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	{{short description\|Conformal ~~map~~ in complex analysis}}		{{short description\|Conformal mappings in complex analysis}}

	{{Confusing\|date=January 2019}}
			]
	{{Complex analysis sidebar}}		{{Complex analysis sidebar}}
	In ], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that ] the ] to a triangle in the upper half plane having lines or circular arcs for edges. ~~Let~~ ~~''πα'',~~ ~~''πβ'',~~ ~~and~~ ~~''πγ''~~ be ~~the~~ ~~interior~~ ~~angles~~ at the ~~vertices~~ of ~~the~~ ~~triangle~~. If ~~any~~ of ~~''α,~~ ~~β'',~~ ~~and~~ ~~''γ''~~ ~~are~~ ~~greater~~ ~~than~~ ~~zero~~, ~~then~~ the Schwarz triangle function ~~can~~ be ~~given~~ in ~~terms~~ of ] ~~as:~~		In ], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that ] the ] to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a ], although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a ], the inverse of the Schwarz triangle function is a ] ] for that triangle's ]. More specifically, it is a ].

			==Formula==
	:<math>s(z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>
			Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in ]. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari,{{sfn\|Nehari\|1975\|page=309}} these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of ] as:

			:<math>s(\alpha, \beta, \gamma; z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>
	where ''a'' = (1−α−β−γ)/2, ''b'' = (1−α+β−γ)/2, ''c'' = 1−α, ''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2, ''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and ''c''′ = 2 − ''c'' = 1 + α. This mapping has singular points at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,
			where
			:''a'' = (1−α−β−γ)/2,
			:''b'' = (1−α+β−γ)/2,
			:''c'' = 1−α,
			:''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2,
			:''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and
			:''c''′ = 2 − ''c'' = 1 + α.

			This function maps the upper half-plane to a ] if α + β + γ > 1, or a ] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a'' = 0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the ].
	:<math>\begin{aligned}

	s(0) &= 0, \\
			===Derivation===
			Through the theory of complex ]s with ]s and the ], the triangle function can be expressed as the quotient of two solutions of a ] with real coefficients and singular points at 0, 1 and ∞. By the ], the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the ] of the ordinary differential equation and induces a group of ]s on quotients of ]s.{{sfn\|Nehari\|1975\|pp=198-208}}

			== Singular points ==
			This mapping has ] at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,{{sfn\|Nehari\|1975\|pages=315−316}}
			:<math>\begin{align}
			s(0) &= 0, \\
	s(1) &= \frac		s(1) &= \frac
	{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}		{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}
	{\Gamma(1-a)\Gamma(1-b)\Gamma(c)},~~\,\text{and}~~ \\		{\Gamma(1-a)\Gamma(1-b)\Gamma(c)}, \\
	s(\infty) &= \exp\left(i \pi \alpha \right)\frac		s(\infty) &= \exp\left(i \pi \alpha \right)\frac
	{\Gamma(1-a')\Gamma(b)\Gamma(c')}		{\Gamma(1-a')\Gamma(b)\Gamma(c')}
	{\Gamma(1-a)\Gamma(b')\Gamma(c)}.		{\Gamma(1-a)\Gamma(b')\Gamma(c)},
	\end{~~aligned~~}</math>		\end{align}</math>

			where <math display=inline>\Gamma(x)</math> is the ].
	This formula can be derived using the ].

			Near each singular point, the function may be approximated as
	This function can be used to map the upper half-plane to a ] on the ] if α + β + γ > 1, or a ] on the ] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a'' = 0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the ]. In the special case of ]s, where all the angles are zero, the triangle function yields the ].

			:<math>\begin{align}
	This function was introduced by ] as the inverse function of the ] uniformizing a ]. Applying successive hyperbolic reflections in its sides, such a triangle generates a ] of the upper half plane (or the unit disk after composition with the ]). The conformal mapping of the upper half plane onto the interior of the ] generalizes the ]. By the ], the discrete group generated by hyperbolic reflections in the sides of the triangle induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the ] of the ordinary differential equation and induces a group of ]s on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an ] for this discrete group of Möbius transformations. This is a special case of a general method of ] that associates automorphic forms with ]s with ]s.
			s_0(z) &= z^\alpha (1+O(z)), \\
			s_1(z) &= (1-z)^\gamma (1+O(1-z)), \\
			s_\infty(z) &= z^\beta (1+O(1/z)),
			\end{align}</math>

			where <math>O(x)</math> is ].
	==Hyperboloid and Klein models==
	]

			== Inverse ==
	In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane.<ref>See:
			When ''α, β'', and ''γ'' are rational, the triangle is a Schwarz triangle. When each of ''α, β'', and ''γ'' are either the reciprocal of an integer or zero, the triangle is a ], i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a ].
	*{{harvnb\|Busemann\|1955}}
	*{{harvnb\|Magnus\|1974}}
	*{{harvnb\|Helgason\|2000}}
	*{{harvnb\|Wolf\|2011}}</ref>

			In the spherical case, that modular function is a ]. For Euclidean triangles, the inverse can be expressed using ]s.<ref name=Lee />
	The group ''G'' = SU(1,1) is formed of matrices

			== Ideal triangles ==
	:<math> g = \begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha}\end{pmatrix}
			When ''α'' = 0 the triangle is degenerate, lying entirely on the real line. If either of ''β'' or ''γ'' are non-zero, the angles can be permuted so that the positive value is ''α'', but that is not an option for an ] having all angles zero.
	</math>

			Instead, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the ]:
	with
			:<math>i\frac{K(1-z)}{K(z)}</math>.
			This expression is the inverse of the ].{{sfn\|Nehari\|1975\|pp=316-318}}

			== Extensions ==
	:<math> \|\alpha\|^2 -\|\beta\|^2=1.</math>
			The ] gives the mapping from the upper half-plane to any Euclidean polygon.

			The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an ''n''-sided polygon, the solution has ''n-3'' additional parameters, which are difficult to determine in practice.{{sfn\|Nehari\|1975\|p=202}} See {{slink\|Schwarzian derivative#Conformal mapping of circular arc polygons}} for more details.
	It is a subgroup of ''G''<sub>''c''</sub> = SL(2,'''C'''), the group of complex 2 × 2 matrices with determinant 1.
	The group ''G''<sub>''c''</sub> acts by Möbius transformations on the extended complex plane. The subgroup ''G'' acts as automorphisms of the unit disk ''D'' and the subgroup ''G''<sub>1</sub> = SL(2,'''R''') acts as automorphisms of the ]. If

	:<math>C=\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}</math>

	then

	:<math>G=CG_1C^{-1},</math>

	since the Möbius transformation corresponding ''M'' is the ] carrying the upper half plane onto the unit disk and the real line onto the unit circle.

	The Lie algebra <math>\mathfrak{g}</math> of SU(1,1) consists of matrices

	:<math> X= \begin{pmatrix} ix & w \\ \overline{w} & -ix \end{pmatrix}</math>

	with ''x'' real. Note that ''X''<sup>2</sup> = (\|''w''\|<sup>2</sup> – ''x''<sup>2</sup>) ''I'' and

	:<math> \det X = x^2 -\|w\|^2 =-\tfrac12 \operatorname{Tr} X^2.</math>

	The hyperboloid <math>\mathfrak{H}</math> in <math>\mathfrak{g}</math> is defined by two conditions. The first is that det ''X'' = 1 or equivalently Tr ''X''<sup>2</sup> = –2. By definition this condition is preserved under ] by ''G''. Since ''G'' is connected it leaves the two components with ''x'' > 0 and ''x'' < 0 invariant. The second condition is that ''x'' > 0. For brevity, write ''X'' = (''x'',''w'').

