See Möbius transformation on Wiktionary
{
"etymology_text": "Named for German mathematician and theoretical astronomer August Ferdinand Möbius (1790–1868).",
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"name": "Non-Euclidean geometry",
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"examples": [
{
"text": "2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97,\nJørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete."
},
{
"text": "2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323,\nModular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant."
},
{
"ref": "2013, Angel Cano, Juan Pablo Navarrete, Seade Kuri José Antonio, Complex Kleinian Groups, page 1:",
"text": "Classical Kleinian groups are discrete subgroups of Möbius transformations which act on the Riemann sphere with a nonempty region of discontinuity.",
"type": "quote"
}
],
"glosses": [
"A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line."
],
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"word": "Möbius group"
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"word": "automorphism"
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{
"word": "conformal mapping"
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{
"sense": "linear fractional transformation",
"word": "homography"
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{
"sense": "linear fractional transformation",
"word": "projective transformation"
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"id": "en-Möbius_transformation-en-noun-PN1hJSoo",
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"raw_glosses": [
"(geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line."
],
"synonyms": [
{
"word": "Mobius transformation"
},
{
"word": "Moebius transformation"
}
],
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"translations": [
{
"code": "nl",
"lang": "Dutch",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "Möbius-transformatie"
},
{
"code": "fr",
"lang": "French",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "transformation de Möbius"
},
{
"code": "de",
"lang": "German",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "Möbiustransformation"
},
{
"code": "it",
"lang": "Italian",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "trasformazione di Möbius"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "transformação de Möbius"
},
{
"code": "es",
"lang": "Spanish",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "transformación de Möbius"
}
],
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"word": "Möbius transformation"
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{
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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{
"text": "2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97,\nJørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete."
},
{
"text": "2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323,\nModular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant."
},
{
"ref": "2013, Angel Cano, Juan Pablo Navarrete, Seade Kuri José Antonio, Complex Kleinian Groups, page 1:",
"text": "Classical Kleinian groups are discrete subgroups of Möbius transformations which act on the Riemann sphere with a nonempty region of discontinuity.",
"type": "quote"
}
],
"glosses": [
"A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line."
],
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"(geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line."
],
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"word": "Mobius transformation"
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"translations": [
{
"code": "nl",
"lang": "Dutch",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "Möbius-transformatie"
},
{
"code": "fr",
"lang": "French",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "transformation de Möbius"
},
{
"code": "de",
"lang": "German",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "Möbiustransformation"
},
{
"code": "it",
"lang": "Italian",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "trasformazione di Möbius"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "transformação de Möbius"
},
{
"code": "es",
"lang": "Spanish",
"sense": "transformation of the complex plane",
"tags": [
"feminine"
],
"word": "transformación de Möbius"
}
],
"word": "Möbius transformation"
}
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