Schwarz triangle function

In mathematics, the Schwarz triangle tessellation was introduced by H. A. Schwarz as a way of tessellating the hyperbolic upper half plane by a Schwarz triangle, i.e. a geodesic triangle in the upper half plane with angles which are either 0 or of the form $π$ over a positive integer greater than one. Through successive hyperbolic reflections in its sides, such a triangle generates a tessellation of the upper half plane (or the unit disk after composition with the Cayley transform). The hyperbolic refections generates a discrete group, with an orientation-preserving normal subgroup of index two. Each such tessellation yields a conformal mapping of the upper half plane onto the interior of the geodesic triangle, generalizing the Schwarz–Christoffel mapping, with the upper half plane replacing the complex plane. The corresponding inverse function, first defined by Schwarz, solved the problem of uniformization a hyperbolic triangle and is called the Schwarz triangle function.

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.

As the inverse function of such a quotient, the triangle function is an automorphic function for this discrete group of Möbius transformations. This is a special case of a general scheme of Henri Poincaré that associates automorphic forms with ordinary differential equations with regular singular points. In the special case of ideal triangles, where all the angles are zero, the tessellation corresponds to the Farey tessellation and the triangle function yields the modular lambda function.

Hyperboloid and Beltrami-Klein models

Geometric relations between Poincaré disk model, hyperboloid model and Beltrami-Klein model

In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane.^[1]

The group G = SU(1,1) is formed of matrices

g={\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}

with

|\alpha |^{2}-|\beta |^{2}=1.

It is a subgroup of G_c = SL(2,C), the group of complex 2 × 2 matrices with determinant 1. The group G_c acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D and the subgroup G₁ = SL(2,R) acts as automorphisms of the upper half plane. If

C={\begin{pmatrix}1&i\\i&1\end{pmatrix}}

then

G=CG_{1}C^{-1},

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

The Lie algebra ${\mathfrak {g}}$ of SU(1,1) consists of matrices

X={\begin{pmatrix}ix&w\\{\overline {w}}&-ix\end{pmatrix}}

with x real. Note that X² = (|w|² – x²) I and

\det X=x^{2}-|w|^{2}=-{\tfrac {1}{2}}\operatorname {Tr} X^{2}.

The hyperboloid ${\mathfrak {H}}$ in ${\mathfrak {g}}$ is defined by two conditions. The first is that det X = 1 or equivalently Tr X² = –2.^[2] By definition this condition is preserved under conjugation 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 ${\mathfrak {H}}$ and the points 0 and (1,0) have stabiliser K consisting of matrices

{\begin{pmatrix}\zeta &0\\0&{\overline {\zeta }}\end{pmatrix}}

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

a_{t}={\begin{pmatrix}\cosh t&\sinh t\\\sinh t&\cosh t\end{pmatrix}}.

Both spaces can therefore be identified with the homogeneous space G/K and there is a G-equivariant map f of ${\mathfrak {H}}$ 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/α and

g(1,0)=(|\alpha |^{2}+|\beta |^{2},-2i\alpha \beta ),

so that

{\frac {\beta }{\overline {\alpha }}}={\frac {\alpha \beta }{|\alpha |{}^{2}}}={\frac {2\alpha \beta }{|\alpha |{}^{2}+|\beta |{}^{2}+1}},

recovering the formula

f(x,w)={\frac {iw}{x+1}}.

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

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

This correspondence extends to one between geometric properties of D and ${\mathfrak {H}}$ . Without entering into the correspondence of G-invariant Riemannian metrics,^[a] each geodesic circle in D corresponds to the intersection of 2-planes through the origin, given by equations Tr XY = 0, with ${\mathfrak {H}}$ . 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 Beltrami-Klein model is obtained by using the map F(x,w) = w/x as the correspondence between ${\mathfrak {H}}$ and D. Identifying this disk with (1,v) with |v| < 1, intersections of 2-planes with ${\mathfrak {H}}$ correspond to intersections of the same 2-planes with this disk and so give straight lines. The Poincaré-Klein map given by

K(z)=iF\circ f^{-1}(z)={\frac {2z}{1+|z|{}^{2}}}

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 $π$ (measured anticlockwise) into a similar pair of curves.^[b] In the limiting case, when the angle is $π$ , the curves are tangent and this again is preserved under a diffeomorphism. The map K yields the Beltrami-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. The use of the Beltrami-Klein model corresponds to projective geometry and cross ratios. (Note that on the unit circle the radial derivative of K vanishes, so that the condition on angles no longer applies there.)^[3]^[4]

The group G₁ = SL(2,R) is formed of real matrices

g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}

with $ad-bc=1.$ The action is by Möbius transformations on the upper half-plane. The Lie algebra of G₁ is ${\mathfrak {g}}_{1}$ , the space of 2 x 2 real matrices of trace zero,

X={\begin{pmatrix}x&y-t\\y+t&-x\end{pmatrix}}.