	The group ''G'' acts transitively on ''D'' and <math>\mathfrak{H}</math> and the points 0 and (1,0) have ] ''K'' consisting of matrices

	:<math>\begin{pmatrix}\zeta & 0 \\ 0 & \overline{\zeta}\end{pmatrix}</math>

	with \|ζ\| = 1. Polar decomposition on ''D'' implies the Cartan decomposition ''G'' = ''KAK'' where ''A'' is the group of matrices

	:<math>a_t = \begin{pmatrix}\cosh t & \sinh t \\ \sinh t & \cosh t\end{pmatrix}.</math>

	Both spaces can therefore be identified with the homogeneous space ''G''/''K'' and there is a ''G''-equivariant map ''f'' of <math>\mathfrak{H}</math> onto ''D'' sending (1,0) to 0. To work out the formula for this map and its inverse it suffices to compute ''g''(1,0) and
	''g''(0) where ''g'' is as above. Thus ''g''(0) = β/{{overline\|α}} and

	:<math>g(1,0)=(\|\alpha\|^2 +\|\beta\|^2,-2i\alpha\beta),</math>

	so that

	:<math>\frac\beta\overline{\alpha} = \frac{\alpha\beta} {\|\alpha\|{}^2} = \frac{2\alpha\beta}{\|\alpha\|{}^2 + \|\beta\|{}^2 + 1},</math>

	recovering the formula

	:<math>f(x,w) = \frac{iw}{x + 1}.</math>

	Conversely if ''z'' = ''iw''/(''x'' + 1), then \|''z''\|<sup>2</sup> = (''x'' – 1)/(''x'' + 1), giving the inverse formula

	:<math> f^{-1}(z) = (x,w) = \left( \frac{1 + \|z\|{}^2}{1 - \|z\|{}^2}, \frac{-2iz}{1 - \|z\|{}^2} \right).</math>

	This correspondence extends to one between geometric properties of ''D'' and <math>\mathfrak{H}</math>. Without entering into the correspondence of ''G''-invariant ]s,{{efn\|1=The Poincaré metric on the disk corresponds to the restriction of the ''G''-invariant pseudo-Riemannian metric ''dx''<sup>2</sup> – ''dw''<sup>2</sup> to the hyperboloid.}} each geodesic circle in ''D'' corresponds to the intersection of 2-planes through the origin, given by equations Tr ''XY'' = 0, with <math>\mathfrak{H}</math>. Indeed, this is obvious for rays arg ''z'' = θ through the origin in ''D''—which correspond to the 2-planes arg ''w'' = θ—and follows in general by ''G''-equivariance.

	The Klein model is obtained by using the map ''F''(''x'',''w'') = ''w''/''x'' as the correspondence between <math>\mathfrak{H}</math> and ''D''. Identifying this disk with (1,''v'') with \|''v''\| < 1, intersections of 2-planes with <math>\mathfrak{H}</math> correspond to intersections of the same 2-planes with this disk and so give straight lines. The Poincaré-Klein map given by

	:<math>K(z) = iF \circ f^{-1}(z) = \frac{2z}{1 + \|z\|{}^2}</math>

	thus gives a diffeomorphism from the unit disk onto itself such that Poincaré geodesic circles are carried into straight lines. This diffeomorphism does not preserve angles but preserves orientation and, like all diffeomorphisms, takes smooth curves through a point making an angle less than {{pi}} (measured anticlockwise) into a similar pair of curves.{{efn\|1=The condition on tangent vectors '''x''', '''y''' is given by det ('''x''','''y''') ≥ 0 and is preserved because the determinant of the Jacobian is positive.}} In the limiting case, when the angle is {{pi}}, the curves are tangent and this again is preserved under a diffeomorphism. The map ''K'' yields the ''Klein model'' of hyperbolic geometry. The map extends to a homeomorphism of the unit disk onto itself which is the identity on the unit circle. Thus by continuity the map ''K'' extends to the endpoints of geodesics, so carries the arc of the circle in the disc cutting the unit circle orthogonally at two given points on to the straight line segment joining those two points.
	(Note that on the unit circle the radial derivative of ''K'' vanishes, so that the condition on angles no longer applies there.)

	==Convex polygons==
	]
	]
	In this section the main results on convexity of hyperbolic polygons are deduced from the corresponding results for Euclidean polygons by considering the relation between Poincaré's disk model and the Klein model. A polygon in the unit disk or upper half plane is made up of a collection of a finite set of vertices joined by geodesics, such that none of the geodesics intersect. In the Klein model this corresponds to the same picture in the Euclidean model with straight lines between the vertices. In the Euclidean model the polygon has an interior and exterior (by an elementary version of the ]), so, since this is preserved under homeomorphism, the same is true in the Poincaré picture.

	As a consequence at each vertex there is a well-defined notion of interior angle.

	In the Euclidean plane a polygon with all its angles less than {{pi}} is convex, i.e. the straight line joining interior points of the polygon also lies in the interior of the polygon. Since the Poincaré-Klein map preserves the property that angles are less than {{pi}}, a hyperbolic polygon with interior angles less than {{pi}} is carried onto a Euclidean polygon with the same property; the Euclidean polygon is therefore convex and hence, since hyperbolic geodesics are carried onto straight lines, so is the hyperbolic polygon. By a continuity argument, geodesics between points on the sides also lie in the closure of the polygon.

	A similar convexity result holds for polygons which have some of their vertices on the boundary of the disk or the upper half plane. In fact each such polygon is an increasing union of polygons with angles less than {{pi}}. Indeed, take points on the edges at each ideal vertex tending to the two edges joining those points to the ideal point with the geodesic joining them. Since two interior points of the original polygon will lie in the interior of one of these smaller polygons, each of which is convex, the original polygon must also be convex.<ref>{{harvnb\|Magnus\|1974\|page=37}}</ref>

	==Tessellation by Schwarz triangles==
	In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles without "cusps"—angles equal to zero or equivalently vertices on the real axis—the elementary approach of {{harvtxt\|Carathéodory\|1954}} will be followed. For triangles with one or two cusps, elementary arguments of {{harvtxt\|Evans\|1973}}, simplifying the approach of {{harvtxt\|Hecke\|1935}}, will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a ]. For an ideal triangle in which all angles are zero, so that all vertices lie on the real axis, the existence of the tessellation will be established by relating it to the ] described in {{harvtxt\|Hardy\|Wright\|2008}} and {{harvtxt\|Series\|2015}}. In this case the tessellation can be considered as that associated with three touching circles on the ], a limiting case of configurations associated with three disjoint non-nested circles and their reflection groups, the so-called "]s", described in detail in {{harvtxt\|Mumford\|Series\|Wright\|2015}}. Alternatively—by dividing the ideal triangle into six triangles with angles 0, {{pi}}/2 and {{pi}}/3—the tessellation by ideal triangles can be understood in terms of tessellations by triangles with one or two cusps.

	===Triangles without cusps===
	]
	]
	Suppose that the ] Δ has angles {{pi}}/''a'', {{pi}}/''b'' and {{pi}}/''c'' with ''a'', ''b'', ''c'' integers greater than 1. The hyperbolic area of Δ equals {{pi}} – {{pi}}/''a'' – {{pi}}/''b'' – {{pi}}/''c'', so that

	:<math>\frac1a + \frac1b + \frac1c < 1.</math>

	The construction of a tessellation will first be carried out for the case when ''a'', ''b'' and ''c'' are greater than 2.<ref>{{harvnb\|Carathéodory\|1954\|pages=177–181}}</ref>

	The original triangle Δ gives a convex polygon ''P''<sub>1</sub> with 3 vertices. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2''m'' copies of the triangle where the angle at the vertex is {{pi}}/''m''. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. The union of these new triangles together with the original triangle form a connected shape ''P''<sub>2</sub>. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than or equal to {{pi}} and each side being the edge of a reflected triangle. In the case when angle of Δ equals
	{{pi}}/3, a vertex of ''P''<sub>2</sub> will have an interior angle of {{pi}}, but this does not affect the convexity of ''P''<sub>2</sub>. Even in this degenerate case when an angle of {{pi}} arises, the two collinear edge are still considered as distinct for the purposes of the construction.

