Hyperbolic geometry

Hyperbolic geometry studies spaces of constant negative curvature.

For a right-angled geodesic triangle in curvature minus one, with legs $a,b$ and hypotenuse $c$, the displayed identity replaces the Euclidean theorem. Normalize the right vertex to $i$ in the upper half-plane, one endpoint to $iy$ with $y>1$, and the other to $s+it$ with $s^2+t^2=1$ and $t>0$. The Hyperbolic distance in the Poincare half-plane gives $\cosh a=(y+y^{-1})/2$, $\cosh b=1/t$ and $\cosh c=(1+y^2)/(2yt)$, proving the identity. For small lengths its expansion gives $c^2=a^2+b^2$ to leading order.

A tessellation is a covering by tiles with pairwise disjoint interiors; a locally finite tessellation meets any compact set in only finitely many tiles. In a tessellation by congruent compact hyperbolic polygons, the adjacency graph records side-crossing steps. The area growth $\mathrm{Area}B(r)=2\pi (\cosh r-1)$ and bounded tile diameter imply exponential growth of the number of tiles within $n$ steps: a unit geodesic segment meets a uniformly bounded number of tiles, by packing congruent interior balls, so a metric ball of radius proportional to $n$ lies in the step ball. Conversely the step ball lies in a metric ball of radius proportional to $n$. The regular right-angled pentagons with four around every vertex give the tiling $\{5,4\}$.

A tessellation by rectangular tiles in which no point is a corner of four tiles. In the common lattice version, the tiles are $1\times 2$ rectangles. The local corner restriction makes existence depend on more than the total area.

In the Poincare disc model, a hyperbolic circle in the Poincare disc consists of points at one fixed hyperbolic distance from a center. At center zero it is the Euclidean circle $|z|=\tanh (R/2)$, since radial hyperbolic distance is $2\mathrm{artanh}|z|$. A Poincare disc automorphism sending a general center to zero transports this description. Conversely every Euclidean circle entirely inside the disc is such a circle: the product of inversion in that circle and unit-circle inversion has reciprocal fixed points, one inside; sending the inner fixed point to zero makes the circle concentric with the unit circle.

The complete simply connected Riemannian manifold of dimension $n$ with constant sectional curvature $-1$ is hyperbolic space. In the upper half-space model its metric is $d{s}^2=(d{x}_1^2+\cdots +d{x}_n^2)/x_n^2$ for $x_n>0$. Its ideal boundary in dimension three is the Riemann sphere.

In the positive Hermitian determinant-one model of three-dimensional hyperbolic space, the base point is $I$ and a matrix $g\in SL(2,\mathbb{C})$ sends it to $g{g}^{\ast }$. The distance obeys $2\cosh d(I,g{g}^{\ast })=\mathrm{tr}(g{g}^{\ast })$. Bounded displacement therefore bounds every matrix entry, and the determinant-one condition gives a closed bounded subset of a finite-dimensional real vector space. Passing to $PSL(2,\mathbb{C})$ preserves compactness. Conjugating moves any chosen base point to $I$. A discrete subgroup meets each such compact set in finitely many elements, proving escape of distinct elements and proper discontinuity, including finite stabilizers.

For a nonempty finite set $\{p_1,\dots ,p_m\}$ in finite-dimensional hyperbolic space, the continuous function $R(x)={\max }_{j}d(x,p_j)$ attains its minimum. Indeed, a bounded candidate radius confines all possible improving centers to a closed bounded ball about $p_1$. Such a ball is compact in the Poincare ball model, since its points remain away from the ideal boundary. Minimizing $R$ on it supplies a closed ball of smallest enclosing radius. More generally this argument works for any nonempty bounded set in a proper metric space when the supremum-distance function is finite.

An isometry of hyperbolic space with positive attained translation length has an invariant complete geodesic joining its two ideal fixed points. This is its axis. It translates along the axis by its translation length; in three dimensions it may additionally rotate around it. In upper half-space, $(z,t)\mapsto (kz,|k|t)$ with $|k|\ne 1$ has the vertical axis through zero and length $|\log |k||$. Every connecting path has length at least the change in $\log t$, and a vertical segment attains the bound.

