Schwarz triangle function: Difference between revisions

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[[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==

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== 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-valued]] function, which are [[automorphicmodular function]]s. for the [[triangle group]] of Möbius transformations for the given Möbius triangle.

Depending onIn the geometryspherical ofcase, thethat trianglemodular function is a [[rational function]]. For Euclidean triangles, the inverse function can be expressed using: [[elliptical function]]s.<ref name=Lee />

* Spherical: [[rational function]]s

* Euclidean: [[elliptical function]]s

* Hyperbolic: [[modular function]]s

== Ideal triangles ==

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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 [[conformal map projection]]s onto [[polyhedral map projection|polyhedral]] 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==

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* {{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}}