	The construction of ''P''<sub>2</sub> can be understood more clearly by noting that some triangles or tiles are added twice, the three which have a side in common with the original triangle. The rest have only a vertex in common. A more systematic way of performing the tiling is first to add a tile to each side (the reflection of the triangle in that edge) and then fill in the gaps at each vertex. This results in a total of 3 + (2''a'' – 3) + (2''b'' - 3) + (2''c'' - 3) = 2(''a'' + ''b'' + ''c'') - 6 new triangles. The new vertices are of two types. Those which are vertices of the triangles attached to sides of the original triangle, which are connected to 2 vertices of Δ. Each of these lie in three new triangles which intersect at that vertex. The remainder are connected to a unique vertex of Δ and belong to two new triangles which have a common edge. Thus there are 3 + (2''a'' – 4) + (2''b'' - 4) + (2''c'' - 4) = 2(''a'' + ''b'' + ''c'') - 9 new vertices. By construction there is no overlapping. To see that ''P''<sub>2</sub> is convex, it suffices to see that the angle between sides meeting at a new vertex make an angle less than or equal to {{pi}}. But the new vertices lies in two or three new triangles, which meet at that vertex, so the angle at that vertex is no greater than 2{{pi}}/3 or {{pi}}, as required.

	This process can be repeated for ''P''<sub>2</sub> to get ''P''<sub>3</sub> by first adding tiles to each edge of ''P''<sub>2</sub> and then filling in the tiles round each vertex of ''P''<sub>2</sub>. Then the process can be repeated from ''P''<sub>3</sub>, to get ''P''<sub>4</sub> and so on, successively producing ''P''<sub>''n''</sub> from ''P''<sub>''n'' – 1</sub>. It can be checked inductively that these are all convex polygons, with non-overlapping tiles.
	Indeed, as in the first step of the process there are two types of tile in building ''P''<sub>''n''</sub> from ''P''<sub>''n'' – 1</sub>, those attached to an edge of ''P''<sub>''n'' – 1</sub> and those attached to a single vertex. Similarly there are two types of vertex, one in which two new tiles meet and those in which three tiles meet. So provided that no tiles overlap, the previous argument shows that angles at vertices are no greater than {{pi}} and hence that ''P''<sub>''n''</sub> is a convex polygon.{{efn\|1=As in the case of ''P''<sub>2</sub>, if an angle of Δ equals {{pi}}/3, vertices where the interior angle is {{pi}} stay marked as vertices and colinear edges are not coallesced.}}

	It therefore has to be verified that in constructing ''P''<sub>''n''</sub> from ''P''<sub>''n'' − 1</sub>:<ref>{{harvnb\|Carathéodory\|1954\|pages=178−180}}</ref>

	{{quote box\|align=left\|
	(a) the new triangles do not overlap with ''P''<sub>''n'' − 1</sub> except as already described;

	(b) the new triangles do not overlap with each other except as already described;

	(c) the geodesic from any point in Δ to a vertex of the polygon ''P''<sub>''n'' – 1</sub> makes an angle ≤ 2{{pi}}/3 with each of the edges of the polygon at that vertex.}}
	{{clear}}
	To prove (a), note that by convexity, the polygon ''P''<sub>''n'' − 1</sub> is the intersection of the convex half-spaces defined by the full circular arcs defining its boundary. Thus at a given vertex of ''P''<sub>''n'' − 1</sub> there are two such circular arcs defining two sectors: one sector contains the interior of ''P''<sub>''n'' − 1</sub>, the other contains the interiors of the new triangles added around the given vertex. This can be visualized by using a Möbius transformation to map the upper half plane to the unit disk and the vertex to the origin; the interior of the polygon and each of the new triangles lie in different sectors of the unit disk. Thus (a) is proved.

	Before proving (c) and (b), a Möbius transformation can be applied to map the upper half plane to the unit disk and a fixed point in the interior of Δ to the origin.

	The proof of (c) proceeds by induction. Note that the radius joining the origin to a vertex of the polygon ''P''<sub>''n'' − 1</sub> makes an angle of less than 2{{pi}}/3 with each of the edges of the polygon at that vertex if exactly two triangles of ''P''<sub>''n'' − 1</sub> meet at the vertex, since each has an angle less than or equal to {{pi}}/3 at that vertex. To check this is true when three triangles of ''P''<sub>''n'' − 1</sub> meet at the vertex, ''C'' say, suppose that the middle triangle has its base on a side ''AB'' of ''P''<sub>''n'' − 2</sub>. By induction the radii ''OA'' and ''OB'' makes angles of less than or equal to 2{{pi}}/3 with the edge ''AB''. In this case the region in the sector between the radii ''OA'' and ''OB'' outside the edge ''AB'' is convex as the intersection of three convex regions. By induction the angles at ''A'' and ''B'' are greater than or equal to {{pi}}/3. Thus the geodesics to ''C'' from ''A'' and ''B'' start off in the region; by convexity, the triangle ''ABC'' lies wholly inside the region. The quadrilateral ''OACB'' has all its angles less than {{pi}} (since ''OAB'' is a geodesic triangle), so is convex. Hence the radius ''OC'' lies inside the angle of the triangle ''ABC'' near ''C''. Thus the angles between ''OC'' and the two edges of {{math\|''P''<sub>''n'' – 1</sub>}} meeting at ''C'' are less than or equal to {{pi}}/3 + {{pi}}/3 = 2{{pi}}/3, as claimed.

	To prove (b), it must be checked how new triangles in ''P''<sub>''n''</sub> intersect.

	First consider the tiles added to the edges of ''P''<sub>''n'' – 1</sub>. Adopting similar notation to (c), let ''AB'' be the base of the tile and ''C'' the third vertex. Then the radii ''OA'' and ''OB'' make angles of less than or equal to 2{{pi}}/3 with the edge ''AB'' and the reasoning in the proof of (c) applies to prove that the triangle ''ABC'' lies within the sector defined by the radii ''OA'' and ''OB''. This is true for each edge of ''P''<sub>''n'' – 1</sub>. Since the interiors of sectors defined by distinct edges are disjoint, new triangles of this type only intersect as claimed.

	Next consider the additional tiles added for each vertex of ''P''<sub>''n'' – 1</sub>. Taking the vertex to be ''A'', three are two edges ''AB''<sub>1</sub> and ''AB''<sub>2</sub> of ''P''<sub>''n'' – 1</sub> that meet at ''A''. Let ''C''<sub>1</sub> and ''C''<sub>2</sub> be the extra vertices of the tiles added to these edges. Now the additional tiles added at ''A'' lie in the sector defined by radii ''OB''<sub>1</sub> and ''OB''<sub>2</sub>. The polygon with vertices ''C''<sub>2</sub> ''O'', ''C''<sub>1</sub>, and then the vertices of the additional tiles has all its internal angles less than {{pi}} and hence is convex. It is therefore wholly contained in the sector defined by the radii ''OC''<sub>1</sub> and ''OC''<sub>2</sub>. Since the interiors of these sectors are all disjoint, this implies all the claims about how the added tiles intersect.
	]
	]

	Finally it remains to prove that the tiling formed by the union of the triangles covers the whole of the upper half plane. Any point ''z'' covered by the tiling lies in a polygon ''P''<sub>''n''</sub> and hence a polygon ''P''<sub>''n'' +1 </sub>. It therefore lies in a copy of the original triangle Δ as well as a copy of ''P''<sub>2</sub> entirely contained in ''P''<sub>''n'' +1 </sub>. The hyperbolic distance between Δ and the exterior of ''P''<sub>2</sub> is equal to ''r'' > 0. Thus the hyperbolic distance between ''z'' and points not coverered by the tiling is at least ''r''. Since this applies to all points in the tiling, the set covered by the tiling is closed. On the other hand, the tiling is open since it coincides with the union of the interiors of the polygons ''P''<sub>''n''</sub>. By connectivity, the tessellation must cover the whole of the upper half plane.