The unit ball in Euclidean space with the displayed Riemannian metric models hyperbolic space of curvature minus one. Its ideal boundary is the unit sphere, and its geodesics are diameters or circle arcs orthogonal to that sphere. The metric is invariant under the conformal ball automorphisms.

A Möbius transformation of the ideal two-sphere extends uniquely to an orientation-preserving isometry of three-dimensional hyperbolic space. In upper half-space, translations, complex similarities, and $z\mapsto -1/z$ extend respectively to $(z,t)\mapsto (z+b,t)$, $(\lambda z,|\lambda |t)$, and $(-\overline{z},t)/(|z{|}^2+t^2)$. Each preserves $(|dz{|}^2+d{t}^2)/t^2$; composing these extensions constructs the general case. Ball-origin stabilizers are ordinary rotations, represented on the boundary by $SU(2)$ modulo its central signs.

A six-sided polygon in the hyperbolic plane whose sides are geodesic segments. The especially useful right-angled case is a right-angled hyperbolic hexagon.

A geodesic hexagon in the hyperbolic plane with all six interior angles $\pi /2$. Any three positive alternating side lengths determine it up to isometry. If those sides are ${\ell }_i/2$, the opposite seam satisfies $\cosh s_i=(\cosh ({\ell }_i/2)+\cosh ({\ell }_j/2)\cosh ({\ell }_k/2))/(\sinh ({\ell }_j/2)\sinh ({\ell }_k/2))$. Two congruent such hexagons glued along the other three sides form a pair of pants with boundary lengths ${\ell }_1,{\ell }_2,{\ell }_3$.

The Beltrami-Klein model realizes the hyperbolic plane inside an open Euclidean disk, with geodesics given by straight chords. Its metric differs from the Euclidean metric, and angles are generally distorted. The straight chords make geodesic convexity especially easy to see.

A surface with a complete Riemannian metric of constant Gaussian curvature $-1$. Its local model is the Poincaré half-plane model; the orientable case is a quotient by a free properly discontinuous group action of orientation-preserving isometries.

Suppose a finite simple polygon in the curvature-minus-one hyperbolic plane has hyperbolic geodesic sides paired to produce a closed topological surface. Its metric descends to a smooth hyperbolic metric exactly when paired polygon sides have equal lengths and the sum of internal angles in each vertex equivalence class is $2\pi$. Equal lengths allow isometric gluing at open edges; at vertices the sectors assemble into a hyperbolic disc precisely when their total angle is $2\pi$. Other positive total angles produce a cone singularity of a surface metric. The area of a hyperbolic geodesic polygon and the Gauss-Bonnet theorem then give area $-2\pi \chi$ for the closed surface.

For a marked closed hyperbolic surface, the representation of its fundamental group whose image acts as the deck group on the hyperbolic plane. It is discrete and faithful, determined up to conjugation by the marking and the metric, and gives $X={\mathbb{H}}^2/\rho ({\pi }_1{\Sigma }_g)$. For a closed orientable surface it admits determinant-one lifts. It is different from the tangent-space action of the Riemannian holonomy group.

The unmarked length spectrum of a compact hyperbolic surface is the multiset of lengths of its closed geodesics, with a consistent convention for primitive curves and orientation. The marked version labels lengths by free homotopy classes. The Selberg trace formula relates the unmarked multiset to the Laplace-Beltrami operator spectrum.

For a closed hyperbolic surface with a specified marking of its fundamental group, assign to each nontrivial conjugacy class the length of its unique closed geodesic. Retaining the group label is stronger than retaining only the unmarked length multiset: it also records which lengths belong to intersecting curves or powers of particular curves.

For closed hyperbolic surfaces of curvature $-1$, equality of the Laplace-Beltrami operator spectra, including multiplicities, is equivalent to equality of the unmarked length spectra of closed geodesics. The Selberg trace formula proves the equivalence. Primitive lengths with multiplicities and the multiset including their iterates determine one another, provided the same orientation convention is used.

In a compact smooth family of marked closed hyperbolic surfaces, only finitely many free homotopy classes of closed geodesics have length below a fixed bound somewhere in the family. A uniform bilipschitz equivalence with a fixed metric reduces this to local finiteness of its length spectrum.

