2733:
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2019:
1250:
153:
1402:
geometry, the homographies and the collineations of the projective line that are considered are those obtained by restrictions to the line of collineations and homographies of spaces of higher dimension. This means that the fundamental theorem of projective geometry (see below) remains valid in the one-dimensional setting. A homography of a projective line may also be properly defined by insisting that the mapping preserves
1228:
4157:
824:
2413:
The composition of two central collineations, while still a homography in general, is not a central collineation. In fact, every homography is the composition of a finite number of central collineations. In synthetic geometry, this property, which is a part of the fundamental theory of projective
2535:
As all the projective spaces of the same dimension over the same field are isomorphic, the same is true for their homography groups. They are therefore considered as a single group acting on several spaces, and only the dimension and the field appear in the notation, not the specific projective
1401:
of the points of a projective line is a collineation, since every set of points are collinear. However, if the projective line is embedded in a higher-dimensional projective space, the geometric structure of that space can be used to impose a geometric structure on the line. Thus, in synthetic
805:
2508:, the first part is an easy corollary of the definitions. Therefore, the proof of the first part in synthetic geometry, and the proof of the third part in terms of linear algebra both are fundamental steps of the proof of the equivalence of the two ways of defining projective spaces.
1223:{\displaystyle {\begin{aligned}y_{1}&={\frac {a_{1,0}+a_{1,1}x_{1}+\dots +a_{1,n}x_{n}}{a_{0,0}+a_{0,1}x_{1}+\dots +a_{0,n}x_{n}}}\\&\vdots \\y_{n}&={\frac {a_{n,0}+a_{n,1}x_{1}+\dots +a_{n,n}x_{n}}{a_{0,0}+a_{0,1}x_{1}+\dots +a_{0,n}x_{n}}}\end{aligned}}}
3545:
3833:
105:. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a
397:
of a finite number of perspectivities. It is a part of the fundamental theorem of projective geometry (see below) that this definition coincides with the more algebraic definition sketched in the introduction and detailed below.
612:
1366:
3904:
3092:
3415:
3714:
In older treatments one often sees the requirement of preserving harmonic tetrads (harmonic sets) (four collinear points whose cross-ratio is −1) but this excludes projective lines defined over fields of
829:
617:
3167:
3442:
269:
if extended to projective spaces. Therefore, this notion is normally defined for projective spaces. The notion is also easily generalized to projective spaces of any dimension, over any
93:, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which extended
2489:
is the composition of an automorphic collineation and a homography. In particular, over the reals, every collineation of a projective space of dimension at least two is a homography.
1711:
It follows that, given two frames, there is exactly one homography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is the
3299:
2038:
are related by several central collineations, which are completely specified by choosing a line of fixed points L passing through the intersection of the lines ABCD and A
129:); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
121:
for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (
2496:. In particular, if the dimension of the implied projective space is at least two, every homography is the composition of a finite number of central collineations.
1397:
In the study of collineations, the case of projective lines is special due to the small dimension. When the line is viewed as a projective space in isolation, any
144:
are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.
800:{\displaystyle {\begin{aligned}y_{0}&=a_{0,0}x_{0}+\dots +a_{0,n}x_{n}\\&\vdots \\y_{n}&=a_{n,0}x_{0}+\dots +a_{n,n}x_{n}.\end{aligned}}}
2748:. A, B, C, D and V are points on the image, their separation given in pixels; A', B', C' and D' are in the real world, their separation in metres.
2782:
The cross-ratio of four collinear points is an invariant under the homography that is fundamental for the study of the homographies of the lines.
132:
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative)
1284:
2217:
are those in which the center is not incident with the axis. A central collineation is uniquely defined by its center, its axis, and the image
3851:
3550:
was interested in periodicity when he calculated iterates in 1879. In his review of a brute force approach to periodicity of homographies,
2981:
3361:
4063:
4010:
4182:
3111:
2423:
1394:. These correspond precisely with those bijections of the Riemann sphere that preserve orientation and are conformal.
4142:
4081:
4029:
3992:
3540:{\displaystyle h^{n}={\begin{pmatrix}1&n\\0&1\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}.}
2303:
The geometric view of a central collineation is easiest to see in a projective plane. Given a central collineation
4016:, translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation
180:, and, specifically, the difference in appearance of two plane objects viewed from different points of view.
4177:
3335:
2720:, whose blocks are the sets of points contained in a line, it is common to call the collineation group the
2504:), the third part is simply a definition. On the other hand, if projective spaces are defined by means of
2090:. It is a part of the fundamental theorem of projective geometry that the two definitions are equivalent.
4161:
2932:
137:
62:
asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include
20:
1624:, and this basis is unique up to the multiplication of all its elements by the same nonzero element of
383:
1233:
which generalizes the expression of the homographic function of the next section. This defines only a
480:, may thus be represented by the coordinates of any nonzero point of this line, which are thus called
1480:
are considered in this section, although most results may be generalized to projective spaces over a
247:
3938:
3178:
2456:
394:
82:
2086:, they are traditionally defined as the composition of one or several special homographies called
1391:
4113:
4055:
2897:
2540:
2452:
1780:
1383:
481:
113:(the above definition is based on this version); this construction facilitates the definition of
3716:
3608:
2968:
2745:
141:
114:
2297:
176:
Historically, the concept of homography had been introduced to understand, explain and study
1540:
be the canonical projection that maps a nonzero vector to the vector line that contains it.
4135:
4124:
3959:
2609:
2521:
2300:. It is an elation, if all the eigenvalues are equal and the matrix is not diagonalizable.
570:
This may be written in terms of homogeneous coordinates in the following way: A homography
177:
8:
4047:
3343:
3323:
2756:
The width of the side street, W is computed from the known widths of the adjacent shops.
2741:
2525:
1262:
419:
270:
133:
110:
27:
2942:
2501:
2433:
2083:
1716:
811:
575:
228:
184:
126:
122:
102:
86:
3331:
1278:). With this representation of the projective line, the homographies are the mappings
4138:
4077:
4059:
4025:
4006:
3988:
3433:
3418:
2675:
2613:
1902:(consisting of the elements having only one nonzero entry, which is equal to 1), and
2524:
of two homographies is another, the homographies of a given projective space form a
1855:), results in multiplying the projective coordinates by the same nonzero element of
3951:
3102:
2670:
acts on the eight points in the projective line over the finite field GF(7), while
1422:
1415:
1234:
407:
262:
224:
39:
4121:
3955:
3327:
2765:
2517:
1719:(where projective spaces are defined through axioms). It is sometimes called the
1274:
94:
77:
Historically, homographies (and projective spaces) have been introduced to study
51:
2732:
2529:
2213:
are the central collineations in which the center is incident with the axis and
2209:, but not necessarily pointwise). There are two types of central collineations.
