Hyperbolic Triangle Centers
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Author |
: A.A. Ungar |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 322 |
Release |
: 2010-06-18 |
ISBN-10 |
: 9789048186372 |
ISBN-13 |
: 9048186374 |
Rating |
: 4/5 (72 Downloads) |
Synopsis Hyperbolic Triangle Centers by : A.A. Ungar
After A. Ungar had introduced vector algebra and Cartesian coordinates into hyperbolic geometry in his earlier books, along with novel applications in Einstein’s special theory of relativity, the purpose of his new book is to introduce hyperbolic barycentric coordinates, another important concept to embed Euclidean geometry into hyperbolic geometry. It will be demonstrated that, in full analogy to classical mechanics where barycentric coordinates are related to the Newtonian mass, barycentric coordinates are related to the Einsteinian relativistic mass in hyperbolic geometry. Contrary to general belief, Einstein’s relativistic mass hence meshes up extraordinarily well with Minkowski’s four-vector formalism of special relativity. In Euclidean geometry, barycentric coordinates can be used to determine various triangle centers. While there are many known Euclidean triangle centers, only few hyperbolic triangle centers are known, and none of the known hyperbolic triangle centers has been determined analytically with respect to its hyperbolic triangle vertices. In his recent research, the author set the ground for investigating hyperbolic triangle centers via hyperbolic barycentric coordinates, and one of the purposes of this book is to initiate a study of hyperbolic triangle centers in full analogy with the rich study of Euclidean triangle centers. Owing to its novelty, the book is aimed at a large audience: it can be enjoyed equally by upper-level undergraduates, graduate students, researchers and academics in geometry, abstract algebra, theoretical physics and astronomy. For a fruitful reading of this book, familiarity with Euclidean geometry is assumed. Mathematical-physicists and theoretical physicists are likely to enjoy the study of Einstein’s special relativity in terms of its underlying hyperbolic geometry. Geometers may enjoy the hunt for new hyperbolic triangle centers and, finally, astronomers may use hyperbolic barycentric coordinates in the velocity space of cosmology.
Author |
: Abraham A. Ungar |
Publisher |
: World Scientific |
Total Pages |
: 360 |
Release |
: 2010 |
ISBN-10 |
: 9789814304931 |
ISBN-13 |
: 981430493X |
Rating |
: 4/5 (31 Downloads) |
Synopsis Barycentric Calculus in Euclidean and Hyperbolic Geometry by : Abraham A. Ungar
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers. This unique book on barycentric calculus in Euclidean and hyperbolic geometry provides an introduction to the fascinating and beautiful subject of novel triangle centers in hyperbolic geometry along with analogies they share with familiar triangle centers in Euclidean geometry. As such, the book uncovers magnificent unifying notions that Euclidean and hyperbolic triangle centers share. In his earlier books the author adopted Cartesian coordinates, trigonometry and vector algebra for use in hyperbolic geometry that is fully analogous to the common use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. As a result, powerful tools that are commonly available in Euclidean geometry became available in hyperbolic geometry as well, enabling one to explore hyperbolic geometry in novel ways. In particular, this new book establishes hyperbolic barycentric coordinates that are used to determine various hyperbolic triangle centers just as Euclidean barycentric coordinates are commonly used to determine various Euclidean triangle centers. The hunt for Euclidean triangle centers is an old tradition in Euclidean geometry, resulting in a repertoire of more than three thousand triangle centers that are known by their barycentric coordinate representations. The aim of this book is to initiate a fully analogous hunt for hyperbolic triangle centers that will broaden the repertoire of hyperbolic triangle centers provided here.
Author |
: Abraham Albert Ungar |
Publisher |
: World Scientific |
Total Pages |
: 360 |
Release |
: 2010-08-26 |
ISBN-10 |
: 9789814464956 |
ISBN-13 |
: 9814464953 |
Rating |
: 4/5 (56 Downloads) |
Synopsis Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction by : Abraham Albert Ungar
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers. This unique book on barycentric calculus in Euclidean and hyperbolic geometry provides an introduction to the fascinating and beautiful subject of novel triangle centers in hyperbolic geometry along with analogies they share with familiar triangle centers in Euclidean geometry. As such, the book uncovers magnificent unifying notions that Euclidean and hyperbolic triangle centers share.In his earlier books the author adopted Cartesian coordinates, trigonometry and vector algebra for use in hyperbolic geometry that is fully analogous to the common use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. As a result, powerful tools that are commonly available in Euclidean geometry became available in hyperbolic geometry as well, enabling one to explore hyperbolic geometry in novel ways. In particular, this new book establishes hyperbolic barycentric coordinates that are used to determine various hyperbolic triangle centers just as Euclidean barycentric coordinates are commonly used to determine various Euclidean triangle centers.The hunt for Euclidean triangle centers is an old tradition in Euclidean geometry, resulting in a repertoire of more than three thousand triangle centers that are known by their barycentric coordinate representations. The aim of this book is to initiate a fully analogous hunt for hyperbolic triangle centers that will broaden the repertoire of hyperbolic triangle centers provided here.
