20:
217:
4966:
3548:
2718:
3024:
is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a
2712:
2799:
in both directions. The image represents the projective line as a circle where antipodal points are identified, and shows the two homeomorphisms of a real line to the projective line; as antipodal points are identified, the image of each line is represented as an open half circle, which can be
2522:
3628:
All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are
849:
are the coordinates on the basis of any element of the corresponding equivalence class. These coordinates are commonly denoted , the colons and the brackets being used for distinguishing from usual coordinates, and emphasizing that this is an equivalence class, which is defined
495:
under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed.
3754:
and denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is
2527:
2301:
1896:. 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
3147:
The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an
3195:, there is no difference. However, for dimension two, there are examples that satisfy these axioms that can not be constructed from vector spaces (or even modules over division rings). These examples do not satisfy the
3359:
98:. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.
2264:
2306:
3645:
finite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the
3191:
The structures defined by these axioms are more general than those obtained from the vector space construction given above. If the (projective) dimension is at least three then, by the
1756:, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See
2010:
1884:, 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
2797:
1709:, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than
82:, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In
2188:
1065:
4143:
In algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles. One would like to be able to associate a projective space to
1865:
asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
4397:
As Pappus's theorem implies
Desargues's theorem this eliminates the non-Desarguesian planes and also implies that the space is defined over a field (and not a division ring).
2064:
onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold. A simple
1943:
In this section, all projective spaces are real projective spaces of finite dimension. However everything applies to complex projective spaces, with slight modifications.
1920:); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
3203:. In dimension one, any set with at least three elements satisfies the axioms, so it is usual to assume additional structure for projective lines defined axiomatically.
854:
the multiplication by a non zero constant. That is, if are projective coordinates of a point, then are also projective coordinates of the same point, for any nonzero
4714:
1912:
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 (
1748:
from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which
4160:
1724:
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a
3290:
1803:
The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of
2707:{\displaystyle \varphi _{i}^{-1}\left(\right)=\left({\frac {x_{0}}{x_{i}}},\dots ,{\widehat {\frac {x_{i}}{x_{i}}}},\dots ,{\frac {x_{n}}{x_{i}}}\right),}
4897:
3968:, it can be shown that any algebraic (not necessarily linear) automorphism must be linear, i.e., coming from a (linear) automorphism of the vector space
4388:, pp. 6–7). Pasch was concerned with real projective space and was attempting to introduce order, which is not a concern of the Veblen–Young axiom.
3210:, p. 231) gives such an extension due to Bachmann. To ensure that the dimension is at least two, replace the three point per line axiom above by:
3639:
2517:{\displaystyle {\begin{aligned}\mathbb {\varphi } _{i}:R^{n}&\to U_{i}\\(y_{0},\dots ,{\widehat {y_{i}}},\dots y_{n})&\mapsto ,\end{aligned}}}
1934:
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.
94:
are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the
3029:
1922:
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative)
2879:. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the
1222:, where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.
4806:
4766:
4626:
4603:
4585:
4529:
2204:
2919:
points if one consider complex points in the projective plane, and if one counts the points with their multiplicity. Another example is the
1214:
is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points.
862:. Also, the above definition implies that are projective coordinates of a point if and only if at least one of the coordinates is nonzero.
154:, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a
1778:
1744:
does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in
3646:
3226:
If the six vertices of a hexagon lie alternately on two lines, the three points of intersection of pairs of opposite sides are collinear.
2895:
complex roots. In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider
1765:
While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of
5000:
488:. A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines.
4950:
4726:
4680:
4659:
4442:
4094:, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of
2966:
1808:
2973:
1972:
3206:
It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space.
4890:
4668:
4643:
4367:
Homogeneous required in order that a zero remains a zero when the homogeneous coordinates are multiplied by a nonzero scalar.