602:. The homogeneous coordinates of a point and the coordinates of its image by
4105:
3551:
2950:
2576:
2505:
1387:
227:
was originally introduced by extending the
Euclidean space, that is, by adding
118:
3172:
but otherwise the linear fractional transformation is seen as an equivalence:
818:(projective completion) the preceding formulas become, in affine coordinates,
4171:
3845:
3829:
3547:
3312:
2809:
form a projective frame of this line. There is therefore a unique homography
2493:
2427:
2129:
2082:
In above sections, homographies have been defined through linear algebra. In
1481:
1254:
255:
78:
3347:
2717:
2647:
2105:
1712:
815:
106:
98:
55:
43:
1361:{\displaystyle z\mapsto {\frac {az+b}{cz+d}},{\text{ where }}ad-bc\neq 0,}
3899:{\textstyle \scriptstyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}},}
3576:, then it is elliptic, and no loss of generality occurs by assuming that
2777:
2737:
1403:
1398:
508:
35:
1249:
2376:
that does not belong to ℓ may be constructed in the following way: let
2293:
2285:
2145:
1238:
549:, define the same homography if and only if there is a nonzero element
401:
2018:
172:
are related by a perspectivity, which is a projective transformation.
152:
2455:(over a projective frame) of a point. These collineations are called
2432:
There are collineations besides the homographies. In particular, any
526:
266:
47:
2716:. When the points and lines of the projective space are viewed as a
2074:
of a point E by this collineation is the intersection of the lines A
2682:, is the homography group of the projective line with five points.
19:
This article is about the mathematical notion. For other uses, see
3087:{\displaystyle U{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U.}
1272:
and a point, called the "point at infinity" and denoted by ∞ (see
3640:
3305:
1455:
3410:{\displaystyle h={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}
231:
to it, in order to define the projection for every point except
4156:
4117:
425:
may be defined as the set of the lines through the origin in a
59:
58:. In general, some collineations are not homographies, but the
2424:
Collineation § Fundamental theorem of projective geometry
2128:, but this term may be confusing, having another meaning; see
3558:
A real homography is involutory (of period 2) if and only if
2417:
1918:
595:
2261:) and the intersection with the axis of the line defined by
2078:
I and OE, where I is the intersection of the lines L and AE.
2744:
to measure real-world dimensions of features depicted in a
2492:
Every homography is the composition of a finite number of
2307:, consider a line ℓ that does not pass through the center
1237:
between affine spaces, which is defined only outside the
2712:
of the collineations of a projective space of dimension
2532:
is the homography group of any complex projective line.
4019:
3861:
3855:
3854:
3768:
3503:
3464:
3376:
3008:
2768:, V is visible, the width of only one shop is needed.
2443:
induces a collineation of every projective space over
4022:
Projective
Geometry: From Foundations to Applications
3445:
3364:
3181:
3114:
2984:
2623:
is the group of the products by a nonzero element of
2500:
If projective spaces are defined by means of axioms (
2414:
geometry is taken as the definition of homographies.
1287:
827:
615:
2272:
A central collineation is a homography defined by a
1244:
402:
Definition and expression in homogeneous coordinates
1409:
3898:
3539:
3409:
3293:
3161:
3086:
2470:Given two projective frames of a projective space
1832:. It is not difficult to verify that changing the
1450:of them. A projective frame is sometimes called a
1360:
1222:
799:
261:With these definitions, a perspectivity is only a
2844:to 1. Given a fourth point on the same line, the
810:When the projective spaces are defined by adding
46:from which the projective spaces derive. It is a
4169:
4020:Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
3834:Homographies of associative composition algebras
3162:{\displaystyle z\mapsto {\frac {za+b}{zc+d}}\ ,}
1906:. On this basis, the homogeneous coordinates of
1721:first fundamental theorem of projective geometry
2555:when acting on a projective space of dimension
203:to the intersection (if it exists) of the line
2563:. Above definition of homographies shows that
2478:that maps the first frame onto the second one.
2292:. It is a homology, if the matrix has another
384:Perspectivity § Perspective collineations
3719:two and so is unnecessarily restrictive. See
2953:. Homographies act on a projective line over
2273:
1943:of it, there is one and only one homography
1917:are simply the entries (coefficients) of the
211:. The projection is not defined if the point
2233:and does not belong to the axis. (The image
16:Isomorphism of projective spaces in geometry
3338:can be represented with homographies where
2245:is the intersection of the line defined by
1474:Projective spaces over a commutative field
598:the multiplication by a nonzero element of
4046:
4037:
3804:
3792:
3682:
3587:. Since the characteristic roots are exp(±
2760:
2752:
2466:consists of the three following theorems.
2464:fundamental theorem of projective geometry
2418:Fundamental theorem of projective geometry
379:
60:fundamental theorem of projective geometry
4052:Projective Geometries Over Finite Fields
3353:
2731:
2054:. Let O the intersection of the lines AA
2017:
2013:
1971:. The projective coordinates of a point
1715:. This result is much more difficult in
1443:points such that no hyperplane contains
1248:
183:In three-dimensional Euclidean space, a
151:
4110:Homographies, quaternions and rotations
4071:
3780:
3670:
2654:) then the homography group is written
2485:is at least two, every collineation of
2481:If the dimension of a projective space
147:
4170:
4000:
3985:Linear Algebra and Projective Geometry
3816:
3744:
3658:
3322:. Ring homographies have been used in
3973:
3906:and its connection with the function
3702:
2474:, there is exactly one homography of
1806:on this frame are the coordinates of
215:belongs to the plane passing through
101:by the addition of new points called
4089:
3982:
3756:
3732:
3720:
3698:
3694:
3304:The homography group of the ring of
2511:
1498:be a projective space of dimension
1268:may be identified with the union of
254:of the above projection is called a
2975:) are described by matrix mappings
1983:are the homogeneous coordinates of
1937:of the same dimension, and a frame
1430:of a projective space of dimension
1386:, which can be identified with the
42:, induced by an isomorphism of the
13:
4112:, Oxford Mathematical Monographs,
4099:
4040:Foundations of Projective Geometry
3769:Beutelspacher & Rosenbaum 1998
2803:on a projective line over a field
301:that may be obtained by embedding
199:is the mapping that sends a point
14:
4194:
4149:
3221:
2124:(traditionally these were called
1926:. Given another projective space
1847:, without changing the frame nor
1377:linear fractional transformations
1245:Homographies of a projective line
4155:
4074:Fundamental Concepts of Geometry
3828:
3569:. If it is periodic with period
3105:, the homography may be written
2722:automorphism group of the design
1795:; the projective coordinates of
1784:: every point may be written as
1410:Projective frame and coordinates
525:. Such an isomorphism induces a
356:, with a different center, then
3945:
3839:
3822:
3810:
3798:
3786:
3774:
2627:of the identity matrix of size
2159:), which is fixed pointwise by
1241:where the denominator is zero.
