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This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. 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Hyperbolic Geometry 373 42. Hyperbolic Trigonometry 41. Hilbert\u0026#39;s Arithmetic of Ends 43. Characterization of Hilbert Planes 388 403 415 Chapter 8. Polyhedra 435 44. The Five Regular Solids 436 45. Euler\u0026#39;s and\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PR10\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA5","page_number":"5","snippet_text":"... model of non- Euclidean geometry over a field due to Poincaré . Then we give an axiomatic treatment of hyperbolic geometry based on the axiom of existence of limit- ing parallel ... \u003cb\u003ePoincaré model\u003c/b\u003e over its associated Introduction 5."},{"page_id":"PA6","page_number":"6","snippet_text":"Robin Hartshorne. plane is isomorphic to the \u003cb\u003ePoincaré model\u003c/b\u003e over its associated field . Once again , algebraic methods help us to understand geometry . A note on references : Propositions in Euclid\u0026#39;s Elements are given by book and number\u0026nbsp;..."},{"page_id":"PA70","page_number":"70","snippet_text":"... model satisfying ( 12 ) , ( 13 ) , ( P ) and not ( I1 ) , just take a set of three points and no lines at all . B A ... \u003cb\u003ePoincaré model\u003c/b\u003e over the real numbers ( Ex- ercise 43.2 ) . However , from the point of view of this book , it\u0026nbsp;..."},{"page_id":"PA95","page_number":"95","snippet_text":"... model of all the axioms listed so far . You are probably willing to believe this , but the precise definition of ... \u003cb\u003ePoincaré model\u003c/b\u003e , which we will discuss in Section 39 . Exercises 9.1 ( Difference of angles ) . Suppose we 9\u0026nbsp;..."},{"page_id":"PA182","page_number":"182","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e of non - Euclidean geometry . So we can ask , is it true in any Hilbert plane ? ) 20.6 In a Hilbert plane with ( P ) , given two circles by their centers and one point each , but without being given their intersection\u0026nbsp;..."},{"page_id":"PA341","page_number":"341","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e of non - Euclidean geome- try : It plays an essential role there . In projective geometry the cross - ratio is also important . A projectivity from one line to another is defined as a composition of a finite number of\u0026nbsp;..."},{"page_id":"PA358","page_number":"358","snippet_text":"... \u003cb\u003emodel\u003c/b\u003e as follows . Two P - angles are P - congruent if the Euclidean angles they define are congruent in the usual sense . For line segments , we proceed as follows . Given two P - points , let the P - line joining them be the circle y\u0026nbsp;..."},{"page_id":"PA359","page_number":"359","snippet_text":"... model . Recall from Section 17 that a rigid motion is a transformation of the ge- ometry that preserves the undefined notions of point , line , betweenness , and congruence . In our case , a P - rigid motion will ... \u003cb\u003ePoincaré Model\u003c/b\u003e 359."},{"page_id":"PA360","page_number":"360","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e . Proposition 39.6 Axioms ( C1 ) and ( C6 ) hold in the \u003cb\u003ePoincaré model\u003c/b\u003e . Proof Suppose it is required to find a point B \u0026#39; on a P - ray emanating from a point A \u0026#39; such that A\u0026#39;B \u0026#39; is P - congruent to a given P - segment AB\u0026nbsp;..."},{"page_id":"PA361","page_number":"361","snippet_text":"Robin Hartshorne. In order to discuss ( E ) in the \u003cb\u003ePoincaré model\u003c/b\u003e , we first need to identify what is a P - circle . By definition , of course , it is the set of all P - points B \u0026#39; such that the P - segment A\u0026#39;B ... \u003cb\u003ePoincaré Model\u003c/b\u003e 361."},{"page_id":"PA362","page_number":"362","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over a Euclidean ordered field F. Proof Since P - lines and P - circles are all either usual circles or lines through O , and since betweenness is the same in the P - model as in the ambient Euclidean space , ( E ) in\u0026nbsp;..."},{"page_id":"PA363","page_number":"363","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over a field F , the function μ ( AB ) = ( AB , PQ ) 1 is a multi- plicative distance function with values in the multiplicative group of the field ( F \u0026gt; o , · ) . Proof Because of our convention ... \u003cb\u003ePoincaré Model\u003c/b\u003e 363."},{"page_id":"PA364","page_number":"364","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e . Using a little Euclidean geometry in the ambient Cartesian plane , we can derive a marvelous relationship between the length of a segment and the angle it makes with a limiting parallel . Proposition 39.13 ( Bolyai\u0026#39;s\u0026nbsp;..."},{"page_id":"PA366","page_number":"366","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , whereas in Euclidean geom- etry the choice of unit length is arbitrary . Exercises All exercises take place in the \u003cb\u003ePoincaré model\u003c/b\u003e over a Euclidean ordered field F , unless otherwise noted . Proofs should be based on\u0026nbsp;..."},{"page_id":"PA367","page_number":"367","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , without using rigid motions , as follows . Given a point A , a P - line y , and given a quan- tity be F , 0 \u0026lt; b \u0026lt; 1 , we need to find a point Be ... \u003cb\u003ePoincaré model\u003c/b\u003e of non - Euclidean geometry. 39. The \u003cb\u003ePoincaré Model\u003c/b\u003e 367."