Action of \({\operatorname{PSL}}_2({\mathbb{R}})\) on the Upper Half Plane

g * z: GrpPSL2Elt, SpcHypElt -> SpcHypElt

g * z: GrpPSL2Elt, [SpcHypElt] -> [SpcHypElt]

g * z: GrpPSL2Elt, SetCspElt -> SetCspElt

g * z: RngIntElt, SpcHypElt -> SpcHypElt

g * z: RngIntElt, [SpcHypElt] -> [SpcHypElt]

g * z: RngIntElt, SetCspElt -> SetCspElt

For \(z\) of type SpcHypElt, SetCspElt or [SpcHypElt], and when \(g\) is an element of a projective linear group, returns the image of \(z\) under the action of \(g\). The type of the image is the same as the type of \(z\). If \(g\) is a positive integer, returns return \(az\), which his is equivalent to acting on \(z\) with the matrix \(\begin{pmatrix}a & 0\\ 0 & 1\end{pmatrix}\in {\operatorname{PGL}}_2({\mathbb{R}})\).

FixedPoints(g, H): GrpPSL2Elt, SpcHyp -> SeqEnum

Returns a sequence of points in \(H\) fixed by the action of \(g\).

IsEquivalent(G, a, b): GrpPSL2, SpcHypElt, SpcHypElt -> BoolElt, GrpPSL2Elt

If points \(a,b\) in the upper half plane are equivalent under the action of the group \(G\), returns true, and the matrix \(g\) in \(G\) such that \(g\cdot a = b\). Otherwise returns false and the identity.

EquivalentPoint(x): SpcHypElt -> SpcHypElt, GrpPSL2Elt

For the point x in the upper half plane, returns a point z in the region with \(-1/2 < z \leq 1/2\) and \(|z| \geq 1\), and a matrix \(g\) in \({\operatorname{PSL}}_2({\mathbb{Z}})\) with g*x = z

Stabilizer(a, G): SpcHypElt, GrpPSL2 -> GrpPSL2Elt

Returns a generator of the subgroup of \(G\) stabilizing \(a\).

FixedArc(g, H): GrpPSL2Elt, SpcHyp -> SeqEnum

If \(g\) is an element of \({\operatorname{PSL}}_2({\mathbb{Z}})\) which is an involution, this returns the end points in the real line of the arc fixed by \(g\), with mid point of the arc also fixed by \(g\). Note that for any point \(b\), the arc from \(b\) to \(g\cdot b\) is fixed by \(g\).

Arithmetic

z + a: SpcHypElt, RngIntElt -> SpcHypElt

z + a: SpcHypElt, FldRatElt -> SpcHypElt

z - a: SpcHypElt, RngIntElt -> SpcHypElt

z - a: SpcHypElt, FldRatElt -> SpcHypElt

For any integer \(a\), and element \(z\) in the upper half plane, this returns the element \(z + a\) in the same copy of the upper half plane.

a * z: RngElt, SpcHypElt -> SpcHypElt

z * a: SpcHypElt, RngElt -> SpcHypElt

a * z: RngIntElt, SetCspElt -> SetCspElt

a * z: FldRatElt, SetCspElt -> SetCspElt

a * seq: RngElt, [SpcHypElt] -> [SpcHypElt]

a * z: RngIntElt, [SetCspElt] -> [SetCspElt]

a * z: FldRatElt, [SetCspElt] -> [SetCspElt]

z * a: SpcHypElt, RngIntElt -> SpcHypElt

z * a: SpcHypElt, RngIntElt -> SpcHypElt

z * a: SpcHypElt, RngIntElt -> SpcHypElt

z * a: SpcHypElt, RngIntElt -> SpcHypElt

z * a: SpcHypElt, RngIntElt -> SpcHypElt

z / a: SpcHypElt, RngIntElt -> SpcHypElt

Given an element z (or a sequence of elements) in the upper half plane, and a positive rational number \(a\), this returns the product (or products) in the same copy of the upper half plane.

Distances, Angles and Geodesics

Distance(z, w): SpcHypElt, SpcHypElt -> FldReElt

Precision: RngIntElt                    Default: 

Returns the hyperbolic distance between \(z\) and \(w\).

TangentAngle(x, y): SpcHypElt, SpcHypElt -> FldReElt

Precision: RngIntElt                    Default: 

Returns the angle of the tangent at \(x\) of the geodescic from \(x\) to \(y\), with given precision.

Angle(e1, e2): [SpcHypElt], [SpcHypElt] -> FldReElt

Precision: RngIntElt                    Default: 

Given two sequences \(e_1 = [z_1,z_2]\) and \(e_2 = [z_1,z_3]\), where \(z1, z2, z3\) are elements of the upper half plane, this returns the angle between the geodesics at \(z_1\).

ExtendGeodesic([z1,z2], H): [SpcHypElt], SpcHyp -> [SpcHypElt]

Given elements \(z1, z2\) in the upper half plane \(H\), this extends the geodesic between \(z1\) and \(z2\) to a semicircle with endpoints on the real line, and returns the two real endpoints as elements of \(H\).

GeodesicsIntersection(x1, x2): [SpcHypElt], [SpcHypElt]) -> SeqEnum

GeodesicsIntersection(x1, x2): [SetCspElt], [SetCspElt]) -> SeqEnum

The intersection in the upper half plane of the two geodesics whose endpoints are given by the sequences \(x_1\) and \(x_2\). If the geodesics intersect along a line, the empty sequence is returned.