For fixed genus $g\ge 2$ and systole lower bound $\epsilon >0$, a uniform finite length cutoff $L$ determines the entire unmarked length spectrum of a closed hyperbolic surface in the thick part, including multiplicities. Equality up to $L$ forces equality everywhere. This is a finite-determination statement for spectra and does not claim that all isospectral surfaces are isometric.

Suppose a compact parameter set carries a locally finite countable multiset of positive quantities, and a fixed injective transform of each labelled quantity is a polynomial in finitely many ambient coordinates. Assume only finitely many labels can have bounded value anywhere in the compact set. Then a uniform finite initial multiset determines the entire multiset. For each source label there are finitely many possible matching labels. Equality for finitely many labels in both directions is a finite union of algebraic sets, obtained from injective matchings and polynomial equalities. These algebraic sets decrease as more labels are included, so they stabilize by the Hilbert basis theorem. A compatible infinite matching in each direction follows by the König infinity lemma, and local finiteness makes multiplicities equal. Squared traces of hyperbolic holonomy representations give the application to the Buser finite length spectrum theorem.

For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.

A collar around a simple closed geodesic of length $\ell$ is an embedded annulus with metric $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, where $t\in \mathbb{R}/\mathbb{Z}$. The collar lemma gives half-width $w=\mathrm{arcsinh}(1/\sinh (\ell /2))$. As $\ell$ tends to zero the collar gets wide, forcing crossing geodesics to become long.

In curvature $-1$, its length is $2\mathrm{arcosh}3$. The level-two principal congruence subgroup has traces congruent to two modulo four, so the smallest hyperbolic absolute trace is six, achieved by $\left(\begin{array}{cc}5& 2\\ 2& 1\end{array}\right)$. The hyperbolic translation length formula gives the result; the shortest closed geodesic is nonsimple.

A collection of disjoint essential simple closed curves that cuts a closed genus-$g$ surface into $2g-2$ surfaces of type pair of pants. There are $3g-3$ curves. Geodesic representatives are used for Fenchel–Nielsen coordinates.

An elementary move replaces one cuff inside the four-holed sphere or one-holed torus obtained by uniting its adjacent pants. The replacement is an essential curve meeting the old cuff minimally. It changes the pants decomposition graph without changing the underlying surface; the identity surface map identifies the two descriptions.

The graph of a pants decomposition has one vertex per pair of pants and one edge per glued cuff. It is a connected trivalent multigraph, possibly with loops. A closed surface of genus $g$ gives $2g-2$ vertices and $3g-3$ edges, so its first Betti number is $g$. The graph records a decomposition rather than the homeomorphism class alone, and omits cuff lengths and twists.

For fixed $g\ge 2$, a constant $B_g$ bounds all cuff lengths in some pants decomposition of every closed genus-$g$ hyperbolic surface. Together with Fenchel–Nielsen coordinates this implies Mumford's compactness theorem.

For a fixed pants decomposition, the positive cuff lengths and real twist parameters give coordinates on Teichmüller space. With twists measured in length units, a full Dehn twist changes ${\tau }_i$ by ${\ell }_i$; with angle units it changes the twist by $2\pi$.

After cutting a hyperbolic surface along a simple closed geodesic, reglue its two boundary copies after an arclength displacement $\tau$. The boundary length remains fixed, while the Riemannian metric varies. In marked Fenchel–Nielsen coordinates, the twist is real-valued; changing it by one boundary length corresponds to a Dehn twist in the marking. This construction also applies to a hyperbolic pair of pants with a fixed cone angle and two geodesic boundaries.

For a fixed pants decomposition of a closed genus-$g$ hyperbolic surface, its $3g-3$ cuff lengths determine all its Y-pieces. Two suitably chosen transverse simple closed curves at each cuff, one obtained from the other by a Dehn twist, determine the corresponding twist: hyperbolic seam identities express their lengths through shifted hyperbolic cosines of that twist, removing the sign ambiguity of just one transversal. Thus finitely many marked lengths determine the Fenchel–Nielsen coordinates. The unmarked length spectrum lacks these labels, which is why this fact alone does not imply global spectral uniqueness.