537:), because of the linearity of
4042:, New York: W.A. Benjamin, Inc
4024:, Cambridge University Press,
3762:
3750:
3738:
3726:
3708:
3688:
3676:
3664:
3652:
3285:
3273:
3258:
3246:
3230:
3227:
3218:
3185:
3118:
3078:
3045:
3000:
2988:
2926:
2727:
2539:Homography groups also called
2197:), which is fixed linewise by
2120:that maps lines onto lines. A
1390:, the homographies are called
1291:
487:Given two projective spaces P(
448:may be represented by a point
117:and allows using the tools of
1:
3967:
3294:{\displaystyle U\thicksim U.}
2674:, which is isomorphic to the
2253:and the line passing through
2229:that differs from the center
370:to itself, which is called a
3336:conformal group of spacetime
1955:onto the canonical frame of
380:§ Central collineations
277:Given two projective spaces
7:
3634:
2933:Projective line over a ring
2872:, denoted , is the element
2516:As every homography has an
2144:, such that there exists a
1890:consisting of the image by
1514:-vector space of dimension
1261:The projective line over a
1257:preserve orthogonal circles
495:) of the same dimension, a
444:has been fixed, a point of
21:Homography (disambiguation)
10:
4201:
4072:Meserve, Bruce E. (1983),
4038:Hartshorne, Robin (1967),
2930:
2900:over the projective frame
2775:
2421:
1994:on the canonical frame of
1896:of the canonical basis of
1413:
507:), which is induced by an
324:a central projection onto
191:(the center) onto a plane
18:
4183:Transformation (function)
4132:Geometry: An Introduction
3978:, Interscience Publishers
2575:may be identified to the
2457:automorphic collineations
2349:. The image of any point
594:. This matrix is defined
541:. Two such isomorphisms,
484:of the projective point.
273:, in the following way:
242:, which does not contain
68:projective transformation
3983:Baer, Reinhold (2005) ,
3939:Messenger of Mathematics
3646:
2971:. The homographies on P(
2961:), consisting of points
2541:projective linear groups
1458:in a space of dimension
592:matrix of the homography
374:, when the dimension of
336:is a perspectivity from
138:Pappus's hexagon theorem
4056:Oxford University Press
4001:Berger, Marcel (2009),
2898:homogeneous coordinates
2453:homogeneous coordinates
2357:is the intersection of
2205:is mapped to itself by
2093:In a projective space,
2022:Points A, B, C, D and A
1781:homogeneous coordinates
1576:such that the frame is
1554:, there exists a basis
1384:complex projective line
482:homogeneous coordinates
156:Points A, B, C, D and A
72:projective collineation
4090:Yale, Paul B. (1968),
3900:
3848:(1879) "On the matrix
3541:
3411:
3295:
3163:
3088:
2969:projective coordinates
2785:Three distinct points
2773:
2746:perspective projection
2311:, and its image under
2079:
1776:projective coordinates
1392:Möbius transformations
1362:
1258:
1224:
801:
378:is at least two. (See
330:
309:in a projective space
195:that does not contain
173:
115:projective coordinates
4092:Geometry and Symmetry
3901:
3542:
3412:
3354:Periodic homographies
3296:
3164:
3089:
2890:. In other words, if
2735:
2697:is a subgroup of the
2685:The homography group
2241:) of any other point
2225:) of any given point
2088:central collineations
2021:
2014:Central collineations
1862:The projective space
1436:is an ordered set of
1373:homographic functions
1363:
1252:
1225:
802:
366:is a homography from
348:a perspectivity from
275:
155:
54:to lines, and thus a
4164:at Wikimedia Commons
4136:Wadsworth Publishing
4134:, page 263, Belmont:
4130:Gunter Ewald (1971)
4048:Hirschfeld, J. W. P.