	To see how to handle the case when an angle of Δ is a right angle, note that the inequality

	:<math>\frac1a + \frac1b + \frac1c < 1</math>.

	implies that if one of the angles is a right angle, say ''a'' = 2, then both ''b'' and ''c'' are greater than 2 and one of them, ''b'' say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles {{pi}}/''c'', {{pi}}/''c'' and 2{{pi}}/''b''. If 2{{pi}}/''b'' ≤ {{pi}}/3, i.e. ''b'' is greater than 5, then all the angles of the doubled triangle are less than or equal to {{pi}}/3. In that case the construction of the tessellation above through increasing convex polygons adapts word for word to this case except that around the vertex with angle 2{{pi}}/''b'', only ''b''—and not 2''b''—copies of the triangle are required to tile a neighborhood of the vertex. This is possible because the doubled triangle is isosceles. The tessellation for the doubled triangle yields that for the original triangle on cutting all the larger triangles in half.<ref name="Cara1954">{{harvnb\|Carathéodory\|1954\|pages=181–182}}</ref>

	It remains to treat the case when ''b'' equals 4 or 5. If ''b'' = 4, then ''c'' ≥ 5: in this case if ''c'' ≥ 6, then ''b'' and ''c'' can be switched and the argument above applies, leaving the case ''b'' = 4 and ''c'' = 5. If ''b'' = 5, then ''c'' ≥ 4. The case ''c'' ≥ 6 can be handled by swapping ''b'' and ''c'', so that the only extra case is ''b'' = 5 and ''c'' = 5. This last isosceles triangle is the doubled version of the first exceptional triangle, so only that triangle Δ<sub>1</sub>—with angles {{pi}}/2, {{pi}}/4 and {{pi}}/5 and hyperbolic area {{pi}}/20—needs to be considered (see below). {{harvtxt\|Carathéodory\|1954}} handles this case by a general method which works for all right angled triangles for which the two other angles are less than or equal to {{pi}}/4. The previous method for constructing ''P''<sub>2</sub>, ''P''<sub>3</sub>, ... is modified by adding an extra triangle each time an angle 3{{pi}}/2 arises at a vertex. The same reasoning applies to prove there is no overlapping and that the tiling covers the hyperbolic upper half plane.<ref name="Cara1954" />

	On the other hand, the given configuration gives rise to an arithmetic triangle group. These were first studied in {{harvtxt\|Fricke\|Klein\|1897}}. and have given rise to an extensive literature. In 1977 Takeuchi obtained a complete classification of arithmetic triangle groups (there are only finitely many) and determined when two of them are commensurable. The particular example is related to ] and the arithmetic theory implies that the triangle group for Δ<sub>1</sub> contains the triangle group for the triangle Δ<sub>2</sub> with angles {{pi}}/4, {{pi}}/4 and {{pi}}/5 as a non-normal subgroup of index 6.<ref>See:
	*{{harvnb\|Takeuchi\|1977a}}
	*{{harvnb\|Takeuchi\|1977b}}
	*{{harvnb\|Weber\|2005}}
	</ref>

	Doubling the triangles Δ<sub>1</sub> and Δ<sub>2</sub>, this implies that there should be a relation between 6 triangles Δ<sub>3</sub> with angles {{pi}}/2, {{pi}}/5 and {{pi}}/5 and hyperbolic area {{pi}}/10 and a triangle Δ<sub>4</sub> with angles {{pi}}/5, {{pi}}/5 and {{pi}}/10 and hyperbolic area 3{{pi}}/5. {{harvtxt\|Threlfall\|1932}} established such a relation directly by completely elementary geometric means, without reference to the arithmetic theory: indeed as illustrated in the fifth figure below, the quadrilateral obtained by reflecting across a side of a triangle of type Δ<sub>4</sub> can be tiled by 12 triangles of type Δ<sub>3</sub>. The tessellation by triangles of the type Δ<sub>4</sub> can be handled by the main method in this section; this therefore proves the existence of the tessellation by triangles of type Δ<sub>3</sub> and Δ<sub>1</sub>.<ref>See:
	*{{harvnb\|Threlfall\|1932\|pages=20–22}}, Figure 9
	*{{harvnb\|Weber\|2005}}
	</ref>

	{{gallery\| width=300px
	\|File:H2checkers 255.png\|<small>Tessellation by triangles with angles {{pi}}/2, {{pi}}/5 and {{pi}}/5</small>
	\|File:H2-5-4-rhombic.svg\|<small>Tessellation obtained by coalescing two triangles</small>
	\|File:Threlfall-245.jpeg\|<small>Tiling with pentagons formed from 10 (2,5,5) triangles</small>
	\|File:Threlfall-245-a.jpeg\|<small>Adjusting to tiling by triangles with angles {{pi}}/5, {{pi}}/10, {{pi}}/10</small>
	\|File:Threlfall-245-b.jpeg\|<small>Tiling 2 (5,10,10) triangles with 12 (2,5,5) triangles</small>
	}}
	{{clear}}

	===Triangles with one or two cusps===
	In the case of a Schwarz triangle with one or two cusps, the process of tiling becomes simpler; but it is easier to use a different method going back to ] to prove that these exhaust the hyperbolic upper half plane.

	In the case of one cusp and non-zero angles {{pi}}/''a'', {{pi}}/''b'' with ''a'', ''b'' integers greater than one, the tiling can be envisaged in the unit disk with the vertex having angle {{pi}}/''a'' at the origin. The tiling starts by adding 2''a'' – 1 copies of the triangle at the origin by successive reflections. This results in a polygon ''P''<sub>1</sub> with 2''a'' cusps and between each two 2''a'' vertices each with an angle {{pi}}/''b''. The polygon is therefore convex. For each non-ideal vertex of ''P''<sub>1</sub>, the unique triangle with that vertex can be similar reflected around that vertex, thus adding 2''b'' – 1 new triangles, 2''b'' – 1 new ideal points and 2 ''b'' – 1 new vertices with angle {{pi}}/''a''. The resulting polygon ''P''<sub>2</sub> is thus made up of 2''a''(2''b'' – 1) cusps and the same number of vertices each with an angle of {{pi}}/''a'', so is convex. The process can be continued in this way to obtain convex polygons ''P''<sub>3</sub>, ''P''<sub>4</sub>, and so on. The polygon ''P''<sub>''n''</sub> will have vertices having angles alternating between 0 and {{pi}}/''a'' for ''n'' even and between 0 and {{pi}}/''b'' for ''n'' odd. By construction the triangles only overlap at edges or vertices, so form a tiling.<ref>{{harvnb\|Carathéodory\|1954\|page=183}}</ref>