A simple closed curve that is a geodesic of length $\ell$ on a hyperbolic surface has an embedded collar of half-width $w$ with $\sinh w\sinh (\ell /2)=1$. In circumference-one coordinates its metric is $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, and its conformal modulus of an annulus is $(2/\ell )\arctan (\mathrm{csch}(\ell /2))\sim \pi /\ell$.

The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.

The metric area of a hyperbolic surface. In the Poincaré half-plane model the density is $dxdy/y^2$. A finite-area torsion-free quotient of genus $g$ with $r$ cusps has area $2\pi (2g-2+r)$ by the Gauss-Bonnet theorem.

A hyperbolic metric is a complete Riemannian metric of constant sectional curvature $-1$. On a surface it supplies local charts modeled on the hyperbolic plane.

A connected Riemann surface whose universal cover is the unit disc inherits the complete curvature-minus-one metric displayed on its cover. Möbius transformations preserving the disc are isometries, so deck transformations preserve the metric and it descends to the surface. This uniformization notion must be distinguished from the potential-theoretic existence of a Green function. Holomorphic maps between such surfaces contract their metric by lifting to Schwarz-Pick theorem on the covers.

Under the Cayley transform between the half-plane and disk, a radial segment ending at a boundary point becomes a vertical half-ray. If a point is within hyperbolic distance $k$ of that half-ray, it lies in a fixed angular sector. Indeed, in the complex upper half-plane, $\cosh \rho (w,iy)=(|w{|}^2+y^2)/(2y\mathrm{Im}w)\ge |w|/\mathrm{Im}w$. This converts bounded hyperbolic approach to a boundary point into nontangential approach, allowing normal-family proof of angular boundary convergence to transfer radial limits of bounded holomorphic functions.

For a nonconstant holomorphic map into the unit disc, the displayed density pulls the target hyperbolic metric back to the source Euclidean coordinate. It is not the operator norm between two hyperbolic tangent spaces, which additionally divides by the source metric density. Away from ramification points $\Delta \log H_g=H_g^2$; a ramification point of multiplicity $m$ adds $2\pi m$ times its point mass to the distributional Laplacian.

This logarithmic area integral measures the accumulated pullback hyperbolic metric area of a holomorphic disc map. Its nonnegative kernel makes it increasing in $R$. Schwarz-Pick theorem bounds $H_g(z)$ by $2/(1-|z{|}^2)$, which controls its characteristic through a critical-point Jensen identity.

For $g^{\prime }(0)\ne 0$, apply Jensen's formula with the smooth curvature term to the hyperbolic derivative density. Its distributional Laplacian is $H_g^2+2\pi \sum m_a{\delta }_a$, giving the displayed identity with ramification points defined by $g^{\prime }(a)=0$. If $g^{\prime }(z)=c{z}^m+\cdots$ at zero, use $m\log R$ as the origin counting term and replace the base constant by $\log {\lim }_{z\to 0}{H}_g(z)/|z{|}^m$. Without this regularization the basepoint-critical formula is undefined.

The universal covering map $\tau \mapsto e^{i\tau }$ from the complex upper half-plane to the punctured unit disc pulls back the metric $|dz|/(|z|\log (1/|z|))$ to $|d\tau |/\mathrm{Im}\tau$. This is its complete curvature-minus-one hyperbolic metric, which is different from restricting the ordinary disk metric to the punctured disk.

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The curvature-minus-one hyperbolic metric on the punctured disk assigns distance $2\log n$ along the radial geodesic from $e^{-1}$ to $e^{-n^2}$. Its exponential distances therefore sum to a finite value; the triangle inequality gives the same conclusion for any fixed basepoint. Nevertheless no nonzero bounded holomorphic function has these zeros: the Riemann removable singularity theorem extends it at zero, and the identity theorem then makes it identically zero. Simple connectivity is necessary in the hyperbolic characterization of bounded analytic zero sets.

