Barycentric Calculus in Euclidean and Hyperbolic Geometry A Comparative Introduction 1st Edition by Abraham Albert Ungar 9789814304931 9789814304931

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Product details:

ISBN 10: 981430493X

ISBN 13: 9789814304931

Author: 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.

Table of contents:

Chapter 1: Euclidean Barycentric Coordinates and the Classic Triangle Centers
1.1 Points, Lines, Distance and Isometries
1.2 Vectors, Angles and Triangles
1.3 Euclidean Barycentric Coordinates
1.4 Analogies with Classical Mechanics
1.5 Barycentric Representations are Covariant
1.6 Vector Barycentric Representation
1.7 Triangle Centroid
1.8 Triangle Altitude
1.9 Triangle Orthocenter
1.10 Triangle Incenter
1.11 Triangle Inradius
1.12 Triangle Circumcenter
1.13 Circumradius
1.14 Triangle Incircle and Excircles
1.15 Excircle Tangency Points
1.16 From Triangle Tangency Points to Triangle Centers
1.17 Triangle In-Exradii
1.18 A Step Toward the Comparative Study
1.19 Tetrahedron Altitude
1.20 Tetrahedron Altitude Length
1.21 Exercises

Chapter 2: Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry
2.1 Einstein Addition
2.2 Einstein Gyration
2.3 From Einstein Velocity Addition to Gyrogroups
2.4 First Gyrogroup Theorems
2.5 The Two Basic Equations of Gyrogroups
2.6 Einstein Gyrovector Spaces
2.7 Gyrovector Spaces
2.8 Einstein Points, Gyrolines and Gyrodistance
2.9 Linking Einstein Addition to Hyperbolic Geometry
2.10 Einstein Gyrovectors, Gyroangles and Gyrotriangles
2.11 The Law of Gyrocosines
2.12 The SSS to AAA Conversion Law
2.13 Inequalities for Gyrotriangles
2.14 The AAA to SSS Conversion Law
2.15 The Law of Gyrosines
2.16 The ASA to SAS Conversion Law
2.17 Gyrotriangle Defect
2.18 Right Gyrotriangles
2.19 Einstein Gyrotrigonometry and Gyroarea
2.20 Gyrotriangle Gyroarea Addition Law
2.21 Gyrodistance Between a Point and a Gyroline
2.22 The Gyroangle Bisector Theorem
2.23 Möbius Addition and Möbius Gyrogroups
2.24 Möbius Gyration
2.25 Möbius Gyrovector Spaces
2.26 Möbius Points, Gyrolines and Gyrodistance
2.27 Linking Möbius Addition to Hyperbolic Geometry
2.28 Möbius Gyrovectors, Gyroangles and Gyrotriangles
2.29 Gyrovector Space Isomorphism
2.30 Möbius Gyrotrigonometry
2.31 Exercises

Chapter 3: The Interplay of Einstein Addition and Vector Addition
3.1 Extension of R into Tn+1
3.2 Scalar Multiplication and Addition in Tn+1
3.3 Inner Product and Norm in Tn+1
3.4 Unit Elements of Tn+1
3.5 From Tn+1 back to R

Chapter 4: Hyperbolic Barycentric Coordinates and Hyperbolic Triangle Centers
4.1 Gyrobarycentric Coordinates in Einstein Gyrovector Spaces
4.2 Analogies with Relativistic Mechanics
4.3 Gyrobarycentric Coordinates in Möbius Gyrovector Spaces
4.4 Einstein Gyromidpoint
4.5 Möbius Gyromidpoint
4.6 Einstein Gyrotriangle Gyrocentroid
4.7 Einstein Gyrotetrahedron Gyrocentroid
4.8 Möbius Gyrotriangle Gyrocentroid
4.9 Möbius Gyrotetrahedron Gyrocentroid
4.10 Foot of a Gyrotriangle Gyroaltitude
4.11 Einstein Point to Gyroline Gyrodistance
4.12 Möbius Point to Gyroline Gyrodistance
4.13 Einstein Gyrotriangle Orthogyrocenter
4.14 Möbius Gyrotriangle Orthogyrocenter
4.15 Foot of a Gyrotriangle Gyroangle Bisector
4.16 Einstein Gyrotriangle Ingyrocenter
4.17 Ingyrocenter to Gyrotriangle Side Gyrodistance
4.18 Möbius Gyrotriangle Ingyrocenter
4.19 Einstein Gyrotriangle Circumgyrocenter
4.20 Einstein Gyrotriangle Circumgyroradius
4.21 Möbius Gyrotriangle Circumgyrocenter
4.22 Comparative Study of Gyrotriangle Gyrocenters
4.23 Exercises

Chapter 5: Hyperbolic Incircles and Excircles
5.1 Einstein Gyrotriangle Ingyrocenter and Exgyrocenters
5.2 Einstein Ingyrocircle and Exgyrocircle Tangency Points
5.3 Useful Gyrotriangle Gyrotrigonometric Relations
5.4 The Tangency Points Expressed Gyrotrigonometrically
5.5 Möbius Gyrotriangle Ingyrocenter and Exgyrocenters
5.6 From Gyrotriangle Tangency Points to Gyrotriangle Gyrocenters
5.7 Exercises

Chapter 6: Hyperbolic Tetrahedra
6.1 Gyrotetrahedron Gyroaltitude
6.2 Point Gyroplane Relations
6.3 Gyrotetrahedron Ingyrocenter and Exgyrocenters
6.4 In-Exgyrosphere Tangency Points
6.5 Gyrotrigonometric Gyrobarycentric Coordinates for the Gyrotetrahedron In-Exgyrocenters
6.6 Gyrotetrahedron Circumgyrocenter
6.7 Exercises

Chapter 7: Comparative Patterns
7.1 Gyromidpoints and Gyrocentroids
7.2 Two and Three Dimensional Ingyrocenters
7.3 Two and Three Dimensional Circumgyrocenters
7.4 Tetrahedron Incenter and Excenters
7.5 Comparative Study of the Pythagorean Theorem
7.6 Hyperbolic Heron’s Formula
7.7 Exercises

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Tags: Abraham Albert Ungar, Barycentric, Euclidean, Hyperbolic