Geometry of the three spaces

1. Straight lines and the parallel axiom

An intrinsic straight line is a geodesic: following it means continuing without turning within the space. Every sufficiently short segment is a shortest path. This is a local statement: continuing straight on a sphere can take you all the way around, although you could reach the same endpoint by a shorter route. A free ray follows such a line in all three geometries; its drawing in a coordinate model may be curved.

The familiar dividing question is the parallel axiom in Playfair’s form: given a line ℓ and a point P outside it, how many lines through P never meet ℓ? Here lines and P lie in the same two-dimensional geometry.

Euclidean and hyperbolic geometry share the usual neutral geometry axioms and differ in the parallel postulate. The sphere also changes global incidence and order: antipodal points lie on infinitely many great circles, and lines close. It is therefore not obtained merely by changing the fifth postulate while keeping every other Euclidean axiom. More on this distinction.

A flat drawing can hide this distinction. On a globe, the equator and meridians are straight in the surface geometry, while most latitude circles are not: following a latitude requires continually turning within the surface. “Looks curved on the page” and “turns within the space” are different properties.

These are plane statements. In three dimensions, lines need not lie in a common geodesic plane; for example, distinct great circles of S³ can be disjoint.

2. Circles, areas, and curvature

A circle consists of points at a fixed intrinsic distance r from its centre. Imagine walking outward by r in every direction, then measuring the length around all the resulting endpoints. That length is the circumference C(r); A(r) is the area of the disk inside it. The radius is measured through the space, not as a ruler distance across its drawing. Let R be the curvature radius, and κ the physical curvature (in inverse length squared): 0, −1/R², or +1/R².

The hyperbolic sine and cosine are sinh t = (eᵗ − e⁻ᵗ)/2 and cosh t = (eᵗ + e⁻ᵗ)/2, where t is a dimensionless argument and e is the base of the natural logarithm. Their exponential growth replaces the bounded sine and cosine of spherical geometry.

Hyperbolic circumference eventually grows exponentially. On a sphere it reaches 2πR at the equator (r = πR/2), then shrinks to zero at the antipode (r = πR). At that last radius the disk covers the whole sphere, of area 4πR².

Here O(κ²r⁴) collects terms of order r⁴ and higher for fixed curvature. Small circles look Euclidean in every case. The ratio above measures curvature without looking at an embedding: compare the length around a circle with the distance walked from its centre.

One metric explains the formulas

Set f(r) = r, R sinh(r/R), or R sin(r/R), respectively. The polar coordinates are the distance r from a chosen centre and the direction angle θ, measured in radians. The symbol ds is an infinitesimal length; dr and dθ are small coordinate changes. In these coordinates:

At fixed radius, an angular change dθ has length f(r)dθ. Integrating over a full turn gives the circumference; adding circular rings from radius 0 to r gives the disk area. The variable t in the integral is the running radius.

In three dimensions, θ is the polar angle and φ the azimuthal angle of a direction; both are in radians. The same radial factor gives the area of a geodesic sphere (the surface at distance r) and the volume of its ball:

Thus H³ has exponentially growing volume, while S³ has finite total volume 2π²R³. The sphere is finite without having an edge. The polar formulas are coordinate descriptions; their degeneracy at r = 0 or at a spherical antipode is not a singularity of the space.

3. Triangles and regular tilings

Take a triangle bounded by geodesic segments. Denote its interior angles by α, β, γ, its enclosed area by A, and the surface curvature by κ as above. For a convex spherical triangle, use the shorter arcs within a hemisphere. Its angle sum measures its area:

A regular tiling {p,q} has p sides per face and q faces meeting at each vertex. For ordinary convex faces (p,q ≥ 3), their interior angles are 2π/q:

To see the tiling condition, a Euclidean regular p-gon has angle (p−2)π/p. Exactly q of these angles must fit around a vertex, adding to 2π. In hyperbolic geometry the face’s angle sum is smaller, so more faces fit than the Euclidean case allows. On a sphere the angle sum is larger, and the tiling closes after finitely many faces.

The relevant curvature is that of the surface being tiled. A hyperbolic geodesic plane is hyperbolic, a horosphere is intrinsically Euclidean, and a geodesic sphere has a round spherical metric—even when all three lie in H³.

4. Three models in two dimensions

A model assigns coordinates and a rule for measuring lengths, called a metric. Different models can describe exactly the same space, like two maps of the same landscape. Their pictures can distort lengths or shapes while representing the same intrinsic distances. The metric determines which paths are straight. These diagrams show two-dimensional spaces; the third dimension in the sphere drawing belongs to its embedding.

The half-plane’s boundary is infinitely far away in its metric. A great circle bends in the surrounding Euclidean drawing but never turns within the sphere. The straight chord through the sphere’s interior is not an allowed path on its surface.