On the unit disc, the pseudohyperbolic distance isIt is invariant under disc Möbius transformations. For the curvature-minus-one hyperbolic metric with length element $2|dz|/(1-|z{|}^2)$, $\delta (z,w)=\tanh (\rho (z,w)/2)$ and $e^{-\rho (z,w)}=(1-\delta )/(1+\delta )$. The Schwarz-Pick theorem says that holomorphic functions from the disc into itself contract this distance. A Blaschke factor has modulus $\delta (z,a)$, so the distance also controls the size of a product with prescribed zeros.

The hyperbolic plane is the simply connected two-dimensional geometry of constant curvature minus one.

A geodesic polygon in the hyperbolic plane is bounded by finitely many geodesic segments. For curvature $-1$, its area is its angle defect, as given by hyperbolic polygon area. Pairing equally long edges produces hyperbolic surfaces, with smoothness at glued vertices determined by their total angles.

An ideal hyperbolic quadrilateral is the region bounded by four hyperbolic geodesics whose four consecutive endpoints are distinct points on the ideal boundary. The ideal vertices lie outside the hyperbolic plane. In the hyperbolic disk, the four vertices $1,i,-1,-i$ give a regular example: its sides are the interior arcs of the circles of radius one centered at $\pm 1\pm i$. Although the sides have infinite length, the quadrilateral has finite hyperbolic area $2\pi$ in curvature $-1$.

A hyperbolic disc consists of points within a specified hyperbolic distance of its centre. In the Poincare disc model, the disc centred at zero with radius $\rho$ has Euclidean radius $\tanh (\rho /2)$; an isometry transports this description to any other centre.

A horocycle in the Poincare disc model is a Euclidean circle internally tangent to the boundary circle. In the upper half-plane model, a horocycle is a Euclidean circle tangent to the real axis from above, or a horizontal line $y=h>0$. The common geometric feature is a single ideal centre of a horocycle.

Two horocycles have a unique common orthogonal hyperbolic line when their ideal centres of horocycles differ: it is the line with those two ideal endpoints. If the centres agree, an isometry puts both horocycles on horizontal lines; every vertical hyperbolic line meets both orthogonally, so there are infinitely many. This classification remains valid for intersecting or tangent horocycles.

The ideal centre of a horocycle is its point of tangency to the boundary at infinity; a horizontal horocycle in the upper half-plane model has centre infinity. Sending the centre to infinity by an isometry makes the horocycle horizontal. Orthogonal hyperbolic lines then become vertical, proving that a hyperbolic line meets a horocycle orthogonally exactly when it has the centre as an ideal endpoint.

A hyperbolic circle of centre $o$ and radius $R>0$ is the level set $\{x:d_{\mathbb{H}}(o,x)=R\}$ for the hyperbolic distance. Equally spaced directions from its centre give vertices of a regular geodesic polygon.

In the Poincaré half-plane model, a hyperbolic circle of radius $R$ centred at $u+iv$ is the Euclidean circle $(x-u{)}^2+(y-v\cosh R{)}^2=v^2{\sinh }^{2}R$. Its Euclidean centre is shifted upward from its hyperbolic centre, and its Euclidean radius is $v\sinh R$. The lower height $v{e}^{-R}$ is positive. The formula follows by mapping the centre to zero in the Poincare disc model and using radial distance $2\mathrm{artanh}\rho$.

A hyperbolic line is a complete geodesic in the hyperbolic plane. In the Poincare half-plane model, hyperbolic lines are vertical rays and semicircles orthogonal to the boundary.

An ideal endpoint of a hyperbolic line is one of its two limiting points on the boundary at infinity. In the Poincare disc model these lie on the unit circle; in the upper half-plane model they lie on $\mathbb{R}\cup \{\infty \}$. These are boundary points, not points of the hyperbolic plane. Any two distinct ideal endpoints determine a unique hyperbolic line.

Two distinct hyperbolic lines are parallel, in the limiting-parallel convention, when they do not intersect in the hyperbolic plane and have exactly one common ideal endpoint. In the upper half-plane model, distinct vertical lines are parallel, sharing infinity. Disjoint lines with no common ideal endpoint are instead ultraparallel hyperbolic lines; only these have a common perpendicular of ultraparallel hyperbolic lines.