2962:
2880:
2769:
3573:
2103:
1027:
4569:
914:
95:
5194:
5189:
4870:
3572:
of the space is defined as one less than this common number. For finite projective spaces of dimension at least three,
2016:
to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of
4985:
4564:
4308:
4235:
2972:
An important property of projective spaces and projective varieties is that the image of a projective variety under a
1126:
is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines
3219:
1927:
1315:, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.
4798:
4618:
2766:. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map is
4609:, translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation
4252:
4883:
4330:
4325:
4275:
2928:
1702:
3568:
is a finite set of points. In any finite projective space, each line contains the same number of points and the
3494:
3192:
2920:
181:, which can be distinguished only by their intersections with the line at infinity: two intersection points for
4920:
4245:
2984:). This is a generalization to every ground field of the compactness of the real and complex projective space.
1873:
1766:
1714:
1289:
and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of
874:
602:
4785:
4068:
5128:
5123:
5103:
4053:
3413:
2943:
2868:
2087:
1793:) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as
1770:
244:
236:
205:
48:
24:
2965:
the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by
5113:
5108:
5088:
3483:
3479:
3231:
3200:
2999:
2947:
2884:
1931:
1905:
1729:
1337:
is an ordered set of points in a projective space that allows defining coordinates. More precisely, in an
842:
2900:
5118:
5098:
5093:
4303:
4185:
4178:
The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space
3002:
during the second half of 20th century, allows defining a generalization of algebraic varieties, called
1812:
1733:
1725:
870:
293:
3412:
theory. There is a bijective correspondence between projective spaces and geomodular lattices, namely,
3287:
if that is the largest number for which there is a strictly ascending chain of subspaces of this form:
1736:). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in
389:
in this plane consists of all projective points (which are lines) contained in a plane passing through
4559:
4406:
This restriction allows the real and complex fields to be used (zero characteristic) but removes the
4358:
The correct definition of the multiplicity if not easy and dates only from the middle of 20th century
3978:
3421:
3196:
3003:
1182:
824:
401:, the intersection of two distinct projective lines consists of a single projective point. The plane
253:
139:
4995:
4990:
4313:
4256:
3975:
3796:
3243:
3149:
2872:
1923:
1901:
1834:
1749:
1745:
1698:
1686:
534:
174:
3958:
can be described more concretely. (We deal only with automorphisms preserving the base field
2987:
A projective space is itself a projective variety, being the set of zeros of the zero polynomial.
5169:
5010:
4965:
4517:
4231:
3467:
3180:
3040:
2958:
2939:
2864:
1917:
1913:
1904:(the above definition is based on this version); this construction facilitates the definition of
1893:
1877:
1794:
1786:
1737:
1706:
1219:
240:
221:
83:
64:
470:. Also, the construction can be done by starting with a vector space of any positive dimension.
5005:
4843:
4802:
4762:
4722:
4694:
4676:
4655:
4622:
4599:
4581:
4525:
4438:
4291:
4037:
3965:
3873:
3021:
2738:
2734:
1951:
1947:
1804:
1753:
1718:
625:
621:
617:
555:
492:
466:), the preceding construction is generally done by starting from a vector space and is called
19:
4750:
4381:
3542:
3451:
2977:
2726:
2292:
2065:
1790:
1774:
1758:
1324:
1231:
894:
819:
504:
467:
248:
159:
151:
4827:
4776:
4736:
4690:
4636:
432:
with the corresponding projective point, one can thus say that the projective plane is the
4980:
4925:
4823:
4772:
4758:
4732:
4718:
4686:
4632:
3425:
3417:
3409:
2924:
2904:
2028:
1885:
1854:
1798:
1710:
1087:
890:
227:
As outlined above, projective spaces were introduced for formalizing statements like "two
201:
170:
155:
56:
28:
4265:, which become isomorphic to projective spaces after an extension of the base field
4036:
modulo the matrices that are scalar multiples of the identity. (These matrices form the
