Barycentric calculus in Euclidean and hyperbolic geometry: a comparative introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2010
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | XIV, 344 S. graph. Darst. |
| ISBN: | 9789814304931 981430493X |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV040910727 | ||
| 003 | DE-604 | ||
| 005 | 20130513 | ||
| 007 | t | ||
| 008 | 130325s2010 d||| |||| 00||| eng d | ||
| 020 | |a 9789814304931 |c (hbk.) £57.00 |9 978-981-4304-93-1 | ||
| 020 | |a 981430493X |c (hbk.) £57.00 |9 981-4304-93-X | ||
| 035 | |a (OCoLC)731506564 | ||
| 035 | |a (DE-599)OBVAC08235267 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-83 |a DE-20 |a DE-188 | ||
| 082 | 0 | |a 516.22 | |
| 084 | |a SK 380 |0 (DE-625)143235: |2 rvk | ||
| 084 | |a 51M05 |2 msc | ||
| 084 | |a 51M09 |2 msc | ||
| 100 | 1 | |a Ungar, Abraham A. |e Verfasser |0 (DE-588)141071176 |4 aut | |
| 245 | 1 | 0 | |a Barycentric calculus in Euclidean and hyperbolic geometry |b a comparative introduction |c Abraham Albert Ungar |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2010 | |
| 300 | |a XIV, 344 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Hyperbolische Geometrie |0 (DE-588)4161041-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Spezielle Relativitätstheorie |0 (DE-588)4182215-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Spezielle Relativitätstheorie |0 (DE-588)4182215-8 |D s |
| 689 | 0 | 1 | |a Hyperbolische Geometrie |0 (DE-588)4161041-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | |m DE-601 |u http://www.gbv.de/dms/bowker/toc/9789814304931.pdf |3 Inhaltsverzeichnis | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025890054&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025890054 | ||
Datensatz im Suchindex
| _version_ | 1804150202947338240 |
|---|---|
| adam_text | Titel: Barycentric calculus in Euclidean and hyperbolic geometry
Autor: Ungar, Abraham A
Jahr: 2010
Contents
Preface vii
1. Euclidean Barycentric Coordinates
and the Classic Triangle Centers 1
1.1 Points, Lines, Distance and Isometries............ 2
1.2 Vectors, Angles and Triangles................. 5
1.3 Euclidean Barycentric Coordinates.............. 8
1.4 Analogies with Classical Mechanics.............. 11
1.5 Barycentric Representations are Covariant.......... 12
1.6 Vector Barycentric Representation.............. 14
1.7 Triangle Centroid........................ 17
1.8 Triangle Altitude........................ 19
1.9 Triangle Orthocenter...................... 24
1.10 Triangle Incenter........................ 27
1.11 Triangle Inradius........................ 33
1.12 Triangle Circumcenter..................... 36
1.13Circumradius.......................... 40
1.14 Triangle Incircle and Excircles................. 42
1.15 Excircle Tangency Points ................... 47
1.16 From Triangle Tangency Points to Triangle Centers..... 52
1.17 Triangle In-Exradii....................... 55
1.18 A Step Toward the Comparative Study ........... 57
1.19 Tetrahedron Altitude...................... 58
1.20 Tetrahedron Altitude Length................. 62
1.21 Exercises ............................ 63
Barycentric Calculus
Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry 65
2.1 Einstein Addition........................ 66
2.2 Einstein Gyration........................ 70
2.3 From Einstein Velocity Addition to Gyrogroups....... 73
2.4 First Gyrogroup Theorems .................. 77
2.5 The Two Basic Equations of Gyrogroups........... 82
2.6 Einstein Gyrovector Spaces.................. 86
2.7 Gyrovector Spaces....................... 89
2.8 Einstein Points, Gyrolines and Gyrodistance......... 95
2.9 Linking Einstein Addition to Hyperbolic Geometry..... 99
2.10 Einstein Gyrovectors, Gyroangles and Gyrotriangles .... 101
2.11 The Law of Gyrocosines.................... 106
2.12 The SSS to AAA Conversion Law.............. 108
2.13Inequalities for Gyrotriangles................. 109
2.14 The AAA to SSS Conversion Law.............. 111
2.15 The Law of Gyrosines..................... 115
2.16 The ASA to SAS Conversion Law.............. 115
2.17Gyrotriangle Defect ...................... 116
2.18Right Gyrotriangles ...................... 118
2.19 Einstein Gyrotrigonometry and Gyroarea.......... 120
2.20 Gyrotriangle Gyroarea Addition Law............ 124
2.21 Gyrodistance Between a Point and a Gyroline....... 127
2.22 The Gyroangle Bisector Theorem............... 133
2.23 Möbius Addition and Möbius Gyrogroups.......... 135
2.24 Möbius Gyration........................ 136
2.25 Möbius Gyrovector Spaces................... 138
