2345:
1258:
2337:
769:
3129:
63:
502:
764:{\displaystyle {\begin{aligned}\left.{\frac {\mathrm {d} }{\mathrm {d} {\theta }}}\operatorname {dist} (\langle r,\theta _{1}\rangle ,\langle r,\theta _{1}+\theta \rangle )\right|_{\theta =0}&=\left.{\frac {\mathrm {d} }{\mathrm {d} {\theta }}}\operatorname {arcosh} \,\left(\cosh ^{2}r-\sinh ^{2}r\cos(\theta )\right)\right|_{\theta =0}\\&=\sinh(r)\end{aligned}}}
369:
4049:
2600:
4090:
traditionally is concerned with
Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the
161:
3787:
3118:
3672:
2051:
997:
2390:
2316:
883:
2713:
2819:
1940:
364:{\displaystyle \operatorname {dist} (\langle r_{1},\theta _{1}\rangle ,\langle r_{2},\theta _{2}\rangle )=\operatorname {arcosh} \,\left(\cosh r_{1}\cosh r_{2}-\sinh r_{1}\sinh r_{2}\cos(\theta _{2}-\theta _{1})\right)\,.}
4044:{\displaystyle \left({\frac {\tanh x_{a}}{\sqrt {1-\tanh ^{2}x_{a}-\tanh ^{2}y_{a}}}}\ ,\ {\frac {\tanh y_{a}}{\sqrt {1-\tanh ^{2}x_{a}-\tanh ^{2}y_{a}}}}\ ,\ {\frac {1}{\sqrt {1-\tanh ^{2}x_{a}-\tanh ^{2}y_{a}}}}\right)}
1753:
1672:
2943:
2891:
1866:
1814:
494:
3001:
450:
3495:
2995:
1490:
3464:
2595:{\displaystyle \operatorname {dist} (\langle x_{1},y_{1}\rangle ,\langle x_{2},y_{2}\rangle )=\operatorname {arcosh} \left(\cosh y_{1}\cosh(x_{2}-x_{1})\cosh y_{2}-\sinh y_{1}\sinh y_{2}\right)\,.}
1946:
1354:
893:
507:
2117:
3758:
2384:
will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other). Then the distance between two such points will be
1122:
1591:
1539:
1403:
1075:
781:
3309:
3282:
3235:
3185:
3329:
3255:
3205:
3154:
2615:
1136:
of the quadrant. Due to the study of ratios in physics and economics where the quadrant is the universe of discourse, its points are said to be located by
2732:
1872:
1681:
1600:
4204:, Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp. 1–35, 2009
2344:
3113:{\displaystyle \theta =2\operatorname {arctan} \,\left({\frac {\sinh y}{\sinh x\cosh y+{\sqrt {\cosh ^{2}x\cosh ^{2}y-1}}}}\right)\,.}
2897:
2845:
1820:
1768:
3667:{\displaystyle x_{p}={\frac {x_{b}}{1+{\sqrt {1-x_{b}^{2}-y_{b}^{2}}}}},\ \ y_{p}={\frac {y_{b}}{1+{\sqrt {1-x_{b}^{2}-y_{b}^{2}}}}}}
455:
1257:
377:
2356:-axis) in the hyperbolic plane (with a standardized curvature of -1) and label the points on it by their distance from an origin (
2949:
4228:
2046:{\displaystyle \theta =2\operatorname {arctan} \,\left({\frac {\tanh y}{\tanh x+{\sqrt {\tanh ^{2}x+\tanh ^{2}y}}}}\right)\,.}
1411:
4186:
4138:
3376:
992:{\displaystyle \theta =\theta _{0}\pm {\frac {\pi }{2}}\quad {\text{ or }}\quad \tanh r=\tanh r_{0}\sec(\theta -\theta _{0})}
2311:{\displaystyle x_{l}=x_{a}\ ,\ \tanh(y_{l})=\tanh(y_{a})\cosh(x_{a})\ ,\ \tanh(y_{a})={\frac {\tanh(y_{l})}{\cosh(x_{l})}}}
1275:
2111:
is the distance along the perpendicular of the given point to its foot (positive on one side and negative on the other).
4251:
2835:
The relationship of
Lobachevsky coordinates to polar coordinates (assuming the origin is the pole and that the positive
1132:
of the
Poincaré model carry over to the quadrant; in particular the left or right shifts of the real axis correspond to
1022:
4256:
2336:
1217:
117:
31:
This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane.
3696:
1167:
There are however different coordinate systems for hyperbolic plane geometry. All are based on choosing a real (non
3682:
The
Weierstrass coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the
1758:
The relationship of axial coordinates to polar coordinates (assuming the origin is the pole and that the positive
152:
3356:
1128:) in the upper half-plane. The hyperbolic metric in the quadrant depends on the Poincaré half-plane metric. The
3474:
The
Poincaré coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the
2364:-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates
2348:
Circles about the points (0,0), (0,1), (0,2) and (0,3) of radius 3.5 in the
Lobachevsky hyperbolic coordinates.
2322:
The
Lobachevsky coordinates are useful for integration for length of curves and area between lines and curves.
3475:
1137:
1083:
3352:
1161:
1157:
4213:
3128:
1172:
51:
1547:
1495:
1359:
1152:
do not exist. The sum of the angles of a quadrilateral in hyperbolic geometry is always less than 4
1047:
28:. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.
3340:
2606:
2330:
2326:
1133:
102:
98:
47:
43:
39:
17:
878:{\displaystyle (\mathrm {d} s)^{2}=(\mathrm {d} r)^{2}+\sinh ^{2}r\,(\mathrm {d} \theta )^{2}\,.}
35:
4130:
62:
4178:
4182:
4134:
3683:
1129:
94:
4081:
4070:
1078:
3287:
3260:
3213:
3163:
4232:
4087:
21:
4201:
2340:
Circles about the origin of radius 1, 5 and 10 in the
Lobachevsky hyperbolic coordinates.
2708:{\displaystyle (\mathrm {d} s)^{2}=\cosh ^{2}y\,(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}
4171:
4123:
3314:
3240:
3190:
3139:
1042:
125:
4245:
4226:
Barycentric
Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction
4225:
775:
2814:{\displaystyle \tanh y=A\cosh x+B\sinh x\quad {\text{ when }}\quad A^{2}<1+B^{2}}
2096:-axis to the origin (positive on one side and negative on the other, the same as in
74:. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or
3343:
and take the
Euclidean coordinates inside the model as the hyperbolic coordinates.
1269:
have axial coordinates, but not every pair of real numbers corresponds to a point.