3960:Mathematical Reviews
3852:
3554:gave this analysis:
3443:
3362:
3179:
3112:
2982:
2610:general linear group
2122:central collineation
2112:is a bijection from
1285:
1253:Homographies of the
825:
613:
574:may be defined by a
499:is a mapping from P(
372:central collineation
293:is a bijection from
238:Given another plane
148:Geometric motivation
4178:Projective geometry
4005:, Springer-Verlag,
3344:composition algebra
3324:quaternion analysis
2848:of the four points
2742:projective geometry
2614:invertible matrices
2528:. For example, the
1543:For every frame of
1382:In the case of the
476:), being a line in
393:was defined as the
320:and restricting to
265:, but it becomes a
142:Desargues's theorem
28:projective geometry
3974:Artin, E. (1957),
3896:
3895:
3886:
3537:
3528:
3489:
3407:
3401:
3291:
3159:
3084:
3033:
2815:of this line onto
2774:
2699:collineation group
2502:synthetic geometry
2434:field automorphism
2201:(any line through
2084:synthetic geometry
2080:
1717:synthetic geometry
1358:
1259:
1220:
1218:
812:points at infinity
797:
795:
229:points at infinity
185:central projection
178:visual perspective
174:
127:synthetic geometry
123:incidence geometry
103:points at infinity
87:Euclidean geometry
4160:Media related to
4065:978-0-19-850295-1
4012:978-3-540-11658-5
3976:Geometric Algebra
3421:when the ring is
3281:
3205:
3155:
3151:
3065:
2772:
2771:
2676:alternating group
2512:Homography groups
2296:and is therefore
2132:) is a bijection
1774:allows to define
1628:. Conversely, if
1371:which are called
1332:
1331: where
1324:
1214:
1015:
511:of vector spaces
40:projective spaces
4190:
4159:
4095:
4086:
4068:
4043:
4034:
4015:
3997:
3979:
3962:
3952:H. S. M. Coxeter
3949:
3943:
3935:
3934:
3932:
3931:
3922:
3919:
3905:
3903:
3902:
3897:
3891:
3890:
3843:
3837:
3832:
3826:
3820:
3814:
3808:
3802:
3796:
3790:
3784:
3778:
3772:
3766:
3760:
3754:
3748:
3742:
3736:
3730:
3724:
3712:
3706:
3692:
3686:
3680:
3674:
3668:
3662:
3656:
3629:
3606:
3586:
3575:
3568:
3552:H. S. M. Coxeter
3546:
3544:
3543:
3538:
3533:
3532:
3494:
3493:
3455:
3454:
3434:integers modulo
3416:
3414:
3413:
3408:
3406:
3405:
3328:dual quaternions
3321:
3300:
3298:
3297:
3292:
3279:
3257:
3256:
3203:
3168:
3166:
3165:
3160:
3153:
3152:
3150:
3136:
3122:
3103:commutative ring
3093:
3091:
3090:
3085:
3063:
3038:
3037:
2966:
2922:
2915:
2895:
2889:
2882:
2871:
2865:
2859:
2853:
2843:
2837:
2831:
2827:
2821:
2814:
2808:
2802:
2796:
2790:
2750:
2749:
2711:
2696:
2673:
2669:
2665:
2638:
2607:
2595:
2574:
2554:
2409:
2403:
2396:
2389:
2370:
2364:
2336:
2335:
2326:
2320:
2176:
2103:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2033:
2029:
2025:
2009:
1993:
1982:
1976:
1970:
1954:
1948:
1942:
1936:
1925:
1916:
1905:
1901:
1895:
1885:
1846:
1831:
1811:
1805:
1794:
1778:, also known as
1773:
1708:
1697:
1645:
1623:
1571:
1553:
1539:
1520:
1509:
1503:
1497:
1479:
1470:
1463:
1449:
1442:
1435:
1428:projective basis
1423:projective frame
1416:Projective frame
1367:
1365:
1364:
1359:
1333:
1330:
1325:
1323:
1309:
1295:
1235:partial function
1229:
1227:
1226:
1221:
1219:
1215:
1213:
1212:
1211:
1202:
1201:
1177:
1176:
1167:
1166:
1148:
1147:
1131:
1130:
1129:
1120:
1119:
1095:
1094:
1085:
1084:
1066:
1065:
1049:
1040:
1039:
1020:
1016:
1014:
1013:
1012:
1003:
1002:
978:
977:
968:
967:
949:
948:
932:
931:
930:
921:
920:
896:
895:
886:
885:
867:
866:
850:
841:
840:
806:
804:
803:
798:
796:
789:
788:
779:
778:
754:
753:
744:
743:
721:
720:
701:
697:
696:
687:
686:
662:
661:
652:
651:
629:
628:
587:
566:
524:
467:
440:. If a basis of
439:
408:projective space
365:
319:
263:partial function
225:projective space
223:. The notion of
219:and parallel to
171:
167:
163:
159:
4200:
4199:
4193:
4192:
4191:
4189:
4188:
4187:
4168:
4167:
4152:
4114:Clarendon Press
4102:
4100:Further reading
4084:
4066:
4032:
4013:
3995:
3970:
3965:
3950:
3946:
3923:
3920:
3911:
3910:
3908:
3907:
3885:
3884:
3879:
3873:
3872:
3867:
3857:
3856:
3853:
3850:
3849:
3844:
3840:
3827:
3823:
3815:
3811:
3805:Hirschfeld 1979
3803:
3799:
3793:Hirschfeld 1979
3791:
3787:
3779:
3775:
3767:
3763:
3755:
3751:
3743:
3739:
3731:
3727:
3713:
3709:
3693:
3689:
3683:Hartshorne 1967
3681:
3677:
3669:
3665:
3657:
3653:
3649:
3637:
3612:
3596:
3577:
3570:
3559:
3527:
3526:
3521:
3515:
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3509:
3499:
3498:
3488:
3487:
3482:
3476:
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3470:
3460:
3459:
3450:
3446:
3444:
3441:
3440:
3400:
3399:
3394:
3388:
3387:
3382:
3372:
3371:
3363:
3360:
3359:
3358:The homography
3356:
3315:
3249:
3245:
3180:
3177:
3176:
3137:
3123:
3121:
3113:
3110:
3109:
3032:
3031:
3026:
3020:
3019:
3014:
3004:
3003:
2983:
2980:
2979:
2962:
2935:
2929:
2917:
2901:
2891:
2884:
2873:
2867:
2861:
2855:
2849:
2839:
2833:
2829:
2823:
2816:
2810:
2804:
2798:
2792:
2786:
2780:
2766:vanishing point
2730:
2701:
2686:
2681:
2671:
2667:
2666:. For example,
2655:
2628:
2597:
2579:
2564:
2544:
2518:inverse mapping
2514:
2494:perspectivities
2430:
2420:
2401:
2394:
2391:
2377:
2368:
2362:
2333:
2328:
2318:
2316:
2177:for all points
2164:
2126:perspectivities
2098:
2097:, of dimension
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2016:
2003:
1995:
1984:
1978:
1972:
1964:
1956:
1950:
1944:
1938:
1927:
1921:
1907:
1903:
1897:
1891:
1888:canonical frame
1871:
1863:
1842:
1840:
1829:
1820:
1813:
1807:
1796:
1785:
1771:
1762:
1751:
1738:
1727:
1699:
1695:
1686:
1675:
1662:
1651:
1644:
1635:
1629:
1621:
1612:
1601:
1588:
1577:
1570:
1561:
1555:
1544:
1522:
1515:
1505:
1499:
1488:
1475:
1465:
1459:
1444:
1437:
1431:
1418:
1412:
1329:
1310:
1296:
1294:
1286:
1283:
1282:
1275:Projective line
1247:
1217:
1216:
1207:
1203:
1191:
1187:
1172:
1168:
1156:
1152:
1137:
1133:
1132:
1125:
1121:
1109:
1105:
1090:
1086:
1074:
1070:
1055:
1051:
1050:
1048:
1041:
1035:
1031:
1028:
1027:
1018:
1017:
1008:
1004:
992:
988:
973:
969:
957:
953:
938:
934:
933:
926:
922:
910:
906:
891:
887:
875:
871:
856:
852:
851:
849:
842:
836:
832:
828:
826:
823:
822:
794:
793:
784:
780:
768:
764:
749:
745:
733:
729:
722:
716:
712:
709:
708:
699:
698:
692:
688:
676:
672:
657:
653:
641:
637:
630:
624:
620:
616:
614:
611:
610:
606:are related by
577:
558:
512:
472:. A point of P(
465:
456:
449:
434:
414:) of dimension
404:
357:
314:
169:
165:
161:
157:
150:
136:. Equivalently
89:, and the term
24:
17:
12:
11:
5:
4198:
4197:
4186:
4185:
4180:
4166:
4165:
4151:
4150:External links
4148:
4147:
4146:
4128:
4106:Patrick du Val
4101:
4098:
4097:
4096:
4087:
4082:
4069:
4064:
4044:
4035:
4030:
4017:
4011:
3998:
3993:
3980:
3969:
3966:
3964:
3963:
3956:On periodicity
3944:
3894:
3889:
3883:
3880:
3878:
3875:
3874:
3871:
3868:
3866:
3863:
3862:
3860:
3838:
3821:
3809:
3797:
3785:
3773:
3761:
3749:
3737:
3725:
3717:characteristic
3707:
3687:
3675:
3663:
3650:
3648:
3645:
3644:
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3465:
3463:
3458:
3453:
3449:
3404:
3398:
3395:
3393:
3390:
3389:
3386:
3383:
3381:
3378:
3377:
3375:
3370:
3367:
3355:
3352:
3330:to facilitate
3302:
3301:
3290:
3287:
3284:
3278:
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3269:
3266:
3263:
3260:
3255:
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3095:
3094:
3083:
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3041:
3036:
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3018:
3015:
3013:
3010:
3009:
3007:
3002:
2999:
2996:
2993:
2990:
2987:
2951:group of units
2931:Main article:
2928:
2925:
2776:Main article:
2770:
2769:
2762:
2758:
2757:
2754:
2729:
2726:
2679:
2577:quotient group
2513:
2510:
2506:linear algebra
2498:
2497:
2490:
2479:
2419:
2416:
2337:, the axis of
2298:diagonalizable
2185:) and a point
2015:
2012:
1999:
1960:
1904:(1, 1, ..., 1)
1867:
1836:
1825:
1818:
1767:
1760:
1747:
1736:
1698:is a frame of
1691:
1684:
1671:
1660:
1646:is a basis of
1640:
1633:
1617:
1610:
1597:
1586:
1566:
1559:
1530:∖ {0} →
1414:Main article:
1411:
1408:
1388:Riemann sphere
1369:
1368:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
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1322:
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1290:
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1190:
1186:
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1128:
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1101:
1098:
1093:
1089:
1083:
1080:
1077:
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1054:
1047:
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1042:
1038:
1034:
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1029:
1026:
1023:
1021:
1019:
1011:
1007:
1001:
998:
995:
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987:
984:
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956:
952:
947:
944:
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656:
650:
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644:
640:
636:
633:
631:
627:
623:
619:
618:
461:
454:
429:-vector space
403:
400:
389:Originally, a
207:and the plane
149:
146:
119:linear algebra
15:
9:
6:
4:
3:
2:
4196:
4195:
4184:
4181:
4179:
4176:
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4173:
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4144:
4143:0-534-00034-7
4140:
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4133:
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4126:
4123:
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4093:
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4083:0-486-63415-9
4079:
4075:
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4067:
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4036:
4033:
4031:0-521-48364-6
4027:
4023:
4018:
4014:
4008:
4004:
3999:
3996:
3994:9780486445656
3990:
3986:
3981:
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3948:
3941:
3940:
3930:
3926:
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3892:
3887:
3881:
3876:
3869:
3864:
3858:
3847:
3846:Arthur Cayley
3842:
3835:
3831:
3825:
3818:
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3806:
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3794:
3789:
3782:
3777:
3770:
3765:
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3729:
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3642:
3639:
3638:
3627:
3623:
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3600:
3594:
3590:
3584:
3580:
3573:
3566:
3562:
3557:
3556:
3555:
3553:
3549:
3548:Arthur Cayley
3534:
3529:
3523:
3518:
3511:
3506:
3500:
3495:
3490:
3484:
3479:
3472:
3467:
3461:
3456:
3451:
3447:
3439:) since then
3438:
3437:
3431:
3428:
3424:
3420:
3402:
3396:
3391:
3384:
3379:
3373:
3368:
3365:
3351:
3349:
3348:biquaternions
3345:
3341:
3337:
3333:
3329:
3325:
3319:
3314:
3313:modular group
3310:
3307:
3288:
3282:
3276:
3270:
3267:
3264:
3261:
3253:
3250:
3242:
3239:
3236:
3233:
3224:
3215:
3212:
3209:
3206:
3200:
3197:
3194:
3191:
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3182:
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3173:
3156:
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3141:
3138:
3133:
3130:
3127:
3124:
3115:
3108:
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2615:
2611:
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2601:
2594:
2591:
2587:
2583:
2578:
2572:
2568:
2562:
2559:over a field
2558:
2552:
2548:
2542:
2537:
2533:
2531:
2527:
2523:
2519:
2509:
2507:
2503:
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2469:
2468:
2467:
2465:
2460:
2458:
2454:
2450:
2446:
2442:
2438:
2435:
2429:
2428:Perspectivity
2425:
2415:
2411:
2408:
2404:
2397:
2388:
2384:
2380:
2375:
2371:
2360:
2356:
2352:
2348:
2344:
2341:is some line
2340:
2331:
2324:
2314:
2310:
2306:
2301:
2299:
2295:
2291:
2288:of dimension
2287:
2283:
2281:
2277:
2270:
2268:
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2260:
2256:
2252:
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2244:
2240:
2236:
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2200:
2196:
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2175:
2171:
2167:
2162:
2158:
2154:
2150:
2147:
2143:
2139:
2135:
2131:
2130:Perspectivity
2127:
2123:
2119:
2115:
2111:
2107:
2101:
2096:
2091:
2089:
2085:
2070:. The image E
2020:
2011:
2007:
2002:
1998:
1991:
1987:
1981:
1977:on the frame
1975:
1968:
1963:
1959:
1953:
1947:
1941:
1934:
1930:
1924:
1920:
1914:
1910:
1900:
1894:
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1495:
1491:
1485:
1483:
1482:division ring
1478:
1472:
1468:
1462:
1457:
1454:, although a
1453:
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1297:
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1280:
1279:
1277:
1276:
1271:
1267:
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1256:
1255:complex plane
1251:
1242:
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1198:
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1022:
1009:
1005:
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950:
945:
942:
939:
935:
927:
923:
917:
914:
911:
907:
903:
900:
897:
892:
888:
882:
879:
876:
872:
868:
863:
860:
857:
853:
846:
844:
837:
833:
821:
820:
819:
817:
816:affine spaces
813:
790:
785:
781:
775:
772:
769:
765:
761:
758:
755:
750:
746:
740:
737:
734:
730:
726:
724:
717:
713:
705:
703:
693:
689:
683:
680:
677:
673:
669:
666:
663:
658:
654:
648:
645:
642:
638:
634:
632:
625:
621:
609:
608:
607:
605:
601:
597:
593:
590:, called the
589:
585:
581:
573:
568:
565:
561:
556:
552:
548:
544:
540:
536:
532:
528:
523:
519:
515:
510:
506:
502:
498:
494:
490:
485:
483:
479:
475:
471:
464:
460:
453:
447:
443:
437:
433:of dimension
432:
428:
424:
421:
417:
413:
409:
399:
396:
392:
387:
385:
381:
377:
373:
369:
364:
360:
355:
351:
347:
343:
339:
335:
329:
327:
323:
317:
313:of dimension
312:
308:
304:
300:
296:
292:
291:perspectivity
288:
285:of dimension
284:
280:
274:
272:
268:
264:
259:
257:
256:perspectivity
253:
249:
245:
241:
236:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
187:from a point
186:
181:
179:
154:
145:
143:
139:
135:
130:
128:
124:
120:
116:
112:
109:over a given
108:
104:
100:
99:affine spaces
96:
92:
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
44:vector spaces
41:
37:
33:
29:
22:
4131:
4109:
4094:, Holden-Day
4091:
4073:
4051:
4039:
4021:
4002:
3984:
3975:
3947:
3937:
3928:
3924:
3916:
3912:
3841:
3836:at Wikibooks
3824:
3812:
3800:
3788:
3781:Meserve 1983
3776:
3764:
3752:
3740:
3735:, p. 66
3728:
3710:
3690:
3678:
3671:Meserve 1983
3666:
3654:
3625:
3621:
3617:
3613:
3602:
3598:
3592:
3588:
3582:
3578:
3571:
3564:
3560:
3435:
3429:
3426:
3422:
3357:
3339:
3332:screw theory
3317:
3308:
3303:
3171:
3098:
3096:
2972:
2963:
2958:
2957:, written P(
2954:
2946:
2938:
2936:
2919:
2911:
2907:
2903:
2892:
2885:
2878:
2874:
2868:
2862:
2856:
2850:
2845:
2840:
2834:
2824:
2817:
2811:
2805:
2799:
2793:
2787:
2784:
2781:
2738:cross-ratios
2721:
2718:block design
2713:
2707:
2703:
2698:
2692:
2688:
2684:
2661:
2657:
2651:
2648:Galois field
2643:
2641:
2634:
2630:
2624:
2620:
2617:
2603:
2599:
2592:
2589:
2585:
2581:
2570:
2566:
2560:
2556:
2550:
2546:
2543:are denoted
2538:
2534:
2530:Möbius group
2515:
2499:
2486:
2482:
2475:
2471:
2463:
2461:
2448:
2447:by applying
2444:
2440:
2436:
2431:
2412:
2406:
2399:
2392:
2386:
2382:
2378:
2373:
2366:
2365:. The image
2358:
2354:
2350:
2346:
2342:
2338:
2329:
2322:
2312:
2308:
2304:
2302:
2289:
2284:that has an
2279:
2275:
2271:
2266:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2189:(called the
2186:
2182:
2178:
2173:
2169:
2165:
2160:
2156:
2152:
2151:(called the
2148:
2141:
2137:
2133:
2125:
2121:
2117:
2113:
2109:
2106:collineation
2099:
2094:
2092:
2087:
2081:
2005:
2000:
1996:
1989:
1985:
1979:
1973:
1966:
1961:
1957:
1951:
1945:
1939:
1932:
1928:
1922:
1912:
1908:
1898:
1892:
1887:
1881:
1877:
1873:
1868:
1864:
1861:
1856:
1852:
1848:
1843:
1837:
1833:
1826:
1822:
1815:
1812:on the base
1808:
1801:
1797:
1790:
1786:
1779:
1775:
1768:
1764:
1757:
1753:
1748:
1744:
1740:
1733:
1729:
1726:Every frame
1725:
1720:
1713:identity map
1710:
1704:
1700:
1692:
1688:
1681:
1677:
1672:
1668:
1664:
1657:
1653:
1647:
1641:
1637:
1630:
1625:
1618:
1614:
1607:
1603:
1598:
1594:
1590:
1583:
1579:
1573:
1567:
1563:
1556:
1549:
1545:
1542:
1535:
1531:
1527:
1523:
1516:
1511:
1506:
1500:
1493:
1489:
1486:
1476:
1473:
1466:
1464:has at most
1460:
1451:
1445:
1438:
1432:
1427:
1421:
1419:
1404:cross-ratios
1396:
1381:
1376:
1372:
1370:
1273:
1269:
1265:
1260:
1232:
809:
603:
599:
591:
583:
579:
576:nonsingular
571:
569:
563:
559:
554:
550:
546:
542:
538:
534:
530:
521:
517:
513:
504:
500:
496:
492:
488:
486:
477:
473:
469:
462:
458:
451:
445:
441:
435:
430:
426:
422:
415:
411:
405:
390:
388:
375:
371:
367:
362:
358:
353:
349:
345:
341:
337:
333:
331:
325:
321:
315:
310:
306:
302:
298:
294:
290:
286:
282:
278:
276:
260:
251:
243:
239:
237:
232:
220:
216:
212:
208:
204:
200:
196:
192:
188:
182:
175:
131:
107:vector space
90:
76:
71:
67:
64:projectivity
63:
56:collineation
31:
25:
3819:, chapter 6
3817:Berger 2009
3747:, chapter 6
3745:Berger 2009
3661:, chapter 4
3659:Berger 2009
3326:, and with
2927:Over a ring
2846:cross-ratio
2778:Cross-ratio
2728:Cross-ratio
2522:composition
2439:of a field
2372:of a point
2353:of ℓ under
1399:permutation
509:isomorphism
395:composition
248:restriction
125:, see also
83:projections
79:perspective
36:isomorphism
4172:Categories
4162:Homography
4003:Geometry I
3968:References
3783:, pp. 43–4
3703:Artin 1957
3697:, p. 244,
3673:, pp. 43–4
2838:to 0, and
2822:that maps
2422:See also:
2327:. Setting
2294:eigenvalue
2286:eigenspace
2282:+1) matrix
2215:homologies
2163:(that is,
2146:hyperplane
1471:vertices.