3443:
Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for
5062:
5047:
4707:
4648:
4221:
4025:
3852:
3007:
2730:
1909:
898:
656:
433:
232:
228:
102:
72:
4506:, Grundlehren der mathematischen Wissenschaftern, 96, Berlin: Springer, pp. 76–77
257:). More precisely, the entrance pupil of a camera or of the eye of an observer is the
5183:
5052:
4846:
4815:
4435:
4279:
4201:
3523:
3230:
And, to ensure that the vector space is defined over a field that does not have even
3089:
2995:
2981:
2296:
2061:
2032:
1869:
1773:) being complex numbers. Several major types of more abstract mathematics (including
909:
178:
3576:
implies that the division ring over which the projective space is defined must be a
5072:
5037:
4930:
4217:
4188:, which is parametrizing higher-dimensional subspaces (of some fixed dimension) of
4124:
3937:
3791:
3756:
3577:
2876:
2075:, any vector can be identified with its coordinates on the basis, and any point of
1897:
1889:
1858:
1846:
1312:
1308:
1234:
of projective subspaces is a projective subspace. It follows that for every subset
951:
527:
462:
457:
449:
128:
114:
68:
60:
364:, which are in one to one correspondence with the directions of parallel lines in
1868:
Historically, homographies (and projective spaces) have been introduced to study
1189:
There is exactly one projective line that passes through two different points of
272:
of the space (the intersection of the axes in the figure); the projection plane (
5157:
4940:
3594:(a prime power). A finite projective space defined over such a finite field has
1962:
1842:
1797:. Another topic that developed from axiomatic studies of projective geometry is
1782:
1694:
1076:
783:
646:
478:
304:
with the projection plane. Such an intersection exists if and only if the point
143:
110:
40:
3454:. These are much harder to classify, as not all of them are isomorphic with a
231:
intersect in exactly one point, and this point is at infinity if the lines are
5152:
5032:
4864:
4407:
4080:
3766:
3675:
3653:
3552:
3402:
2809:
1862:
1838:
1826:
1356:
216:
189:, which is tangent to the line at infinity; and no real intersection point of
5133:
5042:
4955:
4906:
4851:
2763:
2717:
1850:
182:
79:
3660:
with 7 points and 7 lines. The smallest 3-dimensional projective spaces is
499:
A third equivalent definition is to define a projective space of dimension
4698:
1238:
of a projective space, there is a smallest projective subspace containing
173:, two distinct lines in a plane intersect in at most one point, while, in
5057:
5020:
4945:
4860:
3889:
3011:
742:, although this notation may be confused with exponentiation). The space
197:
186:
166:
3790:
is non-zero and in the null space. In this case one obtains a so-called
1701:
is the study of geometric properties that are invariant with respect to
869:
is the field of real or complex numbers, a projective space is called a
177:, they intersect in exactly one point. Also, there is only one class of
78:
This definition of a projective space has the disadvantage of not being
5067:
3661:
3601:
points on a line, so the two concepts of order coincide. Notationally,
3547:
3354:{\displaystyle \varnothing =X_{-1}\subset X_{0}\subset \cdots X_{n}=P.}
988:
204:, projective spaces play a fundamental role, being typical examples of
190:
32:
1669:
169:, as allowing simpler statements and simpler proofs. For example, in
1635:. They are only defined up to scaling with a common nonzero factor.
481:(vector subspaces of dimension one) in a vector space of dimension
337:
split in two disjoint subsets: the lines that are not contained in
5024:
3987:. By identifying maps that differ by a scalar, one concludes that
3546:
3482:
and are projective planes over division rings, but there are many
3440:
Dimension 1 (exactly one line): All points lie on the unique line.
2716:
1741:
851:
381:
for clarity) of the projective plane as the lines passing through
215:
18:
4822:, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London,
4349:
The absence of space after the comma is common for this notation.
2800:
identified with the projective line with a single point removed.
142:"being on the same vector line". As a vector line intersects the
1520:