Schwarz triangle function: Difference between revisions

Revision as of 03:51, 3 September 2022

Latest revision as of 00:57, 29 July 2025

(11 intermediate revisions by 3 users not shown)

Line 3:

[[File:Schwarz triangle function.svg|thumb|The upper half-plane, and the image of the upper half-plane transformed by the Schwarz triangle function with various parameters.]]

{{Complex analysis sidebar}}

In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|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 case 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,{{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 [[hypergeometric functions]] 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,

Line 18:

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''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]].

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''. For an [[ideal triangle]] having all angles zero, other maps can be used. A transformed form of the Schwarz triangle function, with ''α''=''β''=0, ''γ''= 1/2, and <math>z = \frac{1}{1-w^2}</math>, maps to an ideal triangle with vertices at ''i'', 1, and ''-i''. Alternately, 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]]:{{sfn|Nehari|1975|pp=316-318}}

:<math>i\frac{K(1-z)}{K(z)}</math>,

which is the inverse of the [[modular lambda function]].

===Derivation===

Line 51 ⟶ 47:

== 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 [[single-valuedmodular function]]. function, which can be expressed using:

* Spherical: [[rational function]]s

* Euclidean: [[elliptical function]]s

* Hyperbolic: [[modular function]]s

These inverse functions are [[automorphic function]]s for the [[triangle group]] of Möbius transformations for the given Möbius triangle.

AsIn mentionedthe earlierspherical case, thethat Schwarz trianglemodular function is nota directly[[rational usefulfunction]]. forFor idealEuclidean triangles, but the [[modular lambda function]]inverse can be usedexpressed tousing express[[elliptical mapsfunction]]s.<ref fromname=Lee ideal triangles to the upper half-plane./>

== 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]]:

:<math>i\frac{K(1-z)}{K(z)}</math>,.

whichThis expression is the inverse of the [[modular lambda function]].{{sfn|Nehari|1975|pp=316-318}}

== 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'' − 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.

== Applications ==

[[L. P. Lee]] used Schwarz triangle functions to derive [[polyhedralconformal map projection]]s that areonto [[conformalpolyhedral map projection|conformalpolyhedral]] surfaces.<ref name=Lee>{{cite book | last = Lee | first = L. P. | author-link = Laurence Patrick Lee | year = 1976 | title = Conformal Projections basedBased on Elliptic Functions |year ___location =1976 Toronto | publisher =University ofB. TorontoV. PressGutsell, 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 [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer'' '''13'''].</ref>

|isbn=9780919870161 |url=https://archive.org/details/conformalproject0000leel |url-access=limited }} Chapters also published in [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer''. '''13''' (1). 1976.]</ref>

==References==

{{reflist|20em30em}}

==Sources==

Line 75 ⟶ 73:

* {{Cite book |last=Ahlfors |first=Lars V. |author-link=Lars Ahlfors |title=Complex analysis: an introduction to the theory of analytic functions of one complex variable |date=1979 |publisher=McGraw-Hill |isbn=0-07-000657-1 |edition=3 |___location=New York |oclc=4036464}}

* {{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}}

* {{Cite book |last=Hille |first=Einar |author-link=Einar Hille |url=https://www.worldcat.org/oclc/36225146 |title=Ordinary differential equations in the complex ___domain |date=1997 |publisher=Dover Publications |isbn=0-486-69620-0 |___location=Mineola, N.Y. |oclc=36225146}}

* {{Cite book |last=Nehari |first=Zeev |author-link=Zeev Nehari |title=Conformal mapping |date=1975 |publisher=Dover Publications |isbn=0-486-61137-X |___location=New York |oclc=1504503}}

* {{cite book |last1=Sansone |first1=Giovanni |author-link=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 |oclc=245996162}}

{{refend}}