2.26 Möbius Points, Gyrolines and Gyrodistance......... 139
2.27 Linking Möbius Addition to Hyperbolic Geometry..... 142
2.28 Möbius Gyrovectors, Gyroangles and Gyrotriangles..... 143
2.29 Gyrovector Space Isomorphism................ 148
2.30 Möbius Gyrotrigonometry................... 153
2.31 Exercises ............................ 155
The Interplay of Einstein Addition and Vector Addition 157
3.1 Extension of Rns into Tns+l................... 157
3.2 Scalar Multiplication and Addition in T£+1......... 162
3.3 Inner Product and Norm in T£+1............... 163
3.4 Unit Elements of T?+1..................... 165
Contents xiii
3.5 From T?+1 back to W8l..................... 173
4. Hyperbolic Barycentric Coordinates
and Hyperbolic Triangle Centers 179
4.1 Gyrobarycentric Coordinates in Einstein Gyrovector
Spaces.............................. 179
4.2 Analogies with Relativistic Mechanics............ 183
4.3 Gyrobarycentric Coordinates in Möbius Gyrovector
Spaces.............................. 184
4.4 Einstein Gyromidpoint..................... 187
4.5 Möbius Gyromidpoint..................... 189
4.6 Einstein Gyrotriangle Gyrocentroid.............. 190
4.7 Einstein Gyrotetrahedron Gyrocentroid ........... 197
4.8 Möbius Gyrotriangle Gyrocentroid.............. 199
4.9 Möbius Gyrotetrahedron Gyrocentroid............ 200
4.10 Foot of a Gyrotriangle Gyroaltitude............. 201
4.11 Einstein Point to Gyroline Gyrodistance........... 205
4.12 Möbius Point to Gyroline Gyrodistance........... 207
4.13 Einstein Gyrotriangle Orthogyrocenter............ 209
4.14 Möbius Gyrotriangle Orthogyrocenter............ 219
4.15 Foot of a Gyrotriangle Gyroangle Bisector.......... 224
4.16 Einstein Gyrotriangle Ingyrocenter.............. 229
4.17 Ingyrocenter to Gyrotriangle Side Gyrodistance....... 237
4.18 Möbius Gyrotriangle Ingyrocenter .............. 240
4.19 Einstein Gyrotriangle Circumgyrocenter........... 244
4.20 Einstein Gyrotriangle Circumgyroradius........... 249
4.21 Möbius Gyrotriangle Circumgyrocenter ........... 250
4.22 Comparative Study of Gyrotriangle Gyrocenters ...... 253
4.23 Exercises ............................ 257
5. Hyperbolic Incircles and Excircles 259
5.1 Einstein Gyrotriangle Ingyrocenter and Exgyrocenters . . . 259
5.2 Einstein Ingyrocircle and Exgyrocircle Tangency Points . . 265
5.3 Useful Gyrotriangle Gyrotrigonometric Relations...... 268
5.4 The Tangency Points Expressed Gyrotrigonometricallv . . . 269
5.5 Möbius Gyrotriangle Ingyrocenter and Exgyrocenters .... 275
5.6 From Gyrotriangle Tangency Points to Gyrotriangle
Gyrocenters........................... 280
xiv Barycentric Calculus
5.7 Exercises ............................ 283
6. Hyperbolic Tetrahedra 285
6.1 Gyrotetrahedron Gyroaltitude................. 285
6.2 Point Gyroplane Relation«................... 294
6.3 Gyrotetrahedron Ingyrocenter and Exgyrocenters...... 296
6.4 In-Exgyrosphere Tangency Points............... 305
6.5 Gyrotrigonometric Gyrobarycentric Coordinates for the
Gyrotetrahedron In-Exgyrocenters.............. 307
6.6 Gyrotetrahedron Circumgyrocenter.............. 316
6.7 Exercises ............................ 320
7. Comparative Patterns 323
7.1 Gyromidpoints and Gyrocentroids.............. 323
7.2 Two and Three Dimensional Ingyrocenters.......... 326
7.3 Two and Three Dimensional Circumgyrocenters....... 328
7.4 Tetrahedron Incenter and Excenters............. 329
7.5 Comparative study of the Pythagorean Theorem...... 331
7.6 Hyperbolic Heron s Formula.................. 333
7.7 Exercises ............................ 334
Notation And Special Symbols 335
Bibliography 337
Index 341
|
| any_adam_object | 1 |
| author | Ungar, Abraham A. |
| author_GND | (DE-588)141071176 |
| author_facet | Ungar, Abraham A. |
| author_role | aut |
| author_sort | Ungar, Abraham A. |
| author_variant | a a u aa aau |
| building | Verbundindex |
| bvnumber | BV040910727 |
| classification_rvk | SK 380 |
| ctrlnum | (OCoLC)731506564 (DE-599)OBVAC08235267 |
| dewey-full | 516.22 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.22 |
| dewey-search | 516.22 |