1241:-axis to the origin (regarded as positive on one side and negative on the other);
1935:{\displaystyle r=\operatorname {artanh} \,({\sqrt {\tanh ^{2}x+\tanh ^{2}y}}\,)}
1266:
1168:
1153:
25:
1220:, the coordinates are found by dropping perpendiculars from the point onto the
4065:
3208:
1149:
3691:
3157:
2352:
Construct a Cartesian-like coordinate system as follows. Choose a line (the
1748:{\displaystyle \operatorname {artanh} \left(\tanh(x_{a})\cosh(y_{a})\right)}
1667:{\displaystyle \operatorname {artanh} \left(\tanh(y_{a})\cosh(x_{a})\right)}
155:, we get that the distance between two points given in polar coordinates is
91:
2718:
In this coordinate system, straight lines are either perpendicular to the
106:
1025:
is closely related to a model of the hyperbolic plane in the quadrant
3489:
The Poincaré coordinates, in terms of the Beltrami coordinates, are:
1160:). Also in hyperbolic geometry there are no equidistant lines (see
3127:
2343:
2335:
1256:
1013:
are the coordinates of the nearest point on the line to the pole.
110:
4129:(Corrected 4. print. ed.). New York, NY: Springer. pp.
4095:
encapsulate the specification of the anglesum being 180 degrees.
2938:{\displaystyle y=\operatorname {arsinh} \,(\sinh r\sin \theta )}
2886:{\displaystyle x=\operatorname {artanh} \,(\tanh r\cos \theta )}
1861:{\displaystyle y=\operatorname {artanh} \,(\tanh r\sin \theta )}
1809:{\displaystyle x=\operatorname {artanh} \,(\tanh r\cos \theta )}
3486:
and the origin is mapped to the centre of the boundary circle.
3367:
and the origin is mapped to the centre of the boundary circle.
489:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} {\theta }}}=0}
4214:
Hyperbolic Triangle Centers: The Special Relativistic Approach
445:{\displaystyle r=r_{1}=r_{2},\theta =\theta _{2}-\theta _{1}}
3136:
Another coordinate system represents each hyperbolic point
2990:{\displaystyle r=\operatorname {arcosh} \,(\cosh x\cosh y)}
627:
512:
2832:
are real parameters which characterize the straight line.
2092:
is the distance from the foot of the perpendicular to the
1250:
is the distance from the foot of the perpendicular on the
1237:
is the distance from the foot of the perpendicular on the
887:
The straight lines are described by equations of the form
1261:
Circles about the origin in hyperbolic axial coordinates.
1485:{\displaystyle \tanh ^{2}(x_{a})+\tanh ^{2}(y_{a})>1}
4125:
The foundations of geometry and the non-Euclidean plane
3459:{\displaystyle x_{b}=\tanh(x_{a}),\ y_{b}=\tanh(y_{a})}
1164:). This all has influences on the coordinate systems.
3790:
3699:
3498:
3379:
3317:
3290:
3263:
3243:
3216:
3193:
3166:
3142:
3004:
2952:
2900:
2848:
2735:
2618:
2393:
2120:
1949:
1875:
1823:
1771:
1684:
1603:
1550:
1498:
1414:
1362:
1349:{\displaystyle \tanh ^{2}(x_{a})+\tanh ^{2}(y_{a})=1}
1278:
1086:
1050:
896:
784:
505:
458:
380:
164:
3237:
along the horocycle between a fixed reference point
3156:
by two real numbers, defined relative to some given
2726:= a constant) or described by equations of the form
2605:
This formula can be derived from the formulas about
2380:
will be the label of the foot of the perpendicular.
4170:
4122:
4043:
3752:
3666:
3458:
3323:
3303:
3276:
3249:
3229:
3199:
3179:
3148:
3112:
2989:
2937:
2885:
2813:
2707:
2594:
2310:
2045:
1934:
1860:
1808:
1747:
1666:
1585:
1533:
1484:
1397:
1348:
1116:
1069:
991:
877:
763:
488:
444:
363:
116:The reference point (analogous to the origin of a
4116:
4114:
4112:
4110:
4108:
128:from the pole in the reference direction is the
66:Points in the polar coordinate system with pole
3760:and the origin is mapped to the point (0,0,1).
3753:{\displaystyle (t\ ,\ 0\ ,\ {\sqrt {t^{2}+1}})}
2079:are found by dropping a perpendicular onto the
24:, each point can be uniquely identified by two
4091:expressions to coincide, the expressions must
3339:Model-based coordinate systems use one of the
3355:of the point when the point is mapped in the
8:
4169:Ramsay, Arlan; Richtmyer, Robert D. (1995).
3351:The Beltrami coordinates of a point are the
3160:. These numbers are the hyperbolic distance
2461:
2435:
2429:
2403:
595:
570:
564:
545:
232:
206:
200:
174:
132:. The distance from the pole is called the
1179:-axis) and after that many choices exist.
4027:
4014:
4001:
3988:
3972:
3954:
3941:
3928:
3915:
3897:
3884:
3866:
3853:
3840:
3827:
3809:
3796:
3789:
3733:
3727:
3698:
3653:
3648:
3635:
3630:
3618:
3605:
3599:
3590:
3566:
3561:
3548:
3543:
3531:
3518:
3512:
3503:
3497:
3447:
3425:
3406:
3384:
3378:
3316:
3311:is the closest point on the horocycle to
3295:
3289:
3268:
3262:
3242:
3221:
3215:
3192:
3171:
3165:
3141:
3106:
3079:
3063:
3057:
3022:
3017:
3003:
2962:
2951:
2910:
2899:
2858:
2847:
2805:
2786:
2776:
2734:
2699:
2687:
2675:
2663:
2659:
2647:
2634:
2622:
2617:
2588:
2577:
2561:
2542:
2523:
2510:
2491:
2455:
2442:
2423:
2410:
2392:
2296:
2272:
2256:
2244:
2213:
2191:
2166:
2138:
2125:
2119:
2039:
2018:
1999:
1993:
1967:
1962:
1948:
1928:
1914:
1895:
1889:
1885:
1874:
1833:
1822:
1781:
1770:
1731:
1709:
1683:
1650:
1628:
1602:
1574:
1561:
1549:
1522:
1509:
1497:
1467:
1451:
1435:
1419:
1413:
1386:
1373:
1361:
1331:
1315:
1299:
1283:
1277:
1104:
1099:
1085:
1057:
1049:
980:
955:
927:
916:
907:
895:
871:
865:
853:
849:
837:
824:
812:
800:
788:
783:
720:
683:
664:
654:
643:
638:
632:
630:
606:
583:
558:
528:
523:
517:
515:
506:
504:
472:
467:
461:
459:
457:
436:
423:
404:
391:
379:
357:
343:
330:
311:
295:
276:
260:
244:
226:
213:
194:
181:
163:
2325:Lobachevsky coordinates are named after
61:
4235:, Abraham Ungar, World Scientific, 2010
4104:
34:In the descriptions below the constant
4177:. New York: Springer-Verlag. pp.