	{{gallery\|width=300px
	\|File:H2checkers 35i.png\|<small>Tessellation by triangle with angles 0, {{pi}}/3, {{pi}}/5</small>
	\|File:H2checkers 25i.png\|<small>Tessellation by triangle with angles 0, {{pi}}/5, {{pi}}/2</small>
	\|File:H2checkers 5ii.png\|<small>Tessellation by triangle with angles 0, 0, {{pi}}/5</small>}}
	The case where the triangle has two cusps and one non-zero angle {{pi}}/''a'' can be reduced to the case of one cusp by observing that the trinale is the double of a triangle with one cusp and non-zero angles {{pi}}/''a'' and {{pi}}/''b'' with ''b'' = 2. The tiling then proceeds as before.<ref>{{harvnb\|Carathéodory\|1954\|page=184}}</ref>

	To prove that these give tessellations, it is more convenient to work in the upper half plane. Both cases can be treated simultaneously, since the case of two cusps is obtained by doubling a triangle with one cusp and non-zero angles {{pi}}/''a'' and {{pi}}/2. So consider the geodesic triangle in the upper half plane with angles 0, {{pi}}/''a'', {{pi}}/''b'' with ''a'', ''b'' integers greater than one. The interior of such a triangle can be realised as the region ''X'' in the upper half plane lying outside the unit disk \|''z''\| ≤ 1 and between two lines parallel to the imaginary axis through points ''u'' and ''v'' on the unit circle. Let Γ be the triangle group generated by the three reflections in the sides of the triangle.

	To prove that the successive reflections of the triangle cover the upper half plane, it suffices to show that for any ''z'' in the upper half plane there is a ''g'' in Γ such that ''g''(''z'') lies in {{overline\|''X''}}. This follows by an argument of {{harvtxt\|Evans\|1973}}, simplified from the theory of ]s. Let λ = Re ''a'' and μ = Re ''b'' so that, without loss of generality, λ < 0 ≤ μ. The three reflections in the sides are given by

	:<math>R_1(z) = \frac1\overline{z},\ R_2(z) = -\overline{z} + \lambda,\ R_3(z)= -\overline{z} + \mu.</math>

	Thus ''T'' = ''R''<sub>3</sub>∘''R''<sub>2</sub> is translation by μ − λ. It follows that for any ''z''<sub>1</sub> in the upper half plane, there is an element ''g''<sub>1</sub> in the subgroup Γ<sub>1</sub> of Γ generated by ''T'' such that ''w''<sub>1</sub> = ''g''<sub>1</sub>(''z''<sub>1</sub>) satisfies λ ≤ Re ''w''<sub>1</sub> ≤ μ, i.e. this strip is a ] for the translation group Γ<sub>1</sub>. If \|''w''<sub>1</sub>\| ≥ 1, then ''w''<sub>1</sub> lies in ''X'' and the result is proved. Otherwise let ''z''<sub>2</sub> = ''R''<sub>1</sub>(''w''<sub>1</sub>) and find ''g''<sub>2</sub>
	Γ<sub>1</sub> such that ''w''<sub>2</sub> = ''g''<sub>2</sub>(''z''<sub>2</sub>) satisfies λ ≤ Re ''w''<sub>2</sub> ≤ μ. If \|''w''<sub>2</sub>\| ≥ 1 then the result is proved. Continuing in this way, either some ''w''<sub>''n''</sub> satisfies \|''w''<sub>''n''</sub>\| ≥ 1, in which case the result is proved; or \|''w''<sub>''n''</sub>\| < 1 for all ''n''. Now since ''g''<sub>''n'' + 1</sub> lies in Γ<sub>1</sub> and \|''w''<sub>''n''</sub>\| < 1,

	:<math>
	\operatorname{Im} g_{n+1}(z_{n+1})
	= \operatorname{Im} z_{n+1}
	= \operatorname{Im} \frac{w_n}{\|w_n\|{}^2}
	= \frac{\operatorname{Im} w_n}{\|w_n\|{}^2}.
	</math>

	In particular

	:<math>\operatorname{Im} w_{n+1} \ge \operatorname{Im} w_n</math>

	and

	:<math>\frac{\operatorname{Im} w_{n+1}}{\operatorname{Im} w_n} = \|w_n\|^{-2} \ge 1.</math>

	Thus, from the inequality above, the points (''w''<sub>''n''</sub>) lies in the compact set \|''z''\| ≤ 1, λ ≤ Re ''z'' ≤ μ and Im ''z'' ≥ Im ''w''<sub> 1</sub>. It follows that \|''w''<sub>''n''</sub>\| tends to 1; for if not, then there would be an ''r'' < 1 such that \|''w''<sub>''m''</sub>\| ≤ ''r'' for inifitely many ''m'' and then the last equation above would imply that Im ''w''<sub>''n''</sub> tends to infinity, a contradiction.

	Let ''w'' be a limit point of the ''w''<sub>''n''</sub>, so that \|''w''\| = 1. Thus ''w'' lies on the arc of the unit circle between ''u'' and ''v''. If ''w'' ≠ ''u'', ''v'', then ''R''<sub>1</sub> ''w''<sub>''n''</sub> would lie in ''X'' for ''n'' sufficiently large, contrary to assumption. Hence ''w'' =''u'' or ''v''. Hence for ''n'' sufficiently large ''w''<sub>''n''</sub> lies close to ''u'' or ''v'' and therefore must lie in one of the reflections of the triangle about the vertex ''u'' or ''v'', since these fill out neighborhoods of ''u'' and ''v''. Thus there is an element ''g'' in Γ such that ''g''(''w''<sub>''n''</sub>) lies in {{overline\|''X''}}. Since by construction ''w''<sub>''n''</sub> is in the Γ-orbit of ''z''<sub>1</sub>, it follows that there is a point in this orbit lying in {{overline\|''X''}}, as required.<ref>See:
	*{{harvnb\|Evans\|1973\|pages=108−109}}
	*{{harvnb\|Berndt\|Knopp\|2008\|pages=16−17}}</ref>

	===Ideal triangles===
	The tessellation for an ] with all its vertices on the unit circle and all its angles 0 can be considered as a special case of the tessellation for a triangle with one cusp and two now zero angles {{pi}}/3 and {{pi}}/2. Indeed, the ideal triange is made of six copies one-cusped triangle obtained by reflecting the smaller triangle about the vertex with angle {{pi}}/3.
	{{gallery\|width=300px
	\|File:H2checkers_23i.png\|<small>Tessellation for triangle with angles 0, {{pi}}/3 and {{pi}}/2</small>
	\|File:H2chess_23ib.png\|<small>Tessellation for ideal triangle</small>
	\|File:Ideal-triangle hyperbolic tiling.svg\|<small>Second realisation of tessellation for ideal triangle</small>
	\|File:Ideal-triangle hyperbolic tiling line-drawing.svg\|<small>Line drawing of tessellation by ideal triangles</small>}}
	]
	]
	Each step of the tiling, however, is uniquely determined by the positions of the new cusps on the circle, or equivalently the real axis; and these points can be understood directly in terms of ] following {{harvtxt\|Series\|2015}}, {{harvtxt\|Hatcher\|2013}} and {{harvtxt\|Hardy\|Wright\|2008}}. This starts from the basic step that generates the tessellation, the reflection of an ideal triangle in one of its sides. Reflection corresponds to the process of inversion in projective geometry and taking the ], which can be defined in terms of the ]. In fact if ''p'', ''q'', ''r'', ''s'' are distinct points in the Riemann sphere, then there is a unique complex Möbius transformation ''g'' sending ''p'', ''q'' and ''s'' to 0, ∞ and 1 respectively. The cross ratio (''p'', ''q''; ''r'', ''s'') is defined to be ''g''(''r'') and is given by the formula

