Elliptic and Spherical Geometry

The third classical geometry is the geometry of positive curvature $K>0$: the obtuse-angle case of the Saccheri hypotheses, in which there are no parallels at all, triangles have angle sum greater than $\pi$, and the whole space is finite but unbounded. Its concrete model is the sphere; its axiomatically clean version is elliptic geometry, the sphere with antipodal points identified. Unlike hyperbolic geometry, this case is not a neutral geometry — it must break one of Euclid's other postulates — and understanding exactly which one is the conceptual heart of this page.

References: Coxeter, Introduction to Geometry, ch. 6; Greenberg, Euclidean and Non-Euclidean Geometries, ch. 10; Berger, Geometry I, ch. 18; do Carmo, Differential Geometry of Curves and Surfaces, §4.

1. Why This Case Is Not Neutral

Recall the Saccheri–Legendre theorem: in any neutral geometry the angle sum is at most $\pi$, which rules out the obtuse hypothesis. So a geometry with angle sum $>\pi$ cannot satisfy all the neutral axioms. The axiom it must sacrifice is Euclid's Postulate 2, the unbounded extension of lines, together with the uniqueness in Postulate 1:

• On a sphere, "lines" (great circles) are finite (length $2\pi R$) though they have no endpoints — unbounded in the sense of endless but not infinite in length. This already breaches Saccheri's use of infinite lines.
• Two "lines" (great circles) meet in two antipodal points, so a pair of points need not determine a unique line (antipodal points are joined by infinitely many great circles) — breaching Postulate 1.

The repair (elliptic geometry). Identify each pair of antipodal points into a single point. The resulting space is the projective plane ${\mathbb{RP}}^2$. Now any two distinct points determine a unique line, and any two lines meet in a unique point. This restores the incidence axioms — at the cost of the parallel postulate, since two lines always meet: there are no parallels.

So there are two closely related geometries to keep distinct:

The rest of this page works on the sphere for concreteness and notes where antipodal identification is needed for the axiomatic elliptic version.

2. Great Circles as Geodesics

On the sphere of radius $R$, the role of "straight line" is played by the great circles — intersections of the sphere with planes through the centre. They are the geodesics: locally shortest paths, and the paths a taut string follows. Key facts:

• Every great circle has circumference $2\pi R$ and is the largest circle on the sphere.
• Any two distinct great circles intersect in exactly two antipodal points; there are no non-intersecting geodesics, i.e. no parallels.
• Through two non-antipodal points there is a unique great circle (the "line" joining them); through antipodal points there are infinitely many.

These are the properties that, transported to ${\mathbb{RP}}^2$, make elliptic geometry a clean model of geometry-without-parallels.

3. Spherical Excess and Area

The positive-curvature analogue of the hyperbolic defect is the spherical excess: the amount by which the angle sum exceeds $\pi$.

Girard's Theorem (1629). A spherical triangle with angles $\alpha ,\beta ,>\gamma$ on a sphere of radius $R$ has area $$\mathrm{Area}(△)=R^2(\alpha +\beta +\gamma -\pi ).$$ The quantity $E=\alpha +\beta +\gamma -\pi >0$ is the spherical excess.

Proof idea. Two great circles crossing at angle $\theta$ cut the sphere into four lunes; a lune of angle $\theta$ has area $2R^2\theta$ (proportional to the angle, total sphere $4\pi R^2$ at $\theta =\pi$). A triangle is an intersection of three lunes; adding the six lunes generated by its three vertices covers the sphere with the triangle and its antipode counted extra, and bookkeeping yields $\mathrm{Area}=R^2(\alpha +\beta +\gamma -\pi )$. $■$

This is the exact mirror of the hyperbolic defect–area law $\mathrm{Area}=R^2(\pi -\alpha -\beta -\gamma )$; the two unify as $\mathrm{Area}=|\alpha +\beta +\gamma -\pi |/|K|$ with $K=\pm 1/R^2$ (see constant-curvature.md). Consequences:

• Angle sum exceeds $\pi$, confirming the obtuse hypothesis; the excess grows with area. A triangle with three right angles (an octant of the sphere) has angle sum $3\pi /2$ and excess $\pi /2$.
• Total area is finite: the whole sphere is $4\pi R^2$, so the geometry is of finite extent — the space is compact.
• Like the hyperbolic plane, spherical geometry has no similar triangles (angles determine area, hence size) and hence a natural length scale $R$.

4. Spherical Trigonometry

With sides $a,b,c$ measured as arc lengths (angles subtended at the centre, so $a=R\hat{a}$) and opposite angles $\alpha ,\beta ,\gamma$, the fundamental relations (setting $R=1$) are

$$\cos c=\cos a\cos b+\sin a\sin b\cos \gamma (\text{law of cosines}),$$ $$\frac{\sin a}{\sin \alpha }=\frac{\sin b}{\sin \beta }=\frac{\sin c}{\sin \gamma }(\text{law of sines}),$$

and the right-triangle spherical Pythagoras

$$\cos c=\cos a\cos b.$$

Compare the hyperbolic forms: they are obtained from these by $\cos \to \cosh$, $\sin \to \sinh$ on the sides — i.e. by the formal substitution $R\to iR$ (Lambert's "sphere of imaginary radius"). Expanding to second order, spherical Pythagoras $\cos c=\cos a\cos b$ gives $a^2+b^2=c^2$ again: Euclidean geometry is the common flat limit $R\to \infty$ of both curved geometries.

5. Isometries and the Group $O(3)$

The rigid motions of the sphere are exactly the linear maps of ${\mathbb{R}}^3$ preserving the sphere — the orthogonal group

$$\mathrm{Isom}(S^2)=O(3),{\mathrm{Isom}}^{+}(S^2)=SO(3),$$

with $SO(3)$ the orientation-preserving rotations. As with the plane, isometries are generated by reflections (in great circles). Passing to elliptic geometry, antipodal identification replaces $O(3)$ by $O(3)/\{\pm I\}\cong SO(3)$ acting on ${\mathbb{RP}}^2$. This is the $K>0$ entry in the family of isometry groups; its structure (a simple, compact group, no normal translation subgroup) reflects that positively-curved space is homogeneous and isotropic but has no translations in the flat sense.

6. The Place of Elliptic Geometry in the Spine

Elliptic/spherical geometry completes the trichotomy opened by the parallel postulate:

Its consistency needs no separate model-building of the hyperbolic kind: the sphere sits in plain sight inside Euclidean ${\mathbb{R}}^3$, so spherical (hence elliptic) geometry is manifestly consistent relative to Euclidean geometry — which is itself the point of the independence argument. The unifying metric picture, in which all three rows of the table are the single formula $\Sigma -\pi =K\cdot \text{Area}$, is developed in curvature.md and constant-curvature.md.

Physically, the sphere is the local model of a positively-curved space, and $SO(3)$ is the rotation group underlying angular momentum in quantum mechanics and the isotropy of space; its double cover $SU(2)$ is the subject of group-theory/README.md.