Two hyperbolic lines are ultraparallel when they are disjoint and have no common ideal endpoint. They possess a unique common perpendicular, which realizes the distance between them.

Hyperbolic distance is the Riemannian distance induced by the hyperbolic metric.

A horoball in the hyperbolic plane is a region bounded by a horocycle. In the upper-half-plane model, a horoball based at infinity is a Euclidean half-plane $\{y>c\}$; quotients by parabolic isometries give cusp neighborhoods.

A noncentral finite-order orientation-preserving hyperbolic isometry fixes an interior point and is elliptic.

The Poincare disc carries the metric $4|dz{|}^2/(1-|z{|}^2{)}^2$ on $|z|<1$.

Transferring the upper half-plane hyperbolic metric $|dw{|}^2/(\mathrm{Im}w{)}^2$ through the Cayley transform gives $d{s}^2=4|dz{|}^2/(1-|z{|}^2{)}^2$ on the unit disc. Radial integration gives distance $2\mathrm{artanh}r$ from zero to $r$. The alternative convention $d{\tilde{s}}^2=|dz{|}^2/(1-|z{|}^2{)}^2$ divides the metric tensor by four and every length by two, giving $\mathrm{artanh}r$. Both have the same isometries and unparametrized geodesics, but their Gaussian curvatures are respectively minus one and minus four. In general scaling a Riemannian metric by $c>0$ scales lengths by $\sqrt{c}$ and curvature by $1/c$.

For $|a|<1$, the Möbius transformation $f_a(z)=(z-a)/(1-\overline{a}z)$ is an isometry of the Poincare disc model sending $a$ to zero. Rotations combined with these transformations describe its orientation-preserving isometries.

In curvature minus one, a hyperbolic disc of radius $R$ has areaIn the Poincare disc this follows by integrating $4r(1-r^2{)}^{-2}drd\theta$ up to Euclidean radius $\tanh (R/2)$.

If a domain of finite hyperbolic area $A$ contains a hyperbolic disc of radius $r$ in curvature $-1$, then $2\pi (\cosh r-1)\le A$, hence $r\le \mathrm{arcosh}(1+A/(2\pi ))$. For an interior point of a hyperbolic triangle, the open disc with radius equal to its distance from the boundary lies inside the triangle. Its area is at most $\pi$ by Gauss-Bonnet theorem, giving the elementary bound $r\le \mathrm{arcosh}(3/2)$; this need not be the sharp triangle inradius bound.

Hyperbolic discs of area $\pi /2$ have radius $\log 2$. Starting at the origin and placing their centers successively along one radial geodesic gives center distance $n\log 4$ and Euclidean coordinate

An isometry sends the centers of a collinear family of equal hyperbolic discs to points on one hyperbolic geodesic. In the upper-half-plane model no geodesic contains three distinct points on a horizontal Euclidean line, since geodesics are vertical lines or semicircles orthogonal to the boundary.

For a piecewise smooth curve $z(t)$ in the Poincare disc, its hyperbolic length is

The geodesics are diameters and arcs of circles orthogonal to the unit circle. A radial segment is length minimizing becausewith equality for constant angle and monotone radius. Hence the distance from the origin to $z$ is $2\mathrm{artanh}|z|$.

For $z_1,z_2$ in the Poincare disc model, putTheir hyperbolic distance is

For a proper arc $I$ of the boundary of the unit disc, $Q(I)$ is the side of the hyperbolic geodesic between its endpoints adjacent to $I$. For short arcs this is a curved boundary cap. For arcs longer than a semicircle it contains the center. With endpoints $e^{\pm i\alpha }$ and $0<\alpha <\pi /2$, the geodesic is the circle with center $\sec \alpha$ and radius $\tan \alpha$, orthogonal to the unit circle. It is a diameter at $\alpha =\pi /2$.