	:<math>(p, q; r, s) = \frac{(p-r)(q-s)}{(p-s)(q-r)}.</math>

	By definition it is invariant under Möbius transformations. If ''a'', ''b'' lie on the real axis, the harmonic conjugate of ''c'' with respect to ''a'' and ''b'' is defined to be the unique real number ''d'' such that (''a'', ''b''; ''c'', ''d'') = −1. So for example if ''a'' = 1 and ''b'' = –1, the conjugate of ''r'' is 1/''r''. In general Möbius invariance can be used to obtain an explicit formula for ''d'' in terms of ''a'', ''b'' and ''c''. Indeed, translating the centre ''t'' = (''a'' + ''b'')/2 of the circle with diameter having endpoints ''a'' and ''b'' to 0, ''d'' – ''t'' is the harmonic conjugate of ''c'' – ''t'' with respect to ''a'' - ''t'' and ''b'' – ''t''. The radius of the circle is ρ = (''b'' – ''a'')/2 so (''d'' - ''t'')/ρ is the harmonic conjugate of {{math\|(''c'' – ''t'')/ρ}} with respect to 1 and -1. Thus

	:<math>\frac{d-t}\rho = \frac\rho{c-t}</math>

	so that

	:<math>d = \frac{\rho^2}{r-t} + t = \frac{(c-a)b + (c-b)a}{(c-a) + (c-b)}.</math>

	It will now be shown that there is a parametrisation of such ideal triangles given by rationals in reduced form

	:<math>a = \frac{p_1}{q_1},\ b = \frac{p_1 + p_2}{q_1 + q_2},\ c = \frac{p_2}{q_2}</math>

	with ''a'' and ''c'' satisfying the "neighbour condition" ''p''<sub>2</sub>''q''<sub>1</sub> − ''q''<sub>2</sub>''p''<sub>1</sub> = 1.

	The middle term ''b'' is called the ''Farey sum'' or '']'' of the outer terms and written

	:<math>b = a \oplus c.</math>

	The formula for the reflected triangle gives

	:<math>d = \frac{p_1 + 2p_2}{q_1 + 2q_2} = a \oplus b.</math>

	Similarly the reflected triangle in the second semicircle gives a new vertex ''b'' ⊕ ''c''. It is immediately verified that ''a'' and ''b'' satisfy the neighbour condition, as do ''b'' and ''c''.

	Now this procedure can be used to keep track of the triangles obtained by successively reflecting the basic triangle Δ with vertices 0, 1 and ∞. It suffices to consider the strip with 0 ≤ Re z ≤ 1, since the same picture is reproduced in parallel strips by applying reflections in the lines Re ''z'' = 0 and 1. The ideal triangle with vertices 0, 1, ∞ reflects in the semicircle with base into the triangle with vertices ''a'' = 0, ''b'' = 1/2, ''c'' = 1. Thus ''a'' = 0/1 and ''c'' = 1/1 are neighbours and ''b'' = ''a'' ⊕ ''c''. The semicircle is split up into two smaller semicircles with bases and . Each of these intervals splits up into two intervals by the same process, resulting in 4 intervals. Continuing in this way, results into subdivisions into 8, 16, 32 intervals, and so on. At the ''n''th stage, there are 2<sup>''n''</sup> adjacent intervals with 2<sup>''n''</sup> + 1 endpoints. The construction above shows that successive endpoints satisfy the neighbour condition so that new endpoints resulting from reflection are given by the Farey sum formula.

	To prove that the tiling covers the whole hyperbolic plane, it suffices to show that every rational in eventually occurs as an endpoint. There are several ways to see this. One of the most elementary methods is described in {{harvtxt\|Graham\|Knuth\|Patashnik\|1994}} in their development—without the use of ]s—of the theory of the ], which codifies the new rational endpoints that appear at the ''n''th stage. They give ] that every rational appears. Indeed, starting with {0/1,1/1}, successive endpoints are introduced at level ''n''+1 by adding Farey sums or mediants {{math\|(''p''+''r'')/(''q''+''s'')}} between all consecutive terms {{math\|''p''/''q''}}, {{math\|''r''/''s''}} at the ''n''th level (as described above). Let {{math\|1=''x'' = ''a''/''b''}} be a rational lying between 0 and 1 with {{math\|''a''}} and {{math\|''b''}} coprime. Suppose that at some level {{math\|''x''}} is sandwiched between successive terms {{math\|''p''/''q'' < ''x'' < ''r''/''s''}}. These inequalities force {{math\|''aq'' – ''bp'' ≥ 1}} and
	{{math\|''br'' – ''as'' ≥ 1}} and hence, since {{math\|1=''rp'' – ''qs'' = 1}},

	:<math>a + b = (r+s)(ap-bq) + (p+q)(br -as) \ge p+q+r+s.</math>

	This puts an upper bound on the sum of the numerators and denominators. On the other hand, the mediant {{math\|(''p''+''r'')/(''q''+''s'')}} can be introduced and either equals {{math\|''x''}}, in which case the rational {{math\|''x''}} appears at this level; or the mediant provides a new interval containing {{math\|''x''}} with strictly larger numerator-and-denominator sum. The process must therefore terminate after at most {{math\|''a'' + ''b''}} steps, thus proving that {{math\|''x''}} appears.<ref>{{harvnb\|Graham\|Knuth\|Patashnik\|1994\|page=118}}</ref>

	A second approach relies on the ] ''G'' = SL(2,'''Z''').<ref>{{harvnb\|Series\|2015}}</ref> The Euclidean algorithm implies that this group is generated by the matrices

	:<math>S=\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix},\,\,\, T=\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}.</math>

	In fact let ''H'' be the subgroup of ''G'' generated by ''S'' and ''T''. Let

	:<math>g=\begin{pmatrix} a & b\\ c & d\end {pmatrix}</math>

	be an element of SL(2,'''Z'''). Thus ''ad'' − ''cb'' = 1, so that ''a'' and ''c'' are coprime. Let

	:<math>v=\begin{pmatrix}a\\ c\end{pmatrix},\,\,\, u= \begin{pmatrix}1\\ 0\end{pmatrix}.</math>

	Applying ''S'' if necessary, it can be assumed that \|''a''\| > \|''c''\|
	(equality is not possible by coprimeness). We write ''a'' = ''mc'' + ''r'' with
	0 ≤ ''r'' ≤ \|''c''\|. But then

	:<math>T^{-m}\begin{pmatrix} a\\ c \end{pmatrix} = \begin{pmatrix} r\\ c\end{pmatrix}.</math>

	This process can be continued until one of the entries is 0, in which case the other is necessarily ±1. Applying a power of ''S'' if necessary, it follows that ''v'' = ''h'' ''u'' for some ''h'' in ''H''. Hence

	:<math>h^{-1}g=\begin{pmatrix}1 & p\\ 0 & q\end{pmatrix}</math>

	with ''p'', ''q'' integers. Clearly ''p'' = 1, so that ''h''<sup>−1</sup>''g'' = ''T''<sup>''q''</sup>. Thus ''g'' = ''h'' ''T''<sup>''q''</sup> lies in ''H'' as required.

	To prove that all rationals in occur, it suffices to show that ''G'' carries Δ onto triangles in the tessellation. This follows by first noting that ''S'' and ''T'' carry Δ on to such a triangle: indeed as Möbius transformations, ''S''(''z'') = –1/''z'' and ''T''(''z'') = ''z'' + 1, so these give reflections of Δ in two of its sides. But then ''S'' and ''T'' conjugate the reflections in the sides of Δ into reflections in the sides of ''S''Δ and ''T''Δ, which lie in Γ. Thus ''G'' normalizes Γ. Since triangles in the tessellation are exactly those of the form ''g''Δ with ''g'' in Γ, it follows that ''S'' and ''T'', and hence all elements of ''G'', permute triangles in the tessellation. Since every rational is of the form ''g''(0) for ''g'' in ''G'', every rational in is the vertex of a triangle in the tessellation.