5. Poincaré upper half-space

Adding one horizontal coordinate gives the three-dimensional model H³. Write h = (u,v,z): u and v are horizontal coordinates, and z > 0 is coordinate height. The three-coordinate Euclidean drawing contains every point above a boundary plane; distances are measured with:

The differentials du, dv, dz are infinitesimal coordinate changes, and ds is the corresponding physical length. The local length scale is R/z. The model preserves angles (it is conformal), but equal coordinate lengths at different heights represent different physical lengths. Geodesics are vertical lines or circular arcs orthogonal to the boundary plane.

For points h = (u,v,z) and h′ = (u′,v′,z′), let d(h,h′) denote their intrinsic distance. The norm |h − h′| below is their Euclidean coordinate separation. For a vertical segment, z₁ and z₂ are its two endpoint heights:

Planes, spheres, and horospheres

These are vertical 2D cross-sections of surfaces in H³. The black lower line is the excluded boundary plane seen edge-on.

A horosphere is the limit of spheres whose centres and radii tend to infinity together while one point on each sphere stays fixed. The centres approach an ideal boundary point, meaning a direction infinitely far away in the space. At a fixed height z₀ > 0, its induced metric is (R/z₀)²(du² + dv²): flat. Paths straight within this surface need not be geodesics of the surrounding H³.

Hypertrace’s original formulation used upper half-space directly. Today it remains a useful chart for patterns; the shared representation below describes the same geometry.

6. Spherical space in four coordinates

The familiar spherical surface has two intrinsic dimensions: S² ⊂ ℝ³. A small creature living on it can move in two independent directions, although describing its position from outside uses three coordinates. One equation constrains those coordinates, leaving two degrees of freedom. The three-dimensional analogue is S³ ⊂ ℝ⁴. The superscript counts dimensions of the space itself, not the number of embedding coordinates.

Thus “a sphere in four coordinates” means a 3-sphere; a mathematical 4-sphere would live in ℝ⁵. For S³, one equation constrains four coordinates, leaving three degrees of freedom. The extra coordinate does not add a direction in which a traveller can leave the space. Locally, a traveller has the usual three independent directions; globally, continuing straight eventually returns to the start.

The letters w,x,y,z are Cartesian coordinates in the embedding; they are constrained by the displayed equation. Here Sⁿ_R means the n-dimensional sphere of embedding radius R, and ℝⁿ is n-dimensional Euclidean coordinate space. For embedding vectors p,q ∈ S³_R, the normalized dot product p·q/R² gives the cosine of their central angle. Multiplying that angle by R gives the shorter great-circle arc:

Every point has an antipode −p. All geodesics from p meet there after πR; continuing another πR returns to p with the original direction. Shorter arcs minimize distance until the antipode; continuing the same straight path afterward no longer gives the shortest route from the starting point.

A geodesic plane in S³ is a great S², formed by intersecting S³ with a three-dimensional linear subspace through the origin. This is the “floor” of the spherical example: a complete spherical plane, not a finite patch ending at an edge.

7. Hyperboloid and shared coordinates

Another model of H³ uses one sheet of a hyperboloid. Unlike the half-space or ball pictures, it makes geodesics and rigid motions into simple operations on four-coordinate vectors. Write P = (w,X) = (w,x,y,z), where X is the spatial triple (x,y,z); write Q for a second point of the same form. These coordinates are normalized to model radius 1, while R restores physical lengths:

The ambient quadratic form has one opposite sign. Its restriction to the hyperboloid’s tangent spaces is positive, giving ordinary positive spatial lengths. A Euclidean drawing of a hyperboloid alone would give the wrong geometry: the metric is essential. Its geodesics are intersections with two-dimensional linear planes through the embedding origin, just as great circles are plane sections of a sphere.

Changing hyperbolic models

The Poincaré ball b and upper-half-space coordinates h describe the same normalized point P. The vector b lies inside the unit Euclidean ball; h lies above its boundary plane. The projection in the diagram compresses an unbounded hyperboloid into a bounded picture without making intrinsic distances finite. Here X = (x,y,z), while h = (h₁,h₂,h₃):

In the two-dimensional version, the ball becomes a disk. Both Poincaré models preserve angles because their metrics multiply the Euclidean metric by a scalar at each point. The boundary circle is infinitely far away; geodesics are diameters or circular arcs meeting that boundary at right angles. Small-looking objects near the boundary need not be physically small.

The coordinates used here

Hypertrace uses four scalar-first coordinates P = (w,X) for all three spaces. Curved coordinates have unit model radius; R converts distances to physical units. The Euclidean convention sets R = 1.

There are still three independent position coordinates in every row. Coordinate conversions change the description, not the space, its straight lines, or measured distances.

8. Geodesics and travelled distance

Write t = s/R for physical travel s in a curved space, and t = s in the Euclidean convention. Define:

For a starting point P₀ and normalized unit tangent V₀, one equation describes the straight path in every case:

The functions C_K and S_K are generalized cosine and sine; C_K is unrelated to the earlier circumference notation C(r). P(s) is the point after travel s, V(s) its normalized tangent, and the subscript 0 marks their starting values. Here V = dP/dt, the rate of change with respect to normalized travel. For K = ±1, use B_K((w,X),(u,Y)) = wu + K X·Y, with B_K(P,V) = 0 and B_K(V,V) = K. The arguments (w,X) and (u,Y) each consist of a scalar and a three-vector. The first condition says the direction is tangent to the space; the second fixes its speed. For K = 0, V = (0,v), where v is a unit Euclidean three-vector, |v| = 1.