For a short arc $I$ of normalized length $\ell$, every point of its hyperbolic geodesic cap lies within Euclidean distance $C\ell$ of its center boundary point. Conversely, for any fixed $b>0$, the region with angular projection in $I$ and depth $1-|z|\le b\ell$ lies in $Q(\kappa I)$ for a fixed enlargement $\kappa$ depending only on $b$, when $I$ is sufficiently short. To verify this, rotate the midpoint to one and write the endpoints as $e^{\pm i\alpha }$. The cap inequality is $2\mathrm{Re}z>(1+|z{|}^2)\cos \alpha$. Its radial depth at the midpoint is $1-\sec \alpha +\tan \alpha$, comparable to $\alpha$ for small $\alpha$; its angular projection lies in $I$. Enlarging $\alpha$ by a sufficiently large fixed factor makes the inequality hold throughout the indicated rectangle. Large arcs use a finite cover of the disc. Thus Carleson boxes and hyperbolic geodesic caps define equivalent Carleson measure conditions.

The rectangle containment can also be checked directly. Write $z=r{e}^{i\theta }$ with $|\theta |\le \pi \ell$ and $1-r\le b\ell$. Its membership in the enlarged cap is equivalent toFor small enough $\ell$, $r\ge 1/2$. The right side is at most $b^2{\ell }^2$, while the left side is at least $({\kappa }^2-1){\pi }^2{\ell }^2/8$, using the sine product formula and $\sin x\ge x/2$ for small positive $x$. Thus $\kappa =4+4b/\pi$ suffices.

If $P$ is at hyperbolic distance $\rho$ from the origin, the Poincare-disc geodesic through $P$ perpendicular to $OP$ is represented by a Euclidean circle of radius $\mathrm{csch}\rho$ whose centre is at Euclidean distance $\coth \rho$ from the origin.

The upper half-plane carries the hyperbolic metric $(d{x}^2+d{y}^2)/y^2$.

For a hyperbolic Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$, the infimum of $d(z,Az)$ is attained on its axis and equals $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. It is the length of the corresponding closed geodesic in a quotient.

For a determinant-one lift of a hyperbolic holonomy representation, the squared matrix trace of each fixed group word is a polynomial in the finitely many generator-matrix entries. The displayed identity identifies equal positive lengths with equal squared traces. Lift signs have no effect. Finite choices of length matchings therefore give finite unions of affine algebraic sets.

For $z,w$ in the upper half-plane,

For points $ib$ and $a+ib$ with $a,b>0$, the hyperbolic distance is $2\mathrm{arsinh}(a/(2b))$. It equals their Euclidean distance $a$ exactly when the displayed relation holds. The function of $a$ decreases continuously from one to zero: its derivative has the sign of $\sinh (a/2)-(a/2)\cosh (a/2)<0$. Hence every height $b\in (0,1)$ has exactly one positive matching horizontal separation.

The geodesics in the upper half-plane are vertical lines and semicircles centred on the boundary axis. Indeed, the length integrand $\sqrt{1+y^{\prime 2}}/y$ has the Beltrami identitywhich integrates to $(x-c{)}^2+y^2=R^2$. Swapping the coordinate axes gives circles centred on the vertical boundary for the metric $(d{x}^2+d{y}^2)/x^2$.

In the upper half-plane $y>0$, the hyperbolic length element is $ds/y$. The function $\Phi =\mathrm{arsinh}(x/y)$ has gradient $(1,-x/y)/\sqrt{x^2+y^2}$ and norm $1/y$. The Cauchy-Schwarz inequality gives $\int ds/y\ge \Phi (B)-\Phi (A)$ for any oriented path from $A$ to $B$. Equality holds along a semicircle $x^2+y^2=R^2$ traversed with increasing $x$, because its tangent is parallel to that gradient. This is a calibration proof of global minimality, beyond the stationary-path conclusion of the Beltrami identity. Translating $x$ gives the analogous proof for semicircles with other boundary centers.

Two disjoint hyperbolic lines with four distinct ideal endpoints have a unique common perpendicular, and the segment of that perpendicular between them realizes their minimum distance. An upper-half-plane isometry can make the two lines concentric semicircles; their common perpendicular then becomes the imaginary axis.

Three pairwise ultraparallel hyperbolic lines may have disjoint common perpendiculars or coincident ones. In the upper-half-plane model, three semicircles of radius $1/2$ centred at $-5/2,0,5/2$ have mutually disjoint common perpendiculars. Three concentric semicircles instead have the same vertical common perpendicular. Thus pairwise ultraparallelism alone does not determine the incidence of their perpendiculars.