	The reflection group and tessellation for an ideal triangle can also be regarded as a limiting case of the ] for three disjoint unnested circles on the Riemann sphere. Again this group is generated by hyperbolic reflections in the three circles. In both cases the three circles have a common circle which cuts them orthogonally. Using a Möbius transformation, it may be assumed to be the unit circle or equivalently the real axis in the upper half plane.<ref>See:
	*{{harvnb\|McMullen\|1998}}
	*{{harvnb\|Mumford\|Series\|Wright\|2015}}</ref>

	===Approach of Siegel===
	In this subsection the approach of ] to the tessellation theorem for triangles is outlined. Siegel's less elementary approach does not use convexity, instead relying on the theory of ]s, ]s and a version of the ] for coverings. It has been generalized to give proofs of the more general Poincaré polygon theorem. (Note that the special case of tiling by regular ''n''-gons with interior angles 2{{pi}}/''n'' is an immediate consequence of the tessellation by Schwarz triangles with angles {{pi}}/''n'', {{pi}}/''n'' and {{pi}}/2.)<ref>{{harvnb\|Siegel\|1971\|pages=85–87}}</ref><ref>For proofs of Poincaré's polygon theorem, see
	*{{harvnb\|Maskit\|1971}}
	*{{harvnb\|Beardon\|1983\|pages=242–249}}
	*{{harvnb\|Iversen\|1992\|pages=200–208}}
	*{{harvnb\|Berger\|2010\|pages=616–617}}</ref>

	Let Γ be the ] '''Z'''<sub>2</sub> ∗ '''Z'''<sub>2</sub> ∗ '''Z'''<sub>2</sub>. If Δ = ''ABC'' is a Schwarz triangle with angles {{pi}}/''a'', {{pi}}/''b'' and {{pi}}/''c'', where ''a'', ''b'', ''c'' ≥ 2, then there is a natural map of Γ onto the group generated by reflections in the sides of Δ. Elements of Γ are described by a product of the three generators where no two adjacent generators are equal. At the vertices ''A'', ''B'' and ''C'' the product of reflections in the sides meeting at the vertex define rotations by angles 2{{pi}}/''a'', 2{{pi}}/''b'' and 2{{pi}}/''c''; Let ''g''<sub>''A''</sub>, ''g''<sub>''B''</sub> and ''g''<sub>''C''</sub> be the corresponding products of generators of Γ = '''Z'''<sub>2</sub> ∗ '''Z'''<sub>2</sub> ∗ '''Z'''<sub>2</sub>. Let Γ<sub>0</sub> be the normal subgroup of index 2 of Γ, consisting of elements that are the product of an even number of generators; and let Γ<sub>1</sub> be the normal subgroup of Γ generated by (''g''<sub>''A''</sub>)<sup>''a''</sup>, (''g''<sub>''B''</sub>)<sup>''b''</sup> and (''g''<sub>''C''</sub>)<sup>''c''</sup>. These act trivially on Δ. Let {{overline\|Γ}} = Γ/Γ<sub>1</sub> and {{overline\|Γ}}<sub>0</sub> = Γ<sub>0</sub>/Γ<sub>1</sub>.

	The disjoint union of copies of {{overline\|Δ}} indexed by elements of {{overline\|Γ}} with edge identifications has the natural structure of a Riemann surface Σ. At an interior point of a triangle there is an obvious chart. As a point of the interior of an edge the chart is obtained by reflecting the triangle across the edge. At a vertex of a triangle with interior angle {{pi}}/''n'', the chart is obtained from the 2''n'' copies of the triangle obtained by reflecting it successively around that vertex. The group {{overline\|Γ}} acts by deck transformations of Σ, with elements in {{overline\|Γ}}<sub>0</sub> acting as holomorphic mappings and elements not in {{overline\|Γ}}<sub>0</sub> acting as antiholomorphic mappings.

	There is a natural map ''P'' of Σ into the hyperbolic plane. The interior of the triangle with label ''g'' in {{overline\|Γ}} is taken onto ''g''(Δ), edges are taken to edges and vertices to vertices. It is also easy to verify that a neighbourhood of an interior point of an edge is taken into a neighbourhood of the image; and similarly for vertices. Thus ''P'' is locally a homeomorphism and so takes open sets to open sets. The image ''P''(Σ), i.e. the union of the translates ''g''({{overline\|Δ}}), is therefore an open subset of the upper half plane. On the other hand, this set is also closed. Indeed, if a point is sufficiently close to {{overline\|Δ}} it must be in a translate of {{overline\|Δ}}. Indeed, a neighbourhood of each vertex is filled out the reflections of {{overline\|Δ}} and if a point lies outside these three neighbourhoods but is still close to {{overline\|Δ}} it must lie on the three reflections of {{overline\|Δ}} in its sides. Thus there is δ > 0 such that if ''z'' lies within a distance less than δ from {{overline\|Δ}}, then ''z'' lies in a {{overline\|Γ}}-translate of {{overline\|Δ}}. Since the hyperbolic distance is {{overline\|Γ}}-invariant, it follows that if ''z'' lies within a distance less than δ from Γ({{overline\|Δ}}) it actually lies in Γ({{overline\|Δ}}), so this union is closed. By connectivity it follows that ''P''(Σ) is the whole upper half plane.

	On the other hand, ''P'' is a local homeomorphism, so a covering map. Since the upper half plane is simply connected, it follows that ''P'' is one-one and hence the translates of Δ tessellate the upper half plane. This is a consequence of the following version of the monodromy theorem for coverings of Riemann surfaces: if ''Q'' is a covering map between Riemann surfaces Σ<sub>1</sub> and Σ<sub>2</sub>, then any path in Σ<sub>2</sub> can be lifted to a path in Σ<sub>1</sub> and any two homotopic paths with the same end points lift to homotopic paths
	with the same end points; an immediate corollary is that if Σ<sub>2</sub> is simply connected, ''Q'' must be a homeomorphism.<ref>{{harvnb\|Beardon\|1984\|pages=106–107, 110–111}}</ref> To apply this, let Σ<sub>1</sub> = Σ, let Σ<sub>2</sub> be the upper half plane and let ''Q'' = ''P''. By the corollary of the monodromy theorem, ''P'' must be one-one.

	It also follows that ''g''(Δ) = Δ if and only if ''g'' lies in Γ<sub>1</sub>, so that the homomorphism of {{overline\|Γ}}<sub>0</sub> into the Möbius group is faithful.

	==Conformal mapping of Schwarz triangles==
	{{Expand section\|date=October 2021}}
	In this section Schwarz's explicit conformal mapping from the unit disc or the upper half plane to the interior of a Schwarz triangle will be constructed as the ratio of solutions of a hypergeometric ordinary differential equation, following {{harvtxt\|Carathéodory\|1954}}, {{harvtxt\|Nehari\|1975}} and {{harvtxt\|Hille\|1976}}.