},{"page_id":"PA368","page_number":"368","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e made inside a circle I in the Cartesian plane over F , we have seen that any Euclidean circle y entirely contained inside I is a P - circle ( Proposition 39.8 ) . ( a ) If y is a Euclidean circle inside. Congruent\u0026nbsp;..."},{"page_id":"PA369","page_number":"369","snippet_text":"... model . ( b ) If y is a Euclidean circle that cuts r at points P , Q , let I be the P - line having the endpoints P , Q. Show that the points of y inside I form a curve of points equidistant from the P - line 1 ... \u003cb\u003ePoincaré Model\u003c/b\u003e 369."},{"page_id":"PA371","page_number":"371","snippet_text":"... model from the \u003cb\u003ePoincaré model\u003c/b\u003e . Let △ be a circle of radius 1 centered at the origin , and in the Cartesian 3 - space , place a sphere of radius 1 on the plane , with its south pole at the origin ( cf. Exer- cise 37.1 ) . Let r be the\u0026nbsp;..."},{"page_id":"PA372","page_number":"372","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over a field that need not be Euclidean . Let F be a Pythagorean ordered field , let de F , and let I be the circle x2 + y2 = d , which may be a virtual circle if √d \u0026amp; F ( Exercise 37.17 ) . We define the Poincaré\u0026nbsp;..."},{"page_id":"PA373","page_number":"373","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e in the virtual circle x2 + y2 = √2 over the field F of Exercise 39.30 , not every segment can be the side of an equilateral triangle , as follows . ( a ) If xe F with 0 \u0026lt; x and x2 \u0026lt; √2 , let AB be the segment from ( 0\u0026nbsp;..."},{"page_id":"PA374","page_number":"374","snippet_text":"Robin Hartshorne. geometry is isomorphic to the \u003cb\u003ePoincaré model\u003c/b\u003e over this field ( Section 43 ) . Using coordinates from this field we can develop non - Euclidean analytic geometry and trigonometry ( Section 42 ) . So at this point we\u0026nbsp;..."},{"page_id":"PA388","page_number":"388","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over that field . We start with a hyperbolic plane , as in the previous section , which is a Hilbert plane satisfying the axiom of limiting parallels ( L ) . Proposition 41.1 Let A , B , C be three noncollinear points in\u0026nbsp;..."},{"page_id":"PA416","page_number":"416","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over a field in Section 39 , and have used the ambient Cartesian geometry to investigate some of its properties . Then in Section 41 and Section 42 we have introduced the field of ends into an abstract hyperbolic plane\u0026nbsp;..."},{"page_id":"PA420","page_number":"420","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , reflection is given by circular inversion in the corresponding Cartesian circle ( 39.5 ) . So given a point a on г , let △ be the P - line ( a , ∞o ) , which is a circle orthogonal to г at the points a and co . Let A\u0026nbsp;..."},{"page_id":"PA421","page_number":"421","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e . This is given by circular inversion in a circle A , orthogo- nal to г , passing through the points a and -α . Let A be the center of this circle . Then the angle OAx is equal to - RA in our diagram . Thus OA = -1 / cos\u0026nbsp;..."},{"page_id":"PA422","page_number":"422","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over the field F. Proof Indeed , the field F is Euclidean ( 41.4 ) , and the plane II and the \u003cb\u003ePoincaré model\u003c/b\u003e over F both have isomorphic fields of ends by ( 43.2 ) , so by ( 43.1 ) they are isomorphic planes\u0026nbsp;..."},{"page_id":"PA424","page_number":"424","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over a Euclidean field F , it is the group of positive elements of the field under multiplication ( F \u0026gt; 0 , ) using the multiplicative distance function ( 39.10 ) . We say that a subgroup M of an ordered abelian group G\u0026nbsp;..."},{"page_id":"PA431","page_number":"431","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e over the real numbers R. 43.3 ( a ) Use Theorem 43.7 to show that in a semi - Euclidean plane , Aristotle\u0026#39;s axiom ( Section 33 ) implies ( P ) . ( b ) Now prove the same result without using Theorem 43.7 , by\u0026nbsp;..."},{"page_id":"PA494","page_number":"494","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e . Bolyai , in his Science of Absolute Space ( 1832 ; English translation in Bonola ( 1955 ) ) , Section 43 , studied the area of a circle , and recognized that it could be \u0026quot; squared \u0026quot; or not , depending on the arithmetic\u0026nbsp;..."},{"page_id":"PA503","page_number":"503","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e ( Section 39 ) over a field F is the non - Euclidean geometry on the set of points inside a fixed circle . For other axioms , acronyms , and definitions , see the Index . Index of Euclid\u0026#39;s Propositions Post . 