By transport of construction — conjugating by C — the symmetric bilinear form Tr ad(X)ad(Y) = 4 Tr XY is invariant under conjugation and has signature (2,1). As above X² = (x² + y² – t²) I and

\det X=t^{2}-x^{2}-y^{2}=-{\tfrac {1}{2}}\operatorname {Tr} X^{2}.

By Sylvester's law of inertia,^[5] the Killing form (or Cartan-Killing form) is, up to equivalence, the unique symmetric bilinear form of signature (2,1) with corresponding quadratic form –x² –y² +t²; and G / {±I} or equivalently G₁ / {±I} can be identified with SO(2,1).

Conformal mapping of Schwarz triangles

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 Carathéodory (1954, pp. 129–194), Nehari (1975, pp. 198–209, 308–332) and Hille (1976, pp. 371–401). The classical account of Ince (1944, pp. 389–395) has an account of the monodromy of second order complex order differential equations with regular singular points. The limiting case of ideal triangles, the Farey series and the modular lambda function is explained in Ahlfors (1966, pp. 254–274), Chandrasekharan (1985, pp. 108–121), Mumford, Series & Wright (2015) and Hardy & Wright (2008).

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. 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 hypergeometric functions as:

s(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 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,

{\begin{aligned}s(0)&=0,\\[5mu]s(1)&={\frac {\Gamma (1-a')\Gamma (1-b')\Gamma (c')}{\Gamma (1-a)\Gamma (1-b)\Gamma (c)}},\,{\text{and}}\\[5mu]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}}

This formula can be derived using the Schwarzian derivative.

This function can be used to map the upper half-plane to a spherical triangle on the Riemann sphere if α + β + γ > 1, or a hyperbolic triangle on the Poincaré disk 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. In the special case of ideal triangles, where all the angles are zero, the triangle function yields the modular lambda function.

Notes

^ The Poincaré metric on the disk corresponds to the restriction of the G-invariant pseudo-Riemannian metric dx² – dw² to the hyperboloid.
^ 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.

References

^
See:
^ Note that the Killing form B(X,Y) = Tr ${}_{\mathfrak {g}}$ ad X ad Y = 4 Tr XY.Helgason 1978
^ Iversen 1992
^ Magnus 1974
^ Ratcliffe 2019, pp. 52–56
^ Lee, Laurence (1976). Conformal Projections based on Elliptic Functions. Cartographica Monographs. Vol. 16. University of Toronto Press. ISBN 9780919870161. Chapters also published in The Canadian Cartographer. 13 (1). 1976.