	== Applications ==		== Applications ==
	~~The~~ Schwarz triangle functions ~~are relevant~~ to ]s.<ref name=Lee>{{cite book \|last=Lee \|first=Laurence \|title=Conformal Projections ~~based~~ on Elliptic Functions \|~~year~~=~~1976~~ \|publisher=~~University~~ of ~~Toronto~~ ~~Press~~ \|series=Cartographica Monographs \|volume=16		] used Schwarz triangle functions to derive ]s onto ] surfaces.<ref name=Lee>{{cite book \| last = Lee \| first = L. P. \| author-link = Laurence Patrick Lee \| year = 1976 \| title = Conformal Projections Based on Elliptic Functions \| location = Toronto \| publisher = B. V. Gutsell, York University \| series = Cartographica Monographs \| volume = 16 \| url = https://archive.org/details/conformalproject0000leel \| url-access = limited \| isbn = 0-919870-16-3}} Supplement No. 1 to .</ref>
	\|url=https://archive.org/details/conformalproject0000leel \|url-access=limited }} Chapters also published in </ref>

	==Notes==
	{{notelist}}

	==References==		==References==
Line 331:		Line 70:

	==Sources==		==Sources==
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			* {{cite book \|last=Carathéodory \|first=Constantin \|author-link=Constantin Carathéodory \|title=Theory of functions of a complex variable \|volume=2 \|translator=F. Steinhardt \|publisher=Chelsea \|year=1954\|url=https://books.google.com/books?id=UTTvAAAAMAAJ\|oclc=926250115}}
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	== Further reading ==
	* {{citation \|last=Ahlfors \|first=Lars V. \|authorlink=Lars Ahlfors \|title=Complex Analysis \|publisher=McGraw Hill \|year=1966 \|edition=2nd }}
	* {{citation \|last=Chandrasekharan \|first=K. \|authorlink=K. S. Chandrasekharan \|title=Elliptic functions \|series=Grundlehren der Mathematischen Wissenschaften \|volume=281 \|publisher=Springer \|year=1985 \|isbn=3-540-15295-4 }}
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	* {{citation \|last=de Rham\|first=G.\|authorlink=Georges de Rham\|title=Sur les polygones générateurs de groupes fuchsiens \|journal=] \|volume=17 \|year=1971 \|pages=49–61 }}
	* {{citation \|title=Automorphic Functions \|first=Lester R. \|last=Ford \|authorlink=Lester R. Ford \|publisher=] \|year=1951 \|isbn=0821837419 \|orig-date=1929}}
	* {{citation \|last=Ince \|first=E. L. \|authorlink=E. L. Ince \|title=Ordinary Differential Equations \|publisher=Dover \|year=1944 }}
	* {{citation \|last=Lehner \|first=Joseph \|authorlink=Joseph Lehner \|title=Discontinuous groups and automorphic functions \|series=Mathematical Surveys \|volume=8 \|publisher=American Mathematical Society \|year=1964 }}
	* {{citation \|last1=Sansone \|first1=Giovanni \|authorlink=Giovanni Sansone \|last2=Gerretsen \|first2=Johan \|title=Lectures on the theory of functions of a complex variable. II: Geometric theory \|publisher=Wolters-Noordhoff \|year=1969 }}
	* {{citation \|last=Series \|first=Caroline \|authorlink=Caroline Series \|title=The modular surface and continued fractions \|journal=Journal of the London Mathematical Society \|volume=31 \|year=1985 \|pages=69–80 \|doi=10.1112/jlms/s2-31.1.69 }}
	* {{citation \|last=Thurston \|first=William P. \|authorlink=William Thurston \|title=Three-dimensional geometry and topology. Vol. 1. \|editor=Silvio Levy \|series=Princeton Mathematical Series \|volume=35 \|publisher=Princeton University Press \|year=1997 \|isbn=0-691-08304-5 }}

	]		]
	]
	]		]
	]		]
	]		]
			]
			]

Latest revision as of 08:02, 21 January 2025

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In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a Schwarz triangle, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a Möbius triangle, the inverse of the Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function.

Formula

Let πα, πβ, and πγ be the interior angles at the vertices of the triangle in radians. Each of α, β, and γ may take values between 0 and 1 inclusive. Following Nehari, these angles are in clockwise order, with the vertex having angle πα at the origin and the vertex having angle πγ lying on the real line. The Schwarz triangle function can be given in terms of hypergeometric functions as:

s(\alpha ,\beta ,\gamma ;z)=z^{\alpha }{\frac {_{2}F_{1}\left(a',b';c';z\right)}{_{2}F_{1}\left(a,b;c;z\right)}}

where

a = (1−α−β−γ)/2,

b = (1−α+β−γ)/2,

c = 1−α,

a′ = a − c + 1 = (1+α−β−γ)/2,

b′ = b − c + 1 = (1+α+β−γ)/2, and

c′ = 2 − c = 1 + α.

This function maps the upper half-plane to a spherical triangle if α + β + γ > 1, or a hyperbolic triangle if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: a = 0, $_{2}F_{1}\left(a,b;c;z\right)=1$ , and the formula reduces to that given by the Schwarz–Christoffel transformation.

Derivation

Through the theory of complex ordinary differential equations with regular singular points and the Schwarzian derivative, the triangle function can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. By the Schwarz reflection principle, the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of hypergeometric functions.

Singular points

This mapping has regular singular points at z = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,

{\begin{aligned}s(0)&=0,\\s(1)&={\frac {\Gamma (1-a')\Gamma (1-b')\Gamma (c')}{\Gamma (1-a)\Gamma (1-b)\Gamma (c)}},\\s(\infty )&=\exp \left(i\pi \alpha \right){\frac {\Gamma (1-a')\Gamma (b)\Gamma (c')}{\Gamma (1-a)\Gamma (b')\Gamma (c)}},\end{aligned}}

where ${\textstyle \Gamma (x)}$ is the gamma function.

Near each singular point, the function may be approximated as

{\begin{aligned}s_{0}(z)&=z^{\alpha }(1+O(z)),\\s_{1}(z)&=(1-z)^{\gamma }(1+O(1-z)),\\s_{\infty }(z)&=z^{\beta }(1+O(1/z)),\end{aligned}}

where $O(x)$ is big O notation.

Inverse

When α, β, and γ are rational, the triangle is a Schwarz triangle. When each of α, β, and γ are either the reciprocal of an integer or zero, the triangle is a Möbius triangle, i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a modular function.

In the spherical case, that modular function is a rational function. For Euclidean triangles, the inverse can be expressed using elliptical functions.

Ideal triangles

When α = 0 the triangle is degenerate, lying entirely on the real line. If either of β or γ are non-zero, the angles can be permuted so that the positive value is α, but that is not an option for an ideal triangle having all angles zero.

Instead, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the complete elliptic integral of the first kind:

i{\frac {K(1-z)}{K(z)}}

This expression is the inverse of the modular lambda function.

Extensions

The Schwarz–Christoffel transformation gives the mapping from the upper half-plane to any Euclidean polygon.

The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an n-sided polygon, the solution has n-3 additional parameters, which are difficult to determine in practice. See Schwarzian derivative § Conformal mapping of circular arc polygons for more details.

Applications

L. P. Lee used Schwarz triangle functions to derive conformal map projections onto polyhedral surfaces.

References

Nehari 1975, p. 309.
Nehari 1975, pp. 198–208.
Nehari 1975, pp. 315−316.
^ Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.
Nehari 1975, pp. 316–318.
Nehari 1975, p. 202.

Sources

Ahlfors, Lars V. (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable (3 ed.). New York: McGraw-Hill. ISBN 0-07-000657-1. OCLC 4036464.
Carathéodory, Constantin (1954). Theory of functions of a complex variable. Vol. 2. Translated by F. Steinhardt. Chelsea. OCLC 926250115.
Hille, Einar (1997). Ordinary differential equations in the complex domain. Mineola, N.Y.: Dover Publications. ISBN 0-486-69620-0. OCLC 36225146.
Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. ISBN 0-486-61137-X. OCLC 1504503.
Sansone, Giovanni; Gerretsen, Johan (1969). Lectures on the theory of functions of a complex variable. II: Geometric theory. Wolters-Noordhoff. OCLC 245996162.

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