{"title":"Area and volume in non-Euclidean geometry","authors":"N. Abrosimov, A. Mednykh","doi":"10.4171/196-1/11","DOIUrl":"https://doi.org/10.4171/196-1/11","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"30 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":"125037817","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":"The Gauss–Bonnet theorem and the geometry of surfaces","authors":"Son Lam Ho","doi":"10.4171/196-1/8","DOIUrl":"https://doi.org/10.4171/196-1/8","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"47 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":"127504366","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 theorem of Lambert: medians in spherical and hyperbolic geometries","authors":"Himalaya Senapati","doi":"10.4171/196-1/4","DOIUrl":"https://doi.org/10.4171/196-1/4","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"2 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":"134184034","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":"A theorem on equiareal triangles with a fixed base","authors":"V. Pambuccian","doi":"10.4171/196-1/18","DOIUrl":"https://doi.org/10.4171/196-1/18","url":null,"abstract":"","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121756766","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}
This is a survey article on the infinitesimal rigidity of frameworks in Euclidean, hyperbolic, and spherical geometry. We discuss the equivalence of the static and kinematic formulations of the infinitesimal rigidity, the projective interpretation of statics (representing forces as bivectors), and the infinitesimal Pogorelov maps that establish correspondence between infinitesimal motions of a framework and of its geodesic image. Also we describe the Maxwell-Cremona correspondence between equilibrium loads and polyhedral lifts, both for Euclidean and for non-Euclidean frameworks.
{"title":"Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry","authors":"Ivan Izmestiev","doi":"10.4171/196-1/12","DOIUrl":"https://doi.org/10.4171/196-1/12","url":null,"abstract":"This is a survey article on the infinitesimal rigidity of frameworks in Euclidean, hyperbolic, and spherical geometry. We discuss the equivalence of the static and kinematic formulations of the infinitesimal rigidity, the projective interpretation of statics (representing forces as bivectors), and the infinitesimal Pogorelov maps that establish correspondence between infinitesimal motions of a framework and of its geodesic image. Also we describe the Maxwell-Cremona correspondence between equilibrium loads and polyhedral lifts, both for Euclidean and for non-Euclidean frameworks.","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127108593","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}
This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the Beltrami-Cayley-Klein disk with an affine conic. Non-Euclidean conics have metric properties similar to those of Euclidean conics, and even more due to the polarity that works here better than in the Euclidean plane.
{"title":"Spherical and hyperbolic conics","authors":"Ivan Izmestiev","doi":"10.4171/196-1/15","DOIUrl":"https://doi.org/10.4171/196-1/15","url":null,"abstract":"This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the Beltrami-Cayley-Klein disk with an affine conic. Non-Euclidean conics have metric properties similar to those of Euclidean conics, and even more due to the polarity that works here better than in the Euclidean plane.","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132134787","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}
Since the end of the 19th century, and after the works of F. Klein and H. Poincar'e, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen as a "limit" of both geometries. Then all the geometries that can be obtained in this way. Some of these geometries had a rich development, most remarkably hyperbolic geometry and the Lorentzian geometries of Minkowski, de Sitter and anti-de Sitter spaces, which in higher dimension have had large interest for a long time in mathematical physics and more precisely in General Relativity. Moreover, some degenerate spaces appear naturally in the picture, namely the co-Euclidean space (the space of hyperplanes of the Euclidean space), and the co-Minkowski space (that we will restrict to the space of space-like hyperplanes of Minkowski space), first because of duality reasons, and second because they appear as limits of degeneration of classical spaces. In fact, co-Minkowski space recently regained interest under the name half-pipe geometry. The purpose of the present paper is to provide a survey on the properties of these spaces, especially in dimensions 2 and 3, from the point of view of projective geometry. Even with this perspective, the paper does not aim to be an exhaustive treatment. Instead it is focused on the aspects which concern convex subsets and their duality, degeneration of geometries and some properties of surfaces in three-dimensional spaces. The presentation is intended to be elementary, hence containing no deep proofs of theorems, but trying to proceed by accessible observations and elementary proofs.
{"title":"Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions","authors":"Franccois Fillastre, Andrea Seppi","doi":"10.4171/196-1/16","DOIUrl":"https://doi.org/10.4171/196-1/16","url":null,"abstract":"Since the end of the 19th century, and after the works of F. Klein and H. Poincar'e, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen as a \"limit\" of both geometries. Then all the geometries that can be obtained in this way. Some of these geometries had a rich development, most remarkably hyperbolic geometry and the Lorentzian geometries of Minkowski, de Sitter and anti-de Sitter spaces, which in higher dimension have had large interest for a long time in mathematical physics and more precisely in General Relativity. Moreover, some degenerate spaces appear naturally in the picture, namely the co-Euclidean space (the space of hyperplanes of the Euclidean space), and the co-Minkowski space (that we will restrict to the space of space-like hyperplanes of Minkowski space), first because of duality reasons, and second because they appear as limits of degeneration of classical spaces. In fact, co-Minkowski space recently regained interest under the name half-pipe geometry. The purpose of the present paper is to provide a survey on the properties of these spaces, especially in dimensions 2 and 3, from the point of view of projective geometry. Even with this perspective, the paper does not aim to be an exhaustive treatment. Instead it is focused on the aspects which concern convex subsets and their duality, degeneration of geometries and some properties of surfaces in three-dimensional spaces. The presentation is intended to be elementary, hence containing no deep proofs of theorems, but trying to proceed by accessible observations and elementary proofs.","PeriodicalId":429025,"journal":{"name":"Eighteen Essays in Non-Euclidean Geometry","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115720435","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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