Sources

Abramenko, Peter; Brown, Kenneth S. (2007). Buildings: Theory and Applications. Graduate Texts in Mathematics. Vol. 248. Springer-Verlag. ISBN 978-0-387-78834-0. MR 2439729.
Ahlfors, Lars V. (1966), Complex Analysis (2nd ed.), McGraw Hill
Beardon, Alan F. (1983), "Poincaré's Theorem", The Geometry of Discrete Groups, Graduate Texts in Mathematics, vol. 91, Springer, pp. 242–252, ISBN 0-387-90788-2
Beardon, Alan F. (1984), "A primer on Riemann surfaces", London Mathematical Society Lecture Note Series, 78, Cambridge University Press, ISBN 0521271045
Berger, Marcel (2010), Geometry revealed. A Jacob's ladder to modern higher geometry, translated by Lester Senechal, Springer, ISBN 978-3-540-70996-1
Berndt, Bruce C.; Knopp, Marvin I. (2008), Hecke's theory of modular forms and Dirichlet series, Monographs in Number Theory, vol. 5, World Scientific, ISBN 978-981-270-635-5
Björner, Anders; Brenti, Francesco (2005). Combinatorics of Coxeter groups. Graduate Texts in Mathematics. Vol. 231. Springer-Verlag. ISBN 978-3540-442387. MR 2133266.
Borel, Armand (1997). "I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane". Automorphic forms on SL₂(R). Cambridge Tracts in Mathematics. Vol. 130. Cambridge University Press. ISBN 0-521-58049-8. MR 1482800.
Bourbaki, Nicolas (1968). "Chapitre IV : Groupes de Coxeter et systèmes de Tits • Chapitre V : Groupes engendrés par des réflexions". Groupes et algèbres de Lie. Éléments de mathématique (in French). Paris: Hermann. pp. 1–56, 57–141. MR 0240238. Reprinted by Masson in 1981 as ISBN 2-225-76076-4.
Bridson, Martin R.; Haefliger, André (1999). "I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane". Metric spaces of non-positive curvature (PDF). Grundlehren der mathematischen Wissenschaften. Vol. 319. Springer-Verlag. ISBN 3-540-64324-9. MR 1744486.
Brown, Kenneth S. (1989). Buildings. Springer-Verlag. ISBN 0-387-96876-8. MR 0969123.
Busemann, Herbert (1955), The geometry of geodesics, Academic Press
Carathéodory, Constantin (1954), Theory of functions of a complex variable, vol. 2, translated by F. Steinhardt, Chelsea
Chandrasekharan, K. (1985), Elliptic functions, Grundlehren der Mathematischen Wissenschaften, vol. 281, Springer, ISBN 3-540-15295-4
Davis, Michael W. (2008), "Appendix D. The Geometric Representation", The geometry and topology of Coxeter groups, London Mathematical Society Monographs, vol. 32, Princeton University Press, pp. 439–447, ISBN 978-0-691-13138-2
de la Harpe, Pierre (1991). "An invitation to Coxeter groups". Group theory from a geometrical viewpoint (Trieste, 1990). World Scientific. pp. 193–253. MR 1170367.
Deodhar, Vinay V. (1982). "On the root system of a Coxeter group". Comm. Algebra. 10 (6): 611–630. doi:10.1080/00927878208822738. MR 0647210.
Deodhar, Vinay V. (1986). "Some characterizations of Coxeter groups". Enseign. Math. 32: 111–120. MR 0850554.
de Rham, G. (1971). "Sur les polygones générateurs de groupes fuchsiens". Enseign. Math. (in French). 17: 49–61.
Ellis, Graham (2019). "Triangle groups". An Invitation to Computational Homotopy. Oxford University Press. pp. 441–444. ISBN 978-0-19-883298-0. MR 3971587.
Epstein, David B. A.; Petronio, Carlo (1994). "An exposition of Poincaré's polyhedron theorem". Enseign. Math. 40: 113–170. MR 1279064.
Evans, Ronald (1973), "A fundamental region for Hecke's modular group", Journal of Number Theory, 5 (2): 108–115, Bibcode:1973JNT.....5..108E, doi:10.1016/0022-314x(73)90063-2
Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen (in German), B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01 {{citation}}: ISBN / Date incompatibility (help)
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete mathematics (2nd ed.), Addison-Wesley, pp. 116–118, ISBN 0-201-55802-5
Hardy, G. H.; Wright, E. M. (2008), An introduction to the theory of numbers (6th ed.), Oxford University Press, ISBN 978-0-19-921986-5
Hatcher, Allen (2013). "1. The Farey Diagram". Topology of Numbers (PDF). Cornell University. Retrieved 25 February 2022.
Hecke, E. (1935), "Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", Mathematische Annalen (in German), 112: 664–699, doi:10.1007/bf01565437
Heckman, Gert J. (2018). "Coxeter Groups" (PDF). Radboud University Nijmegen. Retrieved 3 March 2022.
Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5
Helgason, Sigurdur (2000), Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, ISBN 0-8218-2673-5
Hille, Einar (1976), Ordinary differential equations in the complex domain, Wiley-Interscience
Hiller, Howard (1982). Geometry of Coxeter groups. Research Notes in Mathematics. Vol. 54. Pitman. ISBN 0-273-08517-4. MR 0649068.
Howlett, Robert (1996). "Introduction to Coxeter Groups". Australian National University Geometric Group Theory Workshop. Sydney.{{cite book}}: CS1 maint: location missing publisher (link)
Humphreys, James E. (1990). Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press. ISBN 0-521-37510-X. MR 1066460.
Ince, E. L. (1944), Ordinary Differential Equations, Dover