| dewey-sort | 3516.22 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01659nam a2200397 c 4500</leader><controlfield tag="001">BV040910727</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20130513 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">130325s2010 d||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9789814304931</subfield><subfield code="c">(hbk.) £57.00</subfield><subfield code="9">978-981-4304-93-1</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">981430493X</subfield><subfield code="c">(hbk.) £57.00</subfield><subfield code="9">981-4304-93-X</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)731506564</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)OBVAC08235267</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-83</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.22</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 380</subfield><subfield code="0">(DE-625)143235:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">51M05</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">51M09</subfield><subfield code="2">msc</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Ungar, Abraham A.</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)141071176</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Barycentric calculus in Euclidean and hyperbolic geometry</subfield><subfield code="b">a comparative introduction</subfield><subfield code="c">Abraham Albert Ungar</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Singapore [u.a.]</subfield><subfield code="b">World Scientific</subfield><subfield code="c">2010</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XIV, 344 S.</subfield><subfield code="b">graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Hyperbolische Geometrie</subfield><subfield code="0">(DE-588)4161041-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Spezielle Relativitätstheorie</subfield><subfield code="0">(DE-588)4182215-8</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Spezielle Relativitätstheorie</subfield><subfield code="0">(DE-588)4182215-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Hyperbolische Geometrie</subfield><subfield code="0">(DE-588)4161041-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2=" "><subfield code="m">DE-601</subfield><subfield code="u">http://www.gbv.de/dms/bowker/toc/9789814304931.pdf</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025890054&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-025890054</subfield></datafield></record></collection> |
| id | DE-604.BV040910727 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:35:05Z |
| institution | BVB |
| isbn | 9789814304931 981430493X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025890054 |
| oclc_num | 731506564 |
| open_access_boolean | |
| owner | DE-83 DE-20 DE-188 |
| owner_facet | DE-83 DE-20 DE-188 |
| physical | XIV, 344 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Ungar, Abraham A. Verfasser (DE-588)141071176 aut Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction Abraham Albert Ungar Singapore [u.a.] World Scientific 2010 XIV, 344 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hyperbolische Geometrie (DE-588)4161041-6 gnd rswk-swf Spezielle Relativitätstheorie (DE-588)4182215-8 gnd rswk-swf Spezielle Relativitätstheorie (DE-588)4182215-8 s Hyperbolische Geometrie (DE-588)4161041-6 s DE-604 DE-601 http://www.gbv.de/dms/bowker/toc/9789814304931.pdf Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025890054&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ungar, Abraham A. Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction Hyperbolische Geometrie (DE-588)4161041-6 gnd Spezielle Relativitätstheorie (DE-588)4182215-8 gnd |
| subject_GND | (DE-588)4161041-6 (DE-588)4182215-8 |
| title | Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction |
| title_auth | Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction |
| title_exact_search | Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction |
| title_full | Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction Abraham Albert Ungar |
| title_fullStr | Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction Abraham Albert Ungar |
| title_full_unstemmed | Barycentric calculus in Euclidean and hyperbolic geometry a comparative introduction Abraham Albert Ungar |
| title_short | Barycentric calculus in Euclidean and hyperbolic geometry |
| title_sort | barycentric calculus in euclidean and hyperbolic geometry a comparative introduction |
| title_sub | a comparative introduction |
| topic | Hyperbolische Geometrie (DE-588)4161041-6 gnd Spezielle Relativitätstheorie (DE-588)4182215-8 gnd |
| topic_facet | Hyperbolische Geometrie Spezielle Relativitätstheorie |
| url | http://www.gbv.de/dms/bowker/toc/9789814304931.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025890054&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ungarabrahama barycentriccalculusineuclideanandhyperbolicgeometryacomparativeintroduction |
Es ist kein Print-Exemplar vorhanden.
Inhaltsverzeichnis