1239:hyperplane
557:such that
497:homography
391:homography
382:below and
91:homography
50:that maps
32:homography
4076:, Dover,
3987:, Dover,
3757:Yale 1968
3733:Baer 2005
3721:Baer 2005
3701:, p. 50,
3699:Baer 2005
3695:Yale 1968
3595:), where
3251:−
3222:∼
3119:↦
2672:PGL(2, 4)
2668:PGL(2, 7)
1350:≠
1341:−
1292:↦
1182:⋯
1100:⋯
1025:⋮
983:⋯
901:⋯
759:⋯
706:⋮
667:⋯
527:bijection
267:bijection
95:Euclidean
48:bijection
4050:(1979),
3807:, p. 129
3759:, p. 224
3685:, p. 138
3635:See also
3620:= 2 cos(
3419:periodic
3306:integers
2937:Suppose
2633:+ 1) × (
2596:, where
2520:and the
2345:through
2211:Elations
1949:mapping
1763:+ ... +
1739:), ...,
1687:+ ... +
1663:), ...,
1613:+ ... +
1589:), ...,
1526: :
1504:, where
516: :
491:) and P(
4125:0169108
4108:(1964)
3933:
3909:
3795:, p. 30
3771:, p. 96
3723:, p. 76
3705:, p. 88
3641:W-curve
3342:is the
3316:PSL(2,
2949:is its
2916:, then
2736:Use of
2612:of the
2608:is the
2536:space.
2451:to all
2390:, then
2332:= ℓ ∩ ℓ
2278:+1) × (
1821:, ...,
1650:, then
1636:, ...,
1562:, ...,
1456:simplex
1452:simplex
582:+1) × (
533:) to P(
529:from P(
503:) to P(
457:, ...,
418:over a
4141:
4118:Oxford
4080:
4062:
4028:
4009:
3991:
3607:, the
3574:> 2
3334:. The
3280:
3204:
3154:
3064:
2616:, and
2426:, and
2361:with ℓ
2191:center
1886:has a
1521:, and
588:matrix
344:, and
246:, the
70:, and
34:is an
3942:9:104
3647:Notes
3609:trace
3605:) = 1
3432:(the
3101:is a
3097:When
2967:with
2941:is a
2888:∪ {∞}
2820:∪ {∞}
2764:As a
2706:+ 1,
2691:+ 1,
2646:is a
2642:When
2602:+ 1,
2584:+ 1,
2569:+ 1,
2549:+ 1,
2526:group
2136:from
2116:onto
1919:tuple
1510:is a
1263:field
596:up to
420:field
271:field
134:field
111:field
52:lines
4139:ISBN
4078:ISBN
4060:ISBN
4026:ISBN
4007:ISBN
3989:ISBN
2945:and
2943:ring
2896:has
2866:and
2797:and
2702:PΓL(
2687:PGL(
2656:PGL(
2637:+ 1)
2588:) /
2565:PGL(
2545:PGL(
2462:The
2265:and
2249:and
2172:) =
2153:axis
2104:, a
2066:, DD
2062:, CC
2058:, BB
1876:) =
1841:and
1487:Let
545:and
305:and
289:, a
281:and
140:and
97:and
81:and
30:, a
3958:in
3936:",
3611:is
3589:hπi
3585:= 1
3567:= 0
3417:is
3346:of
3311:is
2883:of
2828:to
2761:2.
2753:1.
2740:in
2650:GF(
2598:GL(
2580:GL(
2325:(ℓ)
2269:.)
2193:of
2181:in
2155:of
2140:to
2108:of
2102:≥ 2
2034:, D
2030:, C
2026:, B
1752:),
1676:),
1602:),
1572:of
1519:+ 1
1484:.
1469:+ 1
1448:+ 1
1441:+ 2
1426:or
1375:or
814:to
586:+1)
553:of
468:of
438:+ 1
386:.)
352:to
340:to
332:If
318:+ 1
297:to
250:to
168:, D
164:, C
160:, B
85:in
38:of
26:In
4174::
4122:MR
4120:,
4116:,
4058:,
4054:,
3954:,
3927:+
3925:cx
3915:+
3913:ax
3622:hπ
3616:+
3601:,
3583:bc
3581:−
3579:ad
3563:+
3350:.
2923:.
2918:=
2910:,
2906:,
2860:,
2854:,
2832:,
2791:,
2724:.
2660:,
2639:.
2459:.
2410:.
2407:OB
2405:∩
2400:SA
2398:=
2385:∩
2383:AB
2381:=
2359:OA
2321:=
2315:,
2010:.
1859:.
1772:))
1723:.
1696:))
1622:))
1420:A
1406:.
1379:.
567:.
564:af
562:=
520:→
410:P(
406:A
361:⋅
258:.
235:.
205:OA
74:.
66:,
4145:.
4127:.
3929:d
3921:/
3917:b
3893:,
3888:)
3882:d
3877:c
3870:b
3865:a
3859:(
3630:.
3628:)
3626:m
3624:/
3618:d
3614:a
3603:m
3599:h
3597:(
3593:m
3591:/
3572:n
3565:d
3561:a
3535:.
3530:)
3524:1
3519:0
3512:0
3507:1
3501:(
3496:=
3491:)
3485:1
3480:0
3473:n
3468:1
3462:(
3457:=
3452:n
3448:h
3436:n
3430:Z
3427:n
3425:/
3423:Z
3403:)
3397:1
3392:0
3385:1
3380:1
3374:(
3369:=
3366:h
3340:A
3320:)
3318:Z
3309:Z
3289:.
3286:]
3283:1
3277:,
3274:)
3271:b
3268:+
3265:a
3262:z
3259:(
3254:1
3247:)
3243:d
3240:+
3237:c
3234:z
3231:(
3228:[
3225:U
3219:]
3216:d
3213:+
3210:c
3207:z
3201:,
3198:b
3195:+
3192:a
3189:z
3186:[
3183:U
3157:,
3148:d
3145:+
3142:c
3139:z
3134:b
3131:+
3128:a
3125:z
3116:z
3099:A
3082:.
