{"title":"The area formula for hyperbolic triangles","authors":"E. Frenkel, W. Su","doi":"10.4171/196-1/2","DOIUrl":"https://doi.org/10.4171/196-1/2","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130591409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Area preserving maps from the sphere to the Euclidean plane","authors":"C. Charitos","doi":"10.4171/196-1/10","DOIUrl":"https://doi.org/10.4171/196-1/10","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"92 17","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113940588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Monotonicity in spherical and hyperbolic triangles","authors":"Himalaya Senapati","doi":"10.4171/196-1/6","DOIUrl":"https://doi.org/10.4171/196-1/6","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121922136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Inscribing a triangle in a circle in spherical geometry","authors":"Himalaya Senapati","doi":"10.4171/196-1/5","DOIUrl":"https://doi.org/10.4171/196-1/5","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131004114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Notes on Eduard Study’s paper “Contributions to non-Euclidean geometry I”","authors":"A. A'Campo-Neuen, A. Papadopoulos","doi":"10.4171/196-1/14","DOIUrl":"https://doi.org/10.4171/196-1/14","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133497826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the non-existence of a perfect map from the 2-sphere to the Euclidean plane","authors":"C. Charitos, I. Papadoperakis","doi":"10.4171/196-1/9","DOIUrl":"https://doi.org/10.4171/196-1/9","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123297016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We start by recalling the classical theorem of Girard on the area of a spherical triangle in terms of its angle sum, and its analogue in hyperbolic geometry. We then use a formula of Euler for the area of a spherical triangle in terms of side lengths and its analogue in hyperbolic geometry in order to give an equality for the distance between the midpoints of two sides of a spherical (respectively hyperbolic) triangle, in terms of the third side. These equalities give quantitative versions of the positivity (respectively negativity) of the curvature in the sense of Busemann. We present several other results related to area in non-Euclidean geometry.
{"title":"Area in non-Euclidean geometry","authors":"N. A'campo, A. Papadopoulos","doi":"10.4171/196-1/1","DOIUrl":"https://doi.org/10.4171/196-1/1","url":null,"abstract":"We start by recalling the classical theorem of Girard on the area of a spherical triangle in terms of its angle sum, and its analogue in hyperbolic geometry. We then use a formula of Euler for the area of a spherical triangle in terms of side lengths and its analogue in hyperbolic geometry in order to give an equality for the distance between the midpoints of two sides of a spherical (respectively hyperbolic) triangle, in terms of the third side. These equalities give quantitative versions of the positivity (respectively negativity) of the curvature in the sense of Busemann. We present several other results related to area in non-Euclidean geometry.","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124425421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a problem of Schubert in hyperbolic geometry","authors":"Vincent Alberge, E. Frenkel","doi":"10.4171/196-1/3","DOIUrl":"https://doi.org/10.4171/196-1/3","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132399275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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