2097:
2372:by dropping a perpendicular onto the
7:
3763:The point P with axial coordinates (
2612:The corresponding metric tensor is:
4173:Introduction to hyperbolic geometry
3207:to the horocycle, and the (signed)
1117:{\displaystyle u=\ln {\sqrt {x/y}}}
4202:Hyperbolic Barycentric Coordinates
4076:Hyperbolic barycentric coordinates
2688:
2664:
2623:
1144:Cartesian-style coordinate systems
854:
813:
789:
639:
633:
524:
518:
468:
462:
14:
4082:Gyrovector space#Triangle centers
3132:Horocycle-based coordinate system
3124:Horocycle-based coordinate system
1175:) on a chosen directed line (the
4216:, Abraham Ungar, Springer, 2010
3482:-axis is mapped to the segment
3363:-axis is mapped to the segment
2781:
2775:
932:
926:
4158:. Moscow: Mir. pp. 64–68.
3747:
3700:
3690:-axis is mapped to the (half)
3453:
3440:
3412:
3399:
3370:The following equations hold:
3335:Model-based coordinate systems
2984:
2963:
2932:
2911:
2880:
2859:
2696:
2684:
2672:
2660:
2631:
2619:
2529:
2503:
2464:
2400:
2302:
2289:
2278:
2265:
2250:
2237:
2219:
2206:
2197:
2184:
2172:
2159:
1929:
1886:
1855:
1834:
1803:
1782:
1737:
1724:
1715:
1702:
1656:
1643:
1634:
1621:
1586:{\displaystyle P(x_{a},y_{a})}
1580:
1554:
1534:{\displaystyle P(x_{a},y_{a})}
1528:
1502:
1473:
1460:
1441:
1428:
1398:{\displaystyle P(x_{a},y_{a})}
1392:
1366:
1337:
1324:
1305:
1292:
1070:{\displaystyle v={\sqrt {xy}}}
1041:> 0}. For such a point the
986:
967:
862:
850:
821:
809:
797:
785:
754:
748:
707:
701:
598:
542:
349:
323:
235:
171:
140:, and the angle is called the
109:from a reference point and an
1:
4154:Smorgorzhevsky, A.S. (1982).
3686:of the hyperbolic plane, the
3478:of the hyperbolic plane, the
3359:of the hyperbolic plane, the
3341:models of hyperbolic geometry
2839:-axis is the polar axis) is
2061:The Lobachevsky coordinates
1762:-axis is the polar axis) is
1205:are found by constructing a
113:from a reference direction.
1218:Cartesian coordinate system
1209:-axis perpendicular to the
4273:
4121:Martin, George E. (1998).
4063:
2329:one of the discoverers of
1213:-axis through the origin.
1023:Poincaré half-plane model
774:we get the corresponding
153:hyperbolic law of cosines
38:of the plane is −1.
1544:The distance of a point
3678:Weierstrass coordinates
2057:Lobachevsky coordinates
1541:is not a point at all.
1148:In hyperbolic geometry
88:polar coordinate system
58:Polar coordinate system
4156:Lobachevskian geometry
4060:Gyrovector coordinates
4045:
3754:
3668:
3460:
3325:
3305:
3278:
3251:
3231:
3201:
3181:
3150:
3133:
3114:
2991:
2939:
2887:
2815:
2709:
2596:
2349:
2341:
2312:
2047:
1936:
1862:
1810:
1749:
1668:
1587:
1535:
1486:
1399:
1350:
1262:
1138:hyperbolic coordinates
1118:
1071:
993:
879:
765:
490:
446:
365:
83:
4046:
3755:
3669:
3461:
3353:Cartesian coordinates
3326:
3306:
3304:{\displaystyle P_{h}}
3279:
3277:{\displaystyle P_{h}}
3252:
3232:
3230:{\displaystyle y_{h}}
3202:
3182:
3180:{\displaystyle x_{h}}
3151:
3131:
3115:
2992:
2940:
2888:
2816:
2722:-axis (with equation
2710:
2597:
2347:
2339:
2313:
2048:
1937:
1863:
1811:
1750:
1669:
1588:
1536:
1487:
1400:
1351:
1265:Every point and most
1260:
1254:-axis to the origin.
1158:Lambert quadrilateral
1119:
1072:
1017:Quadrant model system
994:
880:
766:
491:
452:, differentiating at
447:
366:
78:. In blue, the point
65:
3788:
3697:
3496:
3470:Poincaré coordinates
3377:
3357:Beltrami–Klein model
3347:Beltrami coordinates
3315:
3288:
3261:
3241:
3214:
3191:
3164:
3140:
3002:
2950:
2898:
2846:
2733:
2616:
2607:hyperbolic triangles
2391:
2118:
1947:
1873:
1821:
1769:
1682:
1601:
1548:
1496:
1412:
1360:
1276:
1134:hyperbolic rotations
1084:
1048:
894:
782:
503:
456:
378:
162:
52:hyperbolic functions
4252:Hyperbolic geometry
3658:
3640:
3571:
3553:
3476:Poincaré disk model
2331:hyperbolic geometry
2327:Nikolai Lobachevsky
1405:is an ideal point.