The hyperbolic metric in the half-plane model is conformal, so hyperbolic and Euclidean angles agree. The line meeting the imaginary axis at $iy$, $y>0$, at signed angle $\alpha \in (0,\pi )$ is represented by the Euclidean circle with centre $y\cot \alpha$ on the real axis and radius $y\csc \alpha$.

PSL2(R) is $S{L}_2(\mathbb{R})/\{\pm I\}$. It acts on the upper half-plane by the Möbius transformationsand is the group of orientation-preserving hyperbolic isometries.

A real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|>2$. It has two fixed ideal boundary points and translates along the geodesic joining them; its hyperbolic translation length is $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. Conjugating by a real Möbius transformation puts it in the form $z\mapsto e^{\ell }z$ with $\ell >0$; its axis is then the imaginary axis.

A real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|<2$. It fixes a point of the Poincaré half-plane model; an integral matrix of this type has trace $0$ or $\pm 1$. In the Poincare disc model, moving its fixed point to zero makes it a rotation about zero.

A nonidentity real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|=2$. It has one ideal boundary fixed point and is conjugate to a nonzero translation on the Poincaré half-plane model.

Every orientation-preserving isometry of the Poincaré half-plane model lies in $PS{L}_2(\mathbb{R})$. Every orientation-reversing isometry is the composition of one of these with the hyperbolic reflection $z\mapsto -\overline{z}$ in the imaginary axis. Equivalently, the full isometry group consists of real Möbius and anti-Möbius transformations, modulo nonzero scalar matrices.

The group $PS{L}_2(\mathbb{R})$ acts transitively on pairs $(\ell ,P)$ consisting of a hyperbolic line and a point on it. Send the two ideal endpoints of $\ell$ to $0$ and $\infty$, then use a positive dilation to move the resulting point on the imaginary axis to $i$.

Reflection in a hyperbolic line is the unique orientation-reversing hyperbolic isometry fixing that line pointwise. The product of reflections in intersecting lines is a rotation through twice their angle.

A reflection in the perpendicular bisector of a point and its image corrects that point. A second bisector reflection through the corrected point can correct another point without moving the first. An isometry fixing two distinct points is either the identity or reflection in their joining geodesic: points with prescribed distances from both centres have at most two possibilities, exchanged by that reflection. Fixing a third noncollinear point then fixes every point. This constructs a factorization with at most three hyperbolic reflections.

The composition of hyperbolic reflections in two ultraparallel hyperbolic lines has infinite order. Map their common perpendicular to the imaginary axis in the Poincaré half-plane model. The two reflecting hyperbolic lines become semicircles centred at zero, with distinct radii $r_1,r_2$. Reflection in the radius-$r$ semicircle is $z\mapsto r^2/\overline{z}$, so their composition is $z\mapsto (r_2/r_1{)}^{2}z$. Every positive power remains nonidentity, because $(r_2/r_1{)}^{2k}\ne 1$ for $k\ge 1$.

A hyperbolic half-turn is the unique orientation-preserving isometry of order two fixing a given point. It exchanges every pair of points lying at equal opposite distances along a geodesic through its centre.

The map $z\mapsto i(1+z)/(1-z)$ is an isometry from the Poincare disc to the upper half-plane.

The hyperboloid model of the curvature-minus-one hyperbolic plane isIts metric is the restriction of the Minkowski bilinear form to tangent planes, and its geodesics are intersections with two-dimensional linear subspaces through the origin.

The three-dimensional hyperboloid model isIt is a regular level set with

This indefinite bilinear form supplies the metric of Minkowski spacetime. On ${\mathbb{R}}^{n+1}$ the Minkowski inner product with mostly-positive convention is

A Lambert quadrilateral is a hyperbolic quadrilateral with three right angles. If its remaining angle is $\theta$ and the two sides incident with that angle have lengths $a,b$, then

A hyperbolic triangle is bounded by three hyperbolic geodesics and may have vertices on the ideal boundary.