On S³, P(s + 2πR) = P(s). The endpoint alone cannot tell how far a ray has travelled. Let φ be its angular phase within one circuit and n the number of complete circuits:

For example, s = 4πR + 0.2R and s = 0.2R reach the same point and tangent. The function arccos returns the angle in [0,π], so the endpoint-distance formula selects a shortest arc. Both journeys have endpoint distance 0.2R, but their travelled lengths differ by two full circuits. Here “ray” means a directed, continued geodesic; on a sphere it is not globally distance-minimizing for arbitrary travel.

9. Distance-preserving transformations

An isometry moves points while preserving distances and angles. It preserves geodesics as well: a shortest local path remains a shortest local path after the motion. The coordinate form depends on the model:

In the table, X and X′ are three-coordinate positions, a is a translation vector, and Q is a 3×3 rotation matrix. P and P′ are four-coordinate positions, and M is a 4×4 matrix. SO(n) denotes matrices preserving Euclidean lengths and orientation (orthogonal matrices of determinant +1). A superscript T denotes transpose. The matrix J records the hyperboloid’s signed quadratic form; diag lists its diagonal entries, with zeros elsewhere, and det is the determinant.

On S³, an embedding rotation can move the observer’s position, not just rotate a frame around it. On the hyperboloid, the analogous motion is a Lorentz boost: it preserves the signed quadratic form and positive sheet. These are spatial isometries here; no time dimension is being simulated.

In the two-dimensional half-plane, put ζ = x + iz. The imaginary unit i satisfies i² = −1; the imaginary part z is positive height. Orientation-preserving hyperbolic motions have a fractional-linear (Möbius) form:

The real coefficients a,b,c,d specify a transformation and satisfy the determinant condition ad − bc = 1. For example, ζ ↦ λζ with scale factor λ > 0 scales the drawing while preserving its hyperbolic metric; along the vertical axis it displaces points by R|ln λ|. Horizontal shifts ζ ↦ ζ + b preserve horocycles z = constant (the two-dimensional counterparts of horospheres) and are parabolic motions, distinct from translation along a geodesic axis.

One algebra for all three cases

Hypertrace represents these motions with quaternion pairs. A quaternion has four real components, one scalar and three vector components; multiplication encodes rotations. Let a,b,c,d now denote ordinary quaternions (independent of the real Möbius coefficients above). Introduce a symbol ε that commutes with them and satisfies ε² = K. Write a motion g = a + εb as the pair (a,b); then:

The first line multiplies two pairs (a,b) and (c,d). In the second line, |a| and |b| are their four-component Euclidean norms and a·b their four-component dot product; these constraints select unit motions. Changing K changes the algebra while retaining the same multiplication rule.

Embed the point P = (w,X) as q(P) = ((w,0),(0,X)), a pair consisting of a real quaternion and a pure vector quaternion. A bar denotes ordinary quaternion conjugation, which keeps the scalar part and negates all three vector components:

Here q(P′) is the transformed point, g⁻¹ undoes the motion, and g_outerg_inner applies the inner motion first. The conjugate g† used in the point action is distinct from the group inverse. For the following two basic motions, n is a unit three-vector interpreted as a pure quaternion. The distance s is a physical displacement along the axis through the origin; θ is a rotation angle in radians around that axis, and t = s/R is normalized travel:

Use R = 1 for Euclidean space. K = −1, 0, +1 gives complex, dual, and split-complex quaternion extensions, respectively. A real scalar in a pair is understood as a quaternion with zero vector part. The same action also maps tangent directions, so position and direction transform together without selecting a different geometric model. See Arizmendi & Pérez-de la Rosa for the algebraic construction; the equations here use this project’s conventions.

10. References and origins

1. Daina Taimina & David W. Henderson. How to Use History to Clarify Common Confusions in Geometry. In particular, the distinction between spherical geometry and replacing only Euclid’s parallel postulate.
2. Anna Felikson. Geometry lecture notes, Durham University. Spherical and hyperbolic geometry, models, triangles, and circle lengths.
3. University of Glasgow. The circumference and area of hyperbolic circles.
4. Francisco José Herranz & Ángel Ballesteros (2006). Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature, SIGMA 2, 010. Section 2 gives ambient coordinates, curvature-dependent trigonometric functions, and geodesic polar metrics.
5. Gerardo Arizmendi & Marco Antonio Pérez-de la Rosa (2019). Some extensions of quaternions and symmetries of simply connected space forms. Unified quaternion description of isometries of E³, H³, and S³.
6. Mrrl (2013). Жизнь на плоскости Лобачевского — Game of Life on the Lobachevsky plane (Russian). The game and geometry article that originally inspired Hypertrace.
7. Alexey Gerasev (2020). Hypertrace: the original project article. Historical half-space formulation and video.