Iversen, Birger (1992), Hyperbolic geometry, London Mathematical Society Student Texts, vol. 25, Cambridge University Press, ISBN 0-521-43508-0
Iwahori, Nagayoshi (1966). "On discrete reflection groups on symmetric Riemannian manifolds". Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965). Tokyo: Nippon Hyoronsha. pp. 57–62. MR 0217741.
Johnson, Norman W. (2018). Geometries and Transformations. Cambridge University Press. ISBN 978-1-107-10340-5. MR 3839612.
Magnus, Wilhelm (1974), Noneuclidean tesselations and their groups, Pure and Applied Mathematics, vol. 61, Academic Press
Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (1976). Combinatorial group theory: Presentations of groups in terms of generators and relations (Second revised ed.). Dover Books. MR 0207802.
Maskit, Bernard (1971), "On Poincaré's theorem for fundamental polygons", Advances in Mathematics, 7 (3): 219–230, doi:10.1016/s0001-8708(71)80003-8
Maskit, Bernard (1988). "Poincaré's theorem". Kleinian groups. Grundlehren der mathematischen Wissenschaften. Vol. 287. Springer-Verlag. ISBN 3-540-17746-9. MR 0959135.
Maxwell, George (1982). "Sphere packings and hyperbolic reflection groups". J. Algebra. 79: 78–97. doi:10.1016/0021-8693(82)90318-0. MR 0679972.
McMullen, Curtis T. (1998), "Hausdorff dimension and conformal dynamics. III. Computation of dimension", American Journal of Mathematics, 120: 691–721, doi:10.1353/ajm.1998.0031, S2CID 15928775
Matsumoto, Hideya (1964). "Générateurs et relations des groupes de Weyl généralisés". C. R. Acad. Sci. Paris (in French). 258: 3419–3422. MR 0183818.
Milnor, John (1975). "On the 3-dimensional Brieskorn manifolds M(p,q,r)". Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox). Ann. of Math. Studies. Vol. 84. Princeton University Press. pp. 175–225. MR 0418127.
Mumford, David; Series, Caroline; Wright, David (2015), Indra's pearls. The vision of Felix Klein, Cambridge University Press, ISBN 978-1-107-56474-9
Nehari, Zeev (1975), Conformal mapping, Dover
Ratcliffe, John G. (2019). Foundations of hyperbolic manifolds. Graduate Texts in Mathematics. Vol. 149 (Third ed.). Springer. ISBN 978-3-030-31597-9. MR 1299730.
Schlag, Wilhelm (2014). "Uniformization". A course in complex analysis and Riemann surfaces. Graduate Studies in Mathematics. Vol. 154. American Mathematical Society. pp. 305–351. ISBN 978-0-8218-9847-5. MR 3186310.
Series, Caroline (2015), Continued fractions and hyperbolic geometry, Loughborough LMS Summer School (PDF), retrieved 15 February 2017
Siegel, C. L. (1971), Topics in complex function theory, vol. II. Automorphic functions and abelian integrals, translated by A. Shenitzer; M. Tretkoff, Wiley-Interscience, pp. 85–87, ISBN 0-471-60843-2
Steinberg, Robert (1968). Endomorphisms of linear algebraic groups. Memoirs of the American Mathematical Society. Vol. 80. American Mathematical Society. MR 0230728.
Szenthe, J. (1993). "A significant property of real hyperbolic spaces". Proceedings of Symposium in Geometry (Cluj-Napoca and Tîrgu Mureş, 1992). pp. 153–161. MR 1333415.
Takeuchi, Kisao (1977a), "Arithmetic triangle groups", Journal of the Mathematical Society of Japan, 29: 91–106, doi:10.2969/jmsj/02910091
Takeuchi, Kisao (1977b), "Commensurability classes of arithmetic triangle groups", Journal of the Faculty of Science, the University of Tokyo, Section IA, Mathematics, 24: 201–212
Threlfall, W. (1932), "Gruppenbilder" (PDF), Abhandlungen der Mathematisch-physischen Klasse der Sachsischen Akademie der Wissenschaften, 41, Hirzel: 1–59
Tits, Jacques (2013). "Groupes et géométries de Coxeter". In F. Buekenhout; B. M. Mühlherr; J-P. Tignol; H. Van Maldeghem (eds.). Œuvres/Collected works, Volume I. Heritage of European Mathematics (in French). Zürich: European Mathematical Society. pp. 803–817. ISBN 978-3-03719-126-2. This manuscript was the foundational text for the theory of Coxeter groups, used for preparing Chapter IV of Bourbaki's Groupes et Algèbres de Lie; it was first published in 2001.
Vinberg, Ernest B. (1971). "Discrete linear groups generated by reflections". Mathematics of the USSR-Izvestiya. 5 (5). Translated by P. Flor: 1083–1119. Bibcode:1971IzMat...5.1083V. doi:10.1070/IM1971v005n05ABEH001203. MR 0302779.
Vinberg, Ernest B. (1985). "Hyperbolic reflection groups". Russian Mathematical Surveys. 40 (1). Translated by J. Wiegold. London Mathematical Society: 31–75. Bibcode:1985RuMaS..40...31V. doi:10.1070/RM1985v040n01ABEH003527.
Vinberg, Ernest B.; Shvartsman, O. V. (1993). "Discrete groups of motions of spaces of constant curvature". Geometry II: Spaces of Constant Curvature. Encyclopaedia Math. Sci. Vol. 29. Springer-Verlag. pp. 139–248. ISBN 3-540-52000-7. MR 1254933.
Weber, Matthias (2005), "Kepler's small stellated dodecahedron as a Riemann surface", Pacific Journal of Mathematics, 220: 167–182, doi:10.2140/pjm.2005.220.167
Wolf, Joseph A. (2011), Spaces of constant curvature (6th ed.), AMS Chelsea, ISBN 978-0-8218-5282-8
Yoshida, Masaaki (1987). Fuchsian differential equations, with special emphasis on the Gauss-Schwarz theory. Aspects of Mathematics. Vol. E11. Friedr. Vieweg & Sohn. ISBN 3-528-08971-7. MR 0986252.