2013/05/21 by Abraham A. Ungar, Ungar, Abraham A.
Mathematics · Physics and Astronomy · #51M10 83A05 #Advanced Mathematical Theories and Applications #Algebraic and Geometric Analysis #FOS: Physical sciences #History and Theory of Mathematics #Mathematical Physics (math-ph) #Mathematics and Applications #math-ph #math.MP #msc:51M10 #msc:83A05
paper · pdf · doi:10.48550/arxiv.1305.4990
78 pages, 26 figures
openalex publication_date 2013/05/21 · arxiv created 2013/05/22 · arxiv updated 2013/05/23 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are \it gyrogroups and \it gyrovector spaces, and Einstein addition being both \it gyrocommutative and \it gyroassociative. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".