1 2 , 18 503\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA503\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA507","page_number":"507","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 369 in semi - Euclidean plane , 319 altitudes of triangle , 51 , 63 , 487 analytic geometry , 118 , 415 , 427. See also hyperbolic analytic geometry analytic proof of altitudes of triangle , 119 of pentagon , 125 of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA507\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA508","page_number":"508","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 366 infinitesimally close to 2RA , 431 angle trisector , 277 angle , 8 , 141 , 326 acute , 141 between circles , 336 cosine function , 403 definition of , 28 , 77 enclosing line of , 378 exterior , 101 in a semicircle\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA508\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA509","page_number":"509","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 357 over an ordered field , 137 used in exterior angle theorem , 101 bicapped pentagonal antiprism , 459 bicapped square antiprism , 456 bidiminished icosahedron , 464 Billingsley , Henry , 1 bilunabirotunda , 469\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA509\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA510","page_number":"510","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 362 circle area in hyperbolic plane , 407 , 413 , 414 area of , 221 , 333 circumscribed , 25 definition of , 27 , 88 , 104 exterior is segment - connected , 116 exterior not segment - connected , 433 given 2 points and\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA510\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA512","page_number":"512","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 363 Dedekind , Richard , 4 , 167 , 488 defect as measure of area , 328 at vertex of polyhedron , 450 , 458 is additive , 311 of triangle , 311 , 324 , 325 , 326 , 327 definitions , 27 , 28 degree measure of angle , 166\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA512\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA513","page_number":"513","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 359 in spherical geometry , 318 exposition , 14 exterior angle theorem , 101 , 376 exterior angle , 36 , 321. See also ( I.16 ) in Index of Euclid\u0026#39;s Propositions exterior of circle , 105 , 116 of polygon , 205 of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA513\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA514","page_number":"514","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 418 of ends , 391 of Laurent series , 163 , 372 of line segments , 168 of rational functions , 158 , 163 of real numbers , 119 of segment arithmetic , 179 , 206 ordered , 2 , 117 , 135-140 Pythagorean , 142 , 145 skew\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA514\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA516","page_number":"516","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 422 law of sines , 411 measure of area function , 407 perpendicular bisectors of triangle , 388 right triangle in , 404 , 406 rigid motions , 391 sine function , 405 squaring of circle , 409 , 414 tangent function , 395\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA516\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA517","page_number":"517","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 366 in field of segment arithmetic , 175 of chord of circle , 124 of side of polygon , 124 Lenstra , Hendrik , 127 less than for angles , 94 for line segments , 85 limit line , 386 limit quadrilateral , 386 , 401 limit\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA517\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA518","page_number":"518","snippet_text":"... model isomorphic , 68 of axiom system , 67 , 95 of betweenness , 78 of congruence axioms , 87 , 89 , 90 over a field , 128 Poincaré ... model , 370 , 371 non - Archimedean model , 161 , 162 \u003cb\u003ePoincaré model\u003c/b\u003e , 356 semielliptic , 318\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA518\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA519","page_number":"519","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 357 in Cartesian plane , 130 parallel lines , 38 by Hilbert\u0026#39;s tools , 103 common orthogonal to , 366 , 377 construction of , 24 cut equal segments , 59 definition , 68 distance between , 298 in Cartesian plane , 119\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA519\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA520","page_number":"520","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 70 , 95 , 295 , 355-373 ( A ) plus ( E ) implies , 426 ( E ) holds , 362 absolute unit of length , 366 addition of ends , 420 analogue of III.36 , 370 angle sum in , 366 betweenness , 357 circle in , 361 congruence\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA520\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA521","page_number":"521","snippet_text":"... models , 446 plane projection of , 449 regular , 446 symmetry group of , 469-480 polynomial cyclotomic , 251 irreducible ... \u003cb\u003ePoincaré model\u003c/b\u003e over , 372 Pythagorean theorem , 8 , 42 , 46 , 203. See also ( 1.47 ) in Index of Euclid\u0026#39;s\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA521\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA522","page_number":"522","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 359 product of , 157 theorem of three , 158 , 389 reflexivity , 82 regular pentagon , 49 exists over 2 , 147 regular polygon , 241 , 250 , 437 constructible , 276 regular polyhedron , 435 , 442 , 443 definition of , 446\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA522\u0026vq=Poincar%C3%A9+model"},{"page_id":"PA526","page_number":"526","snippet_text":"... \u003cb\u003ePoincaré model\u003c/b\u003e , 369 solves cubic equation , 271 with marked ruler , 260 with parabola , 277 trisection of segment , 25 in \u003cb\u003ePoincaré model\u003c/b\u003e , 369 truncated cube , 462 truncated cuboctahedron , 462 truncated dodecahedron , 462 truncated\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA526\u0026vq=Poincar%C3%A9+model"}],"search_query_escaped":"Poincaré model"},{});</script></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>