FORMULA SHEET Hyperbolic Geometry Reference

Every formula in this unit, indexed by topic. The complete working toolkit for the Poincaré disk, the upper half-plane, and their isometries.

The Two Standard Models

Poincaré Disk
$$\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}, \quad ds^2 = \frac{4\,(dx^2 + dy^2)}{(1 - |z|^2)^2}$$
Boundary: unit circle (at infinity). Geodesics: diameters and arcs orthogonal to ∂𝔻.
Upper Half-Plane
$$\mathbb{H} = \{ z \in \mathbb{C} : \mathrm{Im}(z) > 0 \}, \quad ds^2 = \frac{dx^2 + dy^2}{y^2}$$
Boundary: real axis ∪ {∞}. Geodesics: vertical lines and semicircles centered on ℝ.
Cayley transform: 𝔻 ↔ ℍ
$$w = i\,\frac{1 - z}{1 + z}, \qquad z = \frac{i - w}{i + w}$$
A Möbius isomorphism between the disk and half-plane. Use whichever model is more convenient.

Hyperbolic Distance

Disk: arccosh form
$$d_{\mathbb{D}}(z, w) = \mathrm{arcosh}\!\left( 1 + \frac{2\,|z - w|^2}{(1 - |z|^2)(1 - |w|^2)} \right)$$
Most general form. Symmetric in z, w; vanishes iff z = w.
Disk: from origin
$$d_{\mathbb{D}}(0, r) = \ln \frac{1 + r}{1 - r} = 2\,\mathrm{artanh}(r)$$
Diverges as r → 1. Boundary is at infinite distance.
Half-plane
$$d_{\mathbb{H}}(z, w) = \mathrm{arcosh}\!\left( 1 + \frac{|z - w|^2}{2 \,\mathrm{Im}(z)\,\mathrm{Im}(w)} \right)$$
Half-plane vertical
$$d_{\mathbb{H}}(\,(x, y_1),\,(x, y_2)\,) = \left| \ln \frac{y_2}{y_1} \right|$$
For points on the same vertical line, the geodesic is the vertical itself.

Hyperbolic Triangles

Angle defect = area (K = −1)
$$\delta = \pi - (\alpha + \beta + \gamma) = \mathrm{Area}(T)$$
Maximum area: π (ideal triangle, all vertices on the boundary).
Hyperbolic Law of Cosines
$$\cosh c = \cosh a \cosh b - \sinh a \sinh b \cos C$$
Hyperbolic Law of Sines
$$\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$$
Pythagoras (right triangle, C = π/2)
$$\cosh c = \cosh a \cdot \cosh b$$
Watch out: not a² + b² = c². Hyperbolic functions replace polynomials.
Defect angle of dual cosine
$$\cos C = -\cos A \cos B + \sin A \sin B \cosh c$$
Dual to the law of cosines: determines the third angle from the other two angles + the opposite side.

Isometries

Möbius isometry (half-plane)
$$z \;\longmapsto\; \frac{az + b}{cz + d}, \qquad ad - bc = 1, \quad a,b,c,d \in \mathbb{R}$$
Group: PSL(2, ℝ) — orientation-preserving isometries of ℍ.
Classification by trace
$$\begin{aligned} |\,\mathrm{tr}\,| < 2: &\quad \text{elliptic (rotation about interior fixed point)} \\ |\,\mathrm{tr}\,| = 2: &\quad \text{parabolic (one boundary fixed point)} \\ |\,\mathrm{tr}\,| > 2: &\quad \text{hyperbolic (two boundary fixed points, translation along geodesic)} \end{aligned}$$
Disk-model Möbius isometries
$$z \;\longmapsto\; e^{i\theta}\,\frac{z - a}{1 - \bar{a}\,z}, \qquad |a| < 1, \;\theta \in \mathbb{R}$$
Composition of a rotation about origin and a "translation" sending a → 0.

Circles, Horocycles, Equidistants

Hyperbolic circle (radius r)
$$C = 2\pi \sinh r, \qquad A = 2\pi (\cosh r - 1)$$
Both grow exponentially in r — the signature of negative curvature.
Small-r limit (sanity check)
$$C \approx 2\pi r + \frac{\pi r^3}{3} + O(r^5), \qquad A \approx \pi r^2 + \frac{\pi r^4}{12} + O(r^6)$$
Recover Euclidean values to leading order; the next term reveals the negative curvature.

Regular Tilings {p, q}

Existence condition
$$\frac{1}{p} + \frac{1}{q} \;\begin{cases} > 1/2 & \text{spherical} \\ = 1/2 & \text{Euclidean} \\ < 1/2 & \text{hyperbolic} \end{cases}$$
Polygon vertex distance from center
$$\cosh d = \frac{\cos(\pi/q)}{\sin(\pi/p)}$$
Hyperbolic distance from polygon center to a vertex, given Schläfli symbol {p, q}.
Defect of regular p-gon (interior angle 2π/q each)
$$\delta = (p - 2)\pi - p \cdot \frac{2\pi}{q} = \pi\!\left( p - 2 - \frac{2p}{q} \right) > 0$$
Equals the polygon's hyperbolic area.

Special Relativity Bridge

Rapidity ↔ velocity
$$v = c\,\tanh\varphi \qquad\Longleftrightarrow\qquad \varphi = \mathrm{artanh}(v/c)$$
Lorentz boost as hyperbolic rotation
$$\begin{pmatrix} ct' \\ x' \end{pmatrix} = \begin{pmatrix} \cosh\varphi & -\sinh\varphi \\ -\sinh\varphi & \cosh\varphi \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix}$$
Velocity addition
$$\varphi_{12} = \varphi_1 + \varphi_2 \;\Longrightarrow\; v_{12} = \frac{v_1 + v_2}{1 + v_1 v_2 / c^2}$$
Rapidities add linearly; velocities don't. The hyperbolic geometry hides under the algebra.

Useful Hyperbolic-Function Identities

Sum & double-angle
$$\cosh(a+b) = \cosh a \cosh b + \sinh a \sinh b$$
Pythagorean identity
$$\cosh^2 x - \sinh^2 x = 1$$
Inverse formulas
$$\mathrm{arcosh}(x) = \ln(x + \sqrt{x^2 - 1}), \quad \mathrm{artanh}(x) = \tfrac{1}{2}\ln\!\frac{1+x}{1-x}$$