0
0	π/2	π/2
−1/2	2π/3	π/3
−1	π	0

The final row identifies antipodes. In the third row the elliptic path can use the closer representative −q. Explain why this changes the distance without changing the abstract projective point.

Understand

The formula is well-defined because the only unit representatives of a one-dimensional subspace are p and −p. Positivity and symmetry follow immediately. To prove the triangle inequality, use dE([p],[q])=min(dS(p,q),dS(p,−q)). Choose a sign of q attaining the minimum from p, then a sign of r attaining the minimum from that chosen q. The spherical triangle inequality bounds the resulting p-to-r distance by the sum of the two elliptic distances. Taking the smaller allowed p-to-r distance proves the elliptic triangle inequality. Zero distance means representatives agree up to sign, precisely equality of elliptic points.

Incidence uniqueness is linear algebra. Two distinct one-dimensional subspaces span a unique two-dimensional subspace. Two distinct two-dimensional subspaces of R³ intersect in a one-dimensional subspace by the dimension formula. Hence every pair of distinct points has a unique joining line and every pair of distinct lines meets exactly once.

Apply

Repair “Unique joining line implies a unique shortest segment.” Orthogonal classes have distance π/2; either sign of q is equally close to p, producing two different shortest arcs in their elliptic line. A unique complete line can contain more than one minimizing arc between points.

The quotient is locally spherical: sufficiently small neighborhoods do not identify two different nearby points, so local curvature is positive. Globally it is not the sphere. In particular, its lines close up and its total area in the unit-curvature normalization is half the sphere’s area, namely 2π.

For optional depth, RP² is nonorientable, though local angles and distances remain well-defined. A global orientation-dependent argument needs care. The course uses local metric and incidence facts rather than assuming an oriented plane model.

Continue

Keep three statements distinct: spherical representatives, projective incidence classes, and the elliptic metric on those classes. Continue to curvature and triangle area to compare exact area identities across the standard geometries.