105:is determined by a
4257:Coordinate systems
4231:2012-05-19 at the
4041:
3750:
3664:
3644:
3626:
3557:
3539:
3456:
3321:
3301:
3274:
3247:
3227:
3197:
3177:
3146:
3134:
3110:
2987:
2935:
2883:
2811:
2705:
2592:
2350:
2342:
2308:
2043:
1932:
1858:
1806:
1745:
1664:
1583:
1531:
1482:
1395:
1346:
1263:
1187:Axial coordinates
1114:
1067:
989:
875:
761:
759:
486:
442:
361:
142:angular coordinate
84:
36:Gaussian curvature
4034:
4033:
3971:
3965:
3961:
3960:
3883:
3877:
3873:
3872:
3745:
3726:
3720:
3714:
3708:
3684:hyperboloid model
3662:
3659:
3585:
3582:
3575:
3572:
3420:
3324:{\displaystyle P}
3250:{\displaystyle O}
3200:{\displaystyle P}
3149:{\displaystyle P}
3100:
3097:
2779:
2360:=0) point on the
2306:
2230:
2224:
2152:
2146:
2098:axial coordinates
2033:
2030:
1926:
1183:Axial coordinates
1124:produce a point (
1112:
1065:
930:
924:
649:
534:
478:
134:radial coordinate
95:coordinate system
4264:
4236:
4223:
4217:
4211:
4205:
4199:
4193:
4192:
4176:
4166:
4160:
4159:
4151:
4145:
4144:
4128:
4118:
4088:triangle centers
4071:Gyrovector space
4050:
4048:
4047:
4042:
4040:
4036:
4035:
4032:
4031:
4019:
4018:
4006:
4005:
3993:
3992:
3977:
3973:
3969:
3963:
3962:
3959:
3958:
3946:
3945:
3933:
3932:
3920:
3919:
3904:
3903:
3902:
3901:
3885:
3881:
3875:
3874:
3871:
3870:
3858:
3857:
3845:
3844:
3832:
3831:
3816:
3815:
3814:
3813:
3797:
3759:
3757:
3756:
3751:
3746:
3738:
3737:
3728:
3724:
3718:
3712:
3706:
3673:
3671:
3670:
3665:
3663:
3661:
3660:
3657:
3652:
3639:
3634:
3619:
3610:
3609:
3600:
3595:
3594:
3583:
3580:
3576:
3574:
3573:
3570:
3565:
3552:
3547:
3532:
3523:
3522:
3513:
3508:
3507:
3485:
3465:
3463:
3462:
3457:
3452:
3451:
3430:
3429:
3418:
3411:
3410:
3389:
3388:
3366:
3330:
3328:
3327:
3322:
3310:
3308:
3307:
3302:
3300:
3299:
3283:
3281:
3280:
3275:
3273:
3272:
3256:
3254:
3253:
3248:
3236:
3234:
3233:
3228:
3226:
3225:
3206:
3204:
3203:
3198:
3186:
3184:
3183:
3178:
3176:
3175:
3155:
3153:
3152:
3147:
3119:
3117:
3116:
3111:
3105:
3101:
3099:
3098:
3084:
3083:
3068:
3067:
3058:
3034:
3023:
2996:
2994:
2993:
2988:
2944:
2942:
2941:
2936:
2892:
2890:
2889:
2884:
2820:
2818:
2817:
2812:
2810:
2809:
2791:
2790:
2780:
2778: when
2777:
2714:
2712:
2711:
2706:
2704:
2703:
2691:
2680:
2679:
2667:
2652:
2651:
2639:
2638:
2626:
2601:
2599:
2598:
2593:
2587:
2583:
2582:
2581:
2566:
2565:
2547:
2546:
2528:
2527:
2515:
2514:
2496:
2495:
2460:
2459:
2447:
2446:
2428:
2427:
2415:
2414:
2317:
2315:
2314:
2309:
2307:
2305:
2301:
2300:
2281:
2277:
2276:
2257:
2249:
2248:
2228:
2222:
2218:
2217:
2196:
2195:
2171:
2170:
2150:
2144:
2143:
2142:
2130:
2129:
2052:
2050:
2049:
2044:
2038:
2034:
2032:
2031:
2023:
2022:
2004:
2003:
1994:
1979:
1968:
1941:
1939:
1938:
1933:
1927:
1919:
1918:
1900:
1899:
1890:
1867:
1865:
1864:
1859:
1815:
1813:
1812:
1807:
1754:
1752:
1751:
1746:
1744:
1740:
1736:
1735:
1714:
1713:
1673:
1671:
1670:
1665:
1663:
1659:
1655:
1654:
1633:
1632:
1592:
1590:
1589:
1584:
1579:
1578:
1566:
1565:
1540:
1538:
1537:
1532:
1527:
1526:
1514:
1513:
1491:
1489:
1488:
1483:
1472:
1471:
1456:
1455:
1440:
1439:
1424:
1423:
1404:
1402:
1401:
1396:
1391:
1390:
1378:
1377:
1355:
1353:
1352:
1347:
1336:
1335:
1320:
1319:
1304:
1303:
1288:
1287:
1123:
1121:
1120:
1115:
1113:
1108:
1100:
1079:hyperbolic angle
1076:
1074:
1073:
1068:
1066:
1058:
998:
996:
995:
990:
985:
984:
960:
959:
931:
928:
925:
917:
912:
911:
884:
882:
881:
876:
870:
869:
857:
842:
841:
829:
828:
816:
805:
804:
792:
770:
768:
767:
762:
760:
735:
731:
730:
719:
715:
714:
710:
688:
687:
669:
668:
650:
648:
647:
642:
636:
631:
617:
616:
605:
601:
588:
587:
563:
562:
535:
533:
532:
527:
521:
516:
495:
493:
492:
487:
479:
477:
476:
471:
465:
460:
451:
449:
448:
443:
441:
440:
428:
427:
409:
408:
396:
395:
370:
368:
367:
362:
356:
352:
348:
347:
335:
334:
316:
315:
300:
299:
281:
280:
265:
264:
231:
230:
218:
217:
199:
198:
186:
185:
120:) is called the
118:Cartesian system
81:
77:
18:hyperbolic plane
4272:
4271:
4267:
4266:
4265:
4263:
4262:
4261:
4242:
4241:
4240:
4239:
4233:Wayback Machine
4224:
4220:
4212:
4208:
4200:
4196:
4189:
4168:
4167:
4163:
4153:
4152:
4148:
4141:
4120:
4119:
4106:
4101:
4078:
4068:
4062:
4057:
4023:
4010:
3997:
3984:
3950:
3937:
3924:
3911:
3893:
3886:
3862:
3849:
3836:
3823:
3805:
3798:
3795:
3791:
3786:
3785:
3781:) is mapped to
3780:
3771:
3729:
3695:
3694:
3680:
3611:
3601:
3586:
3524:
3514:
3499:
3494:
3493:
3483:
3472:
3443:
3421:
3402:
3380:
3375:
3374:
3364:
3349:
3337:
3313:
3312:
3291:
3286:
3285:
3264:
3259:
3258:
3239:
3238:
3217:
3212:
3211:
3189:
3188:
3167:
3162:
3161:
3138:
3137:
3126:
3075:
3059:
3035:
3024:
3018:
3000:
2999:
2948:
2947:
2896:
2895:
2844:
2843:
2801:
2782:
2731:
2730:
2695:
2671:
2643:
2630:
2614:
2613:
2573:
2557:
2538:
2519:
2506:
2487:
2480:
2476:
2451:
2438:
2419:
2406:
2389:
2388:
2292:
2282:
2268:
2258:
2240:
2209:
2187:
2162:
2134:
2121:
2116:
2115:
2110:
2091:
2078:
2069:
2059:
2014:
1995:
1980:
1969:
1963:
1945:
1944:
1910:
1891:
1871:
1870:
1819:
1818:
1767:
1766:
1727:
1705:
1695:
1691:
1680:
1679:
1646:
1624:
1614:
1610:
1599:
1598:
1570:
1557:
1546:
1545:
1518:
1505:
1494:
1493:
1463:
1447:
1431:
1415:
1410:
1409:
1382:
1369:
1358:
1357:
1327:
1311:
1295:
1279:
1274:
1273:
1249:
1236:
1204:
1195:
1185:
1146:
1082:
1081:
1046:
1045:
1019:
1012:
1008:
976:
951:
903:
892:
891:
861:
833:
820:
796:
780:
779:
758:
757:
733:
732:
679:
660:
659:
655:
637:
629:
626:
625:
618:
579:
554:
522:
514:
511:
510:
501:
500:
466:
454:
453:
432:
419:
400:
387:
376:
375:
339:
326:
307:
291:
272:
256:
249:
245:
222:
209:
190:
177:
160:
159:
92:two-dimensional
79:
75:
70:and polar axis
60:
22:Euclidean plane
12:
11:
5:
4270:
4268:
4260:
4259:
4254:
4244:
4243:
4238:
4237:
4218:
4206:
4194:
4187:
4161:
4146:
4139:
4103:
4102:
4100:
4097:
4077:
4074:
4064:Main article:
4061:
4058:
4056:
4053:
4052:
4051:
4039:
4030:
4026:
4022:
4017:
4013:
4009:
4004:
4000:
3996:
3991:
3987:
3983:
3980:
3976:
3968:
3957:
3953:
3949:
3944:
3940:
3936:
3931:
3927:
3923:
3918:
3914:
3910:
3907:
3900:
3896:
3892:
3889:
3880:
3869:
3865:
3861:
3856:
3852:
3848:
3843:
3839:
3835:
3830:
3826:
3822:
3819:
3812:
3808:
3804:
3801:
3794:
3776:
3767:
3749:
3744:
3741:
3736:
3732:
3723:
3717:
3711:
3705:
3702:
3679:
3676:
3675:
3674:
3656:
3651:
3647:
3643:
3638:
3633:
3629:
3625:
3622:
3617:
3614:
3608:
3604:
3598:
3593:
3589:
3579:
3569:
3564:
3560:
3556:
3551:
3546:
3542:
3538:
3535:
3530:
3527:
3521:
3517:
3511:
3506:
3502:
3484:(−1,0) − (1,0)
3471:
3468:
3467:
3466:
3455:
3450:
3446:
3442:
3439:
3436:
3433:
3428:
3424:
3417:
3414:
3409:
3405:
3401:
3398:
3395:
3392:
3387:
3383:
3365:(−1,0) − (1,0)
3348:
3345:
3336:
3333:
3320:
3298:
3294:
3271:
3267:
3246:
3224:
3220:
3196:
3174:
3170:
3145:
3125:
3122:
3121:
3120:
3109:
3104:
3096:
3093:
3090:
3087:
3082:
3078:
3074:
3071:
3066:
3062:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3033:
3030:
3027:
3021:
3016:
3013:
3010:
3007:
2997:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2961:
2958:
2955:
2945:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2909:
2906:
2903:
2893:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2857:
2854:
2851:
2822:
2821:
2808:
2804:
2800:
2797:
2794:
2789:
2785:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2702:
2698:
2694:
2690:
2686:
2683:
2678:
2674:
2670:
2666:
2662:
2658:
2655:
2650:
2646:
2642:
2637:
2633:
2629:
2625:
2621:
2603:
2602:
2591:
2586:
2580:
2576:
2572:
2569:
2564:
2560:
2556:
2553:
2550:
2545:
2541:
2537:
2534:
2531:
2526:
2522:
2518:
2513:
2509:
2505:
2502:
2499:
2494:
2490:
2486:
2483:
2479:
2475:
2472:
2469:
2466:
2463:
2458:
2454:
2450:
2445:
2441:
2437:
2434:
2431:
2426:
2422:
2418:
2413:
2409:
2405:
2402:
2399:
2396:
2320:
2319:
2304:
2299:
2295:
2291:
2288:
2285:
2280:
2275:
2271:
2267:
2264:
2261:
2255:
2252:
2247:
2243:
2239:
2236:
2233:
2227:
2221:
2216:
2212:
2208:
2205:
2202:
2199:
2194:
2190:
2186:
2183:
2180:
2177:
2174:
2169:
2165:
2161:
2158:
2155:
2149:
2141:
2137:
2133:
2128:
2124:
2106:
2087:
2074:
2065:
2058:
2055:
2054:
2053:
2042:
2037:
2029:
2026:
2021:
2017:
2013:
2010:
2007:
2002:
1998:
1992:
1989:
1986:
1983:
1978:
1975:
1972:
1966:
1961:
1958:
1955:
1952:
1942:
1931:
1925:
1922:
1917:
1913:
1909:
1906:
1903:
1898:
1894:
1888:
1884:
1881:
1878:
1868:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1832:
1829:
1826:
1816:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1780:
1777:
1774:
1743:
1739:
1734:
1730:
1726:
1723:
1720:
1717:
1712:
1708:
1704:
1701:
1698:
1694:
1690:
1687:
1662:
1658:
1653:
1649:
1645:
1642:
1639:
1636:
1631:
1627:
1623:
1620:
1617:
1613:
1609:
1606:
1582:
1577:
1573:
1569:
1564:
1560:
1556:
1553:
1530:
1525:
1521:
1517:
1512:
1508:
1504:
1501:
1481:
1478:
1475:
1470:
1466:
1462:
1459:
1454:
1450:
1446:
1443:
1438:
1434:
1430:
1427:
1422:
1418:
1394:
1389:
1385:
1381:
1376:
1372:
1368:
1365:
1345:
1342:
1339:
1334:
1330:
1326:
1323:
1318:
1314:
1310:
1307:
1302:
1298:
1294:
1291:
1286:
1282:
1245:
1232:
1200:
1191:
1184:
1181:
1145:
1142:
1111:
1107:
1103:
1098:
1095:
1092:
1089:
1064:
1061:
1056:
1053:
1043:geometric mean
1018:
1015:
1010:
1006:
1000:
999:
988:
983:
979:
975:
972:
969:
966:
963:
958:
954:
950:
947:
944:
941:
938:
935:
929: or
923:
920:
915:
910:
906:
902:
899:
874:
868:
864:
860:
856:
852:
848:
845:
840:
836:
832:
827:
823:
819:
815:
811:
808:
803:
799:
795:
791:
787:
772:
771:
756:
753:
750:
747:
744:
741:
738:
736:
734:
729:
726:
723:
718:
713:
709:
706:
703:
700:
697:
694:
691:
686:
682:
678:
675:
672:
667:
663:
658:
653:
646:
641:
635:
628:
624:
621:
619:
615:
612:
609:
604:
600:
597:
594:
591:
586:
582:
578:
575:
572:
569:
566:
561:
557:
553:
550:
547:
544:
541:
538:
531:
526:
520:
513:
509:
508:
485:
482:
475:
470:
464:
439:
435:
431:
426:
422:
418:
415:
412:
407:
403:
399:
394:
390:
386:
383:
372:
371:
360:
355:
351:
346:
342:
338:
333:
329:
325:
322:
319:
314:
310:
306:
303:
298:
294:
290:
287:
284:
279:
275:
271:
268:
263:
259:
255:
252:
248:
243:
240:
237:
234:
229:
225:
221:
216:
212:
208:
205:
202:
197:
193:
189:
184:
180:
176:
173:
170:
167:
97:in which each
59:
56:
13:
10:
9:
6:
4:
3:
2:
4269:
4258:
4255:
4253:
4250:
4249:
4247:
4234:
4230:
4227:
4222:
4219:
4215:
4210:
4207:
4203:
4198:
4195:
4190:
4184:
4180:
4175:
4174:
4165:
4162:
4157:
4150:
4147:
4142:
4136:
4132:
4127:
4126:
4117:
4115:
4113:
4111:
4109:
4105:
4098:
4096:
4094:
4089:
4086:The study of
4084:
4083:
4075:
4073:
4072:
4067:
4059:
4054:
4037:
4028:
4024:
4020:
4015:
4011:
4007:
4002:
3998:
3994:
3989:
3985:
3981:
3978:
3974:
3966:
3955:
3951:
3947:
3942:
3938:
3934:
3929:
3925:
3921:
3916:
3912:
3908:
3905:
3898:
3894:
3890:
3887:
3878:
3867:
3863:
3859:
3854:
3850:
3846:
3841:
3837:
3833:
3828:
3824:
3820:
3817:
3810:
3806:
3802:
3799:
3792:
3784:
3783:
3782:
3779:
3775:
3770:
3766:
3761:
3742:
3739:
3734:
3730:
3721:
3715:
3709:
3703:
3693:
3689:
3685:
3677:
3654:
3649:
3645:
3641:
3636:
3631:
3627:
3623:
3620:
3615:
3612:
3606:
3602:
3596:
3591:
3587:
3577:
3567:
3562:
3558:
3554:
3549:
3544:
3540:
3536:
3533:
3528:
3525:
3519:
3515:
3509:
3504:
3500:
3492:
3491:
3490:
3487:
3481:
3477:
3469:
3448:
3444:
3437:
3434:
3431:
3426:
3422:
3415:
3407:
3403:
3396:
3393:
3390:
3385:
3381:
3373:
3372:
3371:
3368:
3362:
3358:
3354:
3346:
3344:
3342:
3334:
3332:
3318:
3296:
3292:
3269:
3265:
3244:
3222:
3218:
3210:
3194:
3172:
3168:
3159:
3143:
3130:
3123:
3107:
3102:
3094:
3091:
3088:
3085:
3080:
3076:
3072:
3069:
3064:
3060:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3031:
3028:
3025:
3019:
3014:
3011:
3008:
3005:
2998:
2981:
2978:
2975:
2972:
2969:
2966:
2959:
2956:
2953:
2946:
2929:
2926:
2923:
2920:
2917:
2914:
2907:
2904:
2901:
2894:
2877:
2874:
2871:
2868:
2865:
2862:
2855:
2852:
2849:
2842:
2841:
2840:
2838:
2833:
2831:
2827:
2806:
2802:
2798:
2795:
2792:
2787:
2783:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2729:
2728:
2727:
2725:
2721:
2716:
2700:
2692:
2681:
2676:
2668:
2656:
2653:
2648:
2644:
2640:
2635:
2627:
2610:
2608:
2589:
2584:
2578:
2574:
2570:
2567:
2562:
2558:
2554:
2551:
2548:
2543:
2539:
2535:
2532:
2524:
2520:
2516:
2511:
2507:
2500:
2497:
2492:
2488:
2484:
2481:
2477:
2473:
2470:
2467:
2456:
2452:
2448:
2443:
2439:
2432:
2424:
2420:
2416:
2411:
2407:
2397:
2394:
2387:
2386:
2385:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2346:
2338:
2334:
2332:
2328:
2323:
2297:
2293:
2286:
2283:
2273:
2269:
2262:
2259:
2253:
2245:
2241:
2234:
2231:
2225:
2214:
2210:
2203:
2200:
2192:
2188:
2181:
2178:
2175:
2167:
2163:
2156:
2153:
2147:
2139:
2135:
2131:
2126:
2122:
2114:
2113:
2112:
2109:
2105:
2101:
2099:
2095:
2090:
2086:
2082:
2077:
2073:
2068:
2064:
2056:
2040:
2035:
2027:
2024:
2019:
2015:
2011:
2008:
2005:
2000:
1996:
1990:
1987:
1984:
1981:
1976:
1973:
1970:
1964:
1959:
1956:
1953:
1950:
1943:
1923:
1920:
1915:
1911:
1907:
1904:
1901:
1896:
1892:
1882:
1879:
1876:
1869:
1852:
1849:
1846:
1843:
1840:
1837:
1830:
1827:
1824:
1817:
1800:
1797:
1794:
1791:
1788:
1785:
1778:
1775:
1772:
1765:
1764:
1763:
1761:
1756:
1741:
1732:
1728:
1721:
1718:
1710:
1706:
1699:
1696:
1692:
1688:
1685:
1677:
1660:
1651:
1647:
1640:
1637:
1629:
1625:
1618:
1615:
1611:
1607:
1604:
1596:
1575:
1571:
1567:
1562:
1558:
1551:
1542:
1523:
1519:
1515:
1510:
1506:
1499:
1479:
1476:
1468:
1464:
1457:
1452:
1448:
1444:
1436:
1432:
1425:
1420:
1416:
1406:
1387:
1383:
1379:
1374:
1370:
1363:
1343:
1340:
1332:
1328:
1321:
1316:
1312:
1308:
1300:
1296:
1289:
1284:
1280:
1270:
1268:
1259:
1255:
1253:
1248:
1244:
1240:
1235:
1231:
1227:
1223:
1219:
1214:
1212:
1208:
1203:
1199:
1194:
1190:
1182:
1180:
1178:
1174:
1171:) point (the
1170:
1165:
1163:
1159:
1155:
1151:
1143:
1141:
1139:
1135:
1131:
1127:
1109:
1105:
1101:
1096:
1093:
1090:
1087:
1080:
1062:
1059:
1054:
1051:
1044:
1040:
1036:
1032:
1028:
1024:
1016:
1014:
1005:
981:
977:
973:
970:
964:
961:
956:
952:
948:
945:
942:
939:
936:
933:
921:
918:
913:
908:
904:
900:
897:
890:
889:
888:
885:
872:
866:
858:
846:
843:
838:
834:
830:
825:
817:
806:
801:
793:
777:
776:metric tensor
751:
745:
742:
739:
737:
727:
724:
721:
716:
711:
704:
698:
695:
692:
689:
684:
680:
676:
673:
670:
665:
661:
656:
651:
644:
622:
620:
613:
610:
607:
602:
592:
589:
584:
580:
576:
573:
567:
559:
555:
551:
548:
539:
536:
529:
499:
498:
497:
483:
480:
473:
437:
433:
429:
424:
420:
416:
413:
410:
405:
401:
397:
392:
388:
384:
381:
358:
353:
344:
340:
336:
331:
327:
320:
317:
312:
308:
304:
301:
296:
292:
288:
285:
282:
277:
273:
269:
266:
261:
257:
253:
250:
246:
241:
238:
227:
223:
219:
214:
210:
203:
195:
191:
187:
182:
178:
168:
165:
158:
157:
156:
154:
149:
147:
143:
139:
135:
131:
127:
123:
119:
114:
112:
108:
104:
100:
96:
93:
89:
73:
69:
64:
57:
55:
53:
49:
45:
41:
37:
32:
29:
27:
23:
19:
4221:
4209:
4197:
4172:
4164:
4155:
4149:
4124:
4092:
4085:
4079:
4069:
3777:
3773:
3768:
3764:
3762:
3687:
3681:
3488:
3479:
3473:
3369:
3360:
3350:
3338:
3135:
2836:
2834:
2829:
2825:
2823:
2723:
2719:
2717:
2611:
2604:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2353:
2351:
2324:
2321:
2107:
2103:
2102:
2093:
2088:
2084:
2080:
2075:
2071:
2066:
2062:
2060:
1759:
1757:
1678:-axis it is
1675:
1594:
1543:
1407:
1271:
1267:ideal points
1264:
1251:
1246:
1242:
1238:
1233:
1229:
1225:
1221:
1216:Like in the
1215:
1210:
1206:
1201:
1197:
1192:
1188:
1186:
1176:
1166:
1154:right angles
1147:
1125:
1038:
1034:
1030:
1026:
1020:
1003:
1001:
886:
773:
373:
150:
145:
141:
137:
133:
129:
121:
115:
87:
85:
71:
67:
33:
30:
26:real numbers
20:, as in the
15:
1162:hypercycles
146:polar angle
4246:Categories
4188:0387943390
4140:0387906940
4099:References
4066:gyrovector
3209:arc length
1674:. To the
1150:rectangles
1009:and θ
130:polar axis
124:, and the
4021:
4008:−
3995:
3982:−
3948:
3935:−
3922:
3909:−
3891:
3860:
3847:−
3834:
3821:−
3803:
3692:hyperbola
3642:−
3624:−
3555:−
3537:−
3438:
3397:
3158:horocycle
3092:−
3086:
3070:
3049:
3040:
3029:
3006:θ
2979:
2970:
2930:θ
2927:
2918:
2878:θ
2875:
2866:
2770:
2755:
2740:
2654:
2571:
2555:
2549:−
2536:
2517:−
2501:
2485:
2474:
2462:⟩
2436:⟨
2430:⟩
2404:⟨
2398:
2287:
2263:
2235:
2204:
2182:
2157:
2025:
2006:
1985:
1974:
1951:θ
1921:
1902:
1853:θ
1850:
1841:
1801:θ
1798:
1789:
1722:
1700:
1689:
1641:
1619:
1608:
1597:-axis is
1458:
1426:
1322:
1290:
1097:
978:θ
974:−
971:θ
965:
949:
937:
919:π
914:±
905:θ
898:θ
859:θ
844:
746:
722:θ
705:θ
699:
690:
677:−
671:
645:θ
608:θ
596:⟩
593:θ
581:θ
571:⟨
565:⟩
556:θ
546:⟨
540:
530:θ
474:θ
434:θ
430:−
421:θ
414:θ
341:θ
337:−
328:θ
321:
305:
289:
283:−
270:
254:
233:⟩
224:θ
207:⟨
201:⟩
192:θ
175:⟨
169:
151:From the
80:(4, 210°)
4229:Archived
3284:, where
1077:and the
1037:> 0,
107:distance
76:(3, 60°)
4131:447–450
3772:,
2376:-axis.
2083:-axis.
1593:to the
1228:-axes.
1130:motions
16:In the
4185:
4179:97–103
4137:
4055:Others
3970:
3964:
3882:
3876:
3725:
3719:
3713:
3707:
3584:
3581:
3419:
3015:arctan
2960:arcosh
2908:arsinh
2856:artanh
2824:where
2471:arcosh
2229:
2223:
2151:
2145:
1960:arctan
1883:artanh
1831:artanh
1779:artanh
1686:artanh
1605:artanh
1173:Origin
1002:where
652:arcosh
242:arcosh
138:radius
4080:From
3187:from
1492:then
1356:then
1169:ideal
1156:(see
144:, or
111:angle
103:plane
101:on a
99:point
90:is a
4183:ISBN
4135:ISBN
4012:tanh
3986:tanh
3939:tanh
3913:tanh
3888:tanh
3851:tanh
3825:tanh
3800:tanh
3435:tanh
3394:tanh
3257:and
3077:cosh
3061:cosh
3046:cosh
3037:sinh
3026:sinh
2976:cosh
2967:cosh
2915:sinh
2863:tanh
2828:and
2793:<
2767:sinh
2752:cosh
2737:tanh
2645:cosh
2568:sinh
2552:sinh
2533:cosh
2498:cosh
2482:cosh
2395:dist
2368:and
2284:cosh
2260:tanh
2232:tanh
2201:cosh
2179:tanh
2154:tanh
2070:and
2016:tanh
1997:tanh
1982:tanh
1971:tanh
1912:tanh
1893:tanh
1838:tanh
1786:tanh
1719:cosh
1697:tanh
1638:cosh
1616:tanh
1477:>
1449:tanh
1417:tanh
1313:tanh
1281:tanh
1224:and
1196:and
1029:= {(
1021:The
946:tanh
934:tanh
835:sinh
743:sinh
681:sinh
662:cosh
537:dist
374:Let
302:sinh
286:sinh
267:cosh
251:cosh
166:dist
122:pole
86:The
50:are
48:tanh
46:and
44:cosh
40:Sinh
4093:not
2924:sin
2872:cos
2100:).