3079:]
3076:d
3073:+
3070:c
3067:z
3061:,
3058:b
3055:+
3052:a
3049:z
3046:[
3043:U
3040:=
3035:)
3029:d
3024:b
3017:c
3012:a
3006:(
3001:]
2998:1
2995:,
2992:z
2989:[
2986:U
2973:A
2964:U
2959:A
2955:A
2947:U
2939:A
2920:k
2914:)
2912:c
2908:b
2904:a
2902:(
2893:d
2886:F
2881:)
2879:d
2877:(
2875:h
2869:d
2863:c
2857:b
2851:a
2841:c
2835:b
2830:∞
2825:a
2818:F
2812:h
2806:F
2800:c
2794:b
2788:a
2714:n
2710:)
2708:F
2704:n
2695:)
2693:F
2689:n
2680:5
2678:A
2664:)
2662:q
2658:n
2652:q
2644:F
2635:n
2631:n
2629:(
2625:F
2621:I
2618:F
2606:)
2604:F
2600:n
2593:I
2590:F
2586:F
2582:n
2573:)
2571:F
2567:n
2561:F
2557:n
2553:)
2551:F
2547:n
2487:P
2483:P
2476:P
2472:P
2449:σ
2445:F
2441:F
2437:σ
2402:′
2395:′
2393:B
2387:M
2379:S
2374:B
2369:′
2367:B
2363:′
2355:α
2351:A
2347:R
2343:M
2339:α
2334:′
2330:R
2323:α
2319:′
2317:ℓ
2313:α
2309:O
2305:α
2290:n
2280:n
2276:n
2274:(
2267:Q
2263:P
2259:P
2257:(
2255:α
2251:Q
2247:O
2243:Q
2239:Q
2237:(
2235:α
2231:O
2227:P
2223:P
2221:(
2219:α
2207:α
2203:O
2199:α
2195:α
2187:O
2183:H
2179:X
2174:X
2170:X
2168:(
2166:α
2161:α
2157:α
2149:H
2142:P
2138:P
2134:α
2118:P
2114:P
2110:P
2100:n
2095:P
2076:′
2072:′
2068:′
2064:′
2060:′
2056:′
2052:′
2050:D
2048:′
2046:C
2044:′
2042:B
2040:′
2036:′
2032:′
2028:′
2024:′
2008:)
2006:K
2004:(
2001:n
1997:P
1992:)
1990:a
1988:(
1986:h
1980:F
1974:a
1969:)
1967:K
1965:(
1962:n
1958:P
1952:F
1946:h
1940:F
1935:)
1933:V
1931:(
1929:P
1923:v
1915:)
1913:v
1911:(
1909:p
1899:K
1893:p
1884:)
1882:K
1880:(
1878:P
1874:K
1872:(
1869:n
1865:P
1857:K
1853:v
1851:(
1849:p
1844:v
1838:i
1834:e
1830:)
1827:n
1823:e
1819:0
1816:e
1814:(
1809:v
1804:)
1802:v
1800:(
1798:p
1793:)
1791:v
1789:(
1787:p
1769:n
1765:e
1761:0
1758:e
1756:(
1754:p
1749:n
1745:e
1743:(
1741:p
1737:0
1734:e
1732:(
1730:p
1728:(
1707:)
1705:V
1703:(
1701:P
1693:n
1689:e
1685:0
1682:e
1680:(
1678:p
1673:n
1669:e
1667:(
1665:p
1661:0
1658:e
1656:(
1654:p
1652:(
1648:V
1642:n
1638:e
1634:0
1631:e
1626:K
1619:n
1615:e
1611:0
1608:e
1606:(
1604:p
1599:n
1595:e
1593:(
1591:p
1587:0
1584:e
1582:(
1580:p
1578:(
1574:V
1568:n
1564:e
1560:0
1557:e
1552:)
1550:V
1548:(
1546:P
1538:)
1536:V
1534:(
1532:P
1528:V
1524:p
1517:n
1512:K
1507:V
1501:n
1496:)
1494:V
1492:(
1490:P
1477:K
1467:n
1461:n
1446:n
1439:n
1433:n
1356:,
1353:0
1347:c
1344:b
1338:d
1335:a
1327:,
1321:d
1318:+
1315:z
1312:c
1307:b
1304:+
1301:z
1298:a
1289:z
1270:K
1266:K
1209:n
1205:x
1199:n
1196:,
1193:0
1189:a
1185:+
1179:+
1174:1
1170:x
1164:1
1161:,
1158:0
1154:a
1150:+
1145:0
1142:,
1139:0
1135:a
1127:n
1123:x
1117:n
1114:,
1111:n
1107:a
1103:+
1097:+
1092:1
1088:x
1082:1
1079:,
1076:n
1072:a
1068:+
1063:0
1060:,
1057:n
1053:a
1046:=
1037:n
1033:y
1010:n
1006:x
1000:n
997:,
994:0
990:a
986:+
980:+
975:1
971:x
965:1
962:,
959:0
955:a
951:+
946:0
943:,
940:0
936:a
928:n
924:x
918:n
915:,
912:1
908:a
904:+
898:+
893:1
889:x
883:1
880:,
877:1
873:a
869:+
864:0
861:,
858:1
854:a
847:=
838:1
834:y
791:.
786:n
782:x
776:n
773:,
770:n
766:a
762:+
756:+
751:0
747:x
741:0
738:,
735:n
731:a
727:=
718:n
714:y
694:n
690:x
684:n
681:,
678:0
674:a
670:+
664:+
659:0
655:x
649:0
646:,
643:0
639:a
635:=
626:0
622:y
604:φ
600:K
584:n
580:n
578:(
572:φ
560:g
555:K
551:a
547:g
543:f
539:f
535:W
531:V
522:W
518:V
514:f
505:W
501:V
493:W
489:V
478:V
474:V
470:K
466:)
463:n
459:x
455:0
452:x
450:(
446:V
442:V
436:n
431:V
427:K
423:K
416:n
412:V
376:P
368:P
363:f
359:g
354:P
350:Q
346:g
342:Q
338:P
334:f
328:.
326:Q
322:P
316:n
311:R
307:Q
303:P
299:Q
295:P
287:n
283:Q
279:P
252:Q
244:O
240:Q
233:O
221:P
217:O
213:A
209:P
201:A
197:O
193:P
189:O
170:′
166:′
162:′
158:′
23:.
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