cjerdonek authored on Nov. 14, 2020, 8:48 p.m. | | Additions are marked like this |
This article discusses triangles in the hyperbolic plane whose vertices are permitted to correspond to vectors not just with imaginary norm but also real norm when viewed using the Minkowski model of the hyperbolic plane. We call these generalized hyperbolic triangles. Speaking more loosely, these are hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane, including outside the boundary. In terms of planar shapes, these triangles correspond to the hyperbolic polygons one can get when one is allowed to replace each vertex of a usual hyperbolic triangle with an additional side and two right-angled vertices. These additional polygons include-- quadrilaterals with at least two right angles, pentagons with at least four right angles, and all-right hexagons. | This article discusses triangles in the hyperbolic plane whose vertices are permitted to correspond to vectors not just with imaginary norm but also real norm when viewed using the Minkowski model of the hyperbolic plane. We call these generalized hyperbolic triangles. Speaking more loosely, these are hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane, including outside the boundary. These generalized triangles correspond to the hyperbolic polygons one can get when one is allowed to replace each vertex of a hyperbolic triangle in the usual sense with an additional side and two right-angled vertices. The resulting polygons include-- quadrilaterals with at least two right angles, pentagons with at least four right angles, and all-right hexagons. |