1847:sin
1795:cos
1408:If
1272:If
1126:u,v
1033:):
1031:x,y
962:sec
696:cos
318:cos
136:or
126:ray
4248::
4181:.
4133:.
4107:^
3331:.
2715:.
2609:.
2333:.
1755:.
1140:.
1094:ln
778::
496::
148:.
54:.
42:,
4191:.
4143:.
4038:)
4029:a
4025:y
4016:2
4003:a
3999:x
3990:2
3979:1
3975:1
3967:,
3956:a
3952:y
3943:2
3930:a
3926:x
3917:2
3906:1
3899:a
3895:y
3879:,
3868:a
3864:y
3855:2
3842:a
3838:x
3829:2
3818:1
3811:a
3807:x
3793:(
3778:a
3774:y
3769:a
3765:x
3748:)
3743:1
3740:+
3735:2
3731:t
3722:,
3716:0
3710:,
3704:t
3701:(
3688:x
3655:2
3650:b
3646:y
3637:2
3632:b
3628:x
3621:1
3616:+
3613:1
3607:b
3603:y
3597:=
3592:p
3588:y
3578:,
3568:2
3563:b
3559:y
3550:2
3545:b
3541:x
3534:1
3529:+
3526:1
3520:b
3516:x
3510:=
3505:p
3501:x
3480:x
3454:)
3449:a
3445:y
3441:(
3432:=
3427:b
3423:y
3416:,
3413:)
3408:a
3404:x
3400:(
3391:=
3386:b
3382:x
3361:x
3319:P
3297:h
3293:P
3270:h
3266:P
3245:O
3223:h
3219:y
3195:P
3173:h
3169:x
3144:P
3108:.
3103:)
3095:1
3089:y
3081:2
3073:x
3065:2
3055:+
3052:y
3043:x
3032:y
3020:(
3012:2
3009:=
2985:)
2982:y
2973:x
2964:(
2957:=
2954:r
2933:)
2921:r
2912:(
2905:=
2902:y
2881:)
2869:r
2860:(
2853:=
2850:x
2837:x
2830:B
2826:A
2807:2
2803:B
2799:+
2796:1
2788:2
2784:A
2773:x
2764:B
2761:+
2758:x
2749:A
2746:=
2743:y
2724:x
2720:x
2701:2
2697:)
2693:y
2689:d
2685:(
2682:+
2677:2
2673:)
2669:x
2665:d
2661:(
2657:y
2649:2
2641:=
2636:2
2632:)
2628:s
2624:d
2620:(
2590:.
2585:)
2579:2
2575:y
2563:1
2559:y
2544:2
2540:y
2530:)
2525:1
2521:x
2512:2
2508:x
2504:(
2493:1
2489:y
2478:(
2468:=
2465:)
2457:2
2453:y
2449:,
2444:2
2440:x
2433:,
2425:1
2421:y
2417:,
2412:1
2408:x
2401:(
2382:y
2378:x
2374:x
2370:y
2366:x
2362:x
2358:x
2354:x
2318:.
2303:)
2298:l
2294:x
2290:(
2279:)
2274:l
2270:y
2266:(
2254:=
2251:)
2246:a
2242:y
2238:(
2226:,
2220:)
2215:a
2211:x
2207:(
2198:)
2193:a
2189:y
2185:(
2176:=
2173:)
2168:l
2164:y
2160:(
2148:,
2140:a
2136:x
2132:=
2127:l
2123:x
2108:ℓ
2104:y
2094:x
2089:ℓ
2085:x
2081:x
2076:ℓ
2072:y
2067:ℓ
2063:x
2041:.
2036:)
2028:y
2020:2
2012:+
2009:x
2001:2
1991:+
1988:x
1977:y
1965:(
1957:2
1954:=
1930:)
1924:y
1916:2
1908:+
1905:x
1897:2
1887:(
1880:=
1877:r
1856:)
1844:r
1835:(
1828:=
1825:y
1804:)
1792:r
1783:(
1776:=
1773:x
1760:x
1742:)
1738:)
1733:a
1729:y
1725:(
1716:)
1711:a
1707:x
1703:(
1693:(
1676:y
1661:)
1657:)
1652:a
1648:x
1644:(
1635:)
1630:a
1626:y
1622:(
1612:(
1595:x
1581:)
1576:a
1572:y
1568:,
1563:a
1559:x
1555:(
1552:P
1529:)
1524:a
1520:y
1516:,
1511:a
1507:x
1503:(
1500:P
1480:1
1474:)
1469:a
1465:y
1461:(
1453:2
1445:+
1442:)
1437:a
1433:x
1429:(
1421:2
1393:)
1388:a
1384:y
1380:,
1375:a
1371:x
1367:(
1364:P
1344:1
1341:=
1338:)
1333:a
1329:y
1325:(
1317:2
1309:+
1306:)
1301:a
1297:x
1293:(
1285:2
1252:y
1247:a
1243:y
1239:x
1234:a
1230:x
1226:y
1222:x
1211:x
1207:y
1202:a
1198:y
1193:a
1189:x
1177:x
1110:y
1106:/
1102:x
1091:=
1088:u
1063:y
1060:x
1055:=
1052:v
1039:y
1035:x
1027:Q
1011:0
1007:0
1004:r
987:)
982:0
968:(
957:0
953:r
943:=
940:r
922:2
909:0
901:=
873:.
867:2
863:)
855:d
851:(
847:r
839:2
831:+
826:2
822:)
818:r
814:d
810:(
807:=
802:2
798:)
794:s
790:d
786:(
755:)
752:r
749:(
740:=
728:0
725:=
717:|
712:)
708:)
702:(
693:r
685:2
674:r
666:2
657:(
640:d
634:d
623:=
614:0
611:=
603:|
599:)
590:+
585:1
577:,
574:r
568:,
560:1
552:,
549:r
543:(
525:d
519:d
484:0
481:=
469:d
463:d
438:1
425:2
417:=
411:,
406:2
402:r
398:=
393:1
389:r
385:=
382:r
359:.
354:)
350:)
345:1
332:2
324:(
313:2
309:r
297:1
293:r
278:2
274:r
262:1
258:r
247:(
239:=
236:)
228:2
220:,
215:2
211:r
204:,
196:1
188:,
183:1
179:r
172:(
82:.
72:L
68:O
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