Author |
: Michael P. Hitchman |
Publisher |
: Jones & Bartlett Learning |
Total Pages |
: 255 |
Release |
: 2009 |
ISBN-10 |
: 9780763754570 |
ISBN-13 |
: 0763754579 |
Rating |
: 4/5 (70 Downloads) |
Synopsis Geometry with an Introduction to Cosmic Topology by : Michael P. Hitchman
The content of Geometry with an Introduction to Cosmic Topology is motivated by questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have and edge? Is it infinitely big? Dr. Hitchman aims to clarify this fascinating area of mathematics. This non-Euclidean geometry text is organized intothree natural parts. Chapter 1 provides an overview including a brief history of Geometry, Surfaces, and reasons to study Non-Euclidean Geometry. Chapters 2-7 contain the core mathematical content of the text, following the ErlangenProgram, which develops geometry in terms of a space and a group of transformations on that space. Finally chapters 1 and 8 introduce (chapter 1) and explore (chapter 8) the topic of cosmic topology through the geometry learned in the preceding chapters.
Author |
: Abraham Ungar |
Publisher |
: Morgan & Claypool Publishers |
Total Pages |
: 194 |
Release |
: 2009-03-08 |
ISBN-10 |
: 9781598298239 |
ISBN-13 |
: 1598298232 |
Rating |
: 4/5 (39 Downloads) |
Synopsis A Gyrovector Space Approach to Hyperbolic Geometry by : Abraham Ungar
The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry
Author |
: Abraham Albert Ungar |
Publisher |
: CRC Press |
Total Pages |
: 616 |
Release |
: 2014-12-17 |
ISBN-10 |
: 9781482236682 |
ISBN-13 |
: 1482236680 |
Rating |
: 4/5 (82 Downloads) |
Synopsis Analytic Hyperbolic Geometry in N Dimensions by : Abraham Albert Ungar
The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry. Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language t
Author |
: Clark Kimberling |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 212 |
Release |
: 2003 |
ISBN-10 |
: 1931914028 |
ISBN-13 |
: 9781931914024 |
Rating |
: 4/5 (28 Downloads) |
Synopsis Geometry in Action by : Clark Kimberling
Geometry in Action uses Sketchpad? to awaken student creativity through discovery-based learning. It supplements any college geometry course in which The Geometer's Sketchpad is used. All students must have access to The Geometer's Sketchpad.Each book is packaged with a CD-ROM for students that illustrates what is meant by geometry in action. Students explore 27 sketches prepared by the author to demonstrate Sketchpad's capabilities by dragging points to see shifts in graphs, by animating tesselations to create new patterns, and much, much more! Also included on this CD is the Poincare Disk, a Sketchpad file used to dig deeper into non-Euclidean geometry with The Geometer's Sketchpad.
Author |
: Saul Stahl |
Publisher |
: Jones & Bartlett Learning |
Total Pages |
: 320 |
Release |
: 1993 |
ISBN-10 |
: 086720298X |
ISBN-13 |
: 9780867202984 |
Rating |
: 4/5 (8X Downloads) |
Synopsis The Poincaré Half-plane by : Saul Stahl
The Poincare Half-Planeprovides an elementary and constructive development of this geometry that brings the undergraduate major closer to current geometric research. At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor.
Author |
: Gerard A. Venema |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 147 |
Release |
: 2013-12-31 |
ISBN-10 |
: 9781614441113 |
ISBN-13 |
: 1614441111 |
Rating |
: 4/5 (13 Downloads) |
Synopsis Exploring Advanced Euclidean Geometry with GeoGebra by : Gerard A. Venema
This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincare disk model for hyperbolic geometry. The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.
Author |
: Vagn Lundsgaard Hansen |
Publisher |
: World Scientific |
Total Pages |
: 225 |
Release |
: 2024-06-04 |
ISBN-10 |
: 9789811260971 |
ISBN-13 |
: 9811260974 |
Rating |
: 4/5 (71 Downloads) |
Synopsis Shadows Of The Circle: From Conic Sections To Planetary Motion (Second Edition) by : Vagn Lundsgaard Hansen
The ancient Greeks were the first to seriously ask for scientific explanations of the panorama of the heavens based on mathematical ideas. Ever since, mathematics has played a major role for human perception and description of the outside physical world, and in a larger perspective for comprehending the universe. This second edition pays tribute to this line of thought and takes the reader on a journey in the mathematical universe from conic sections to mathematical modelling of planetary systems.In the second edition, the four chapters in the first edition on conic sections (two chapters), isoperimetric problems for plane figures, and non-Euclidean geometry, are treated in four revised chapters with many new exercises added. In three new chapters, the reader is taken through mathematics in curves, mathematics in a Nautilus shell, and mathematics in the panorama of the heavens. In all chapters of the book, the circle plays a prominent role.This book is addressed to undergraduate and graduate students as well as researchers interested in the geometry of conic sections, including the historical background and mathematical methods used. It features selected important results, and proofs that not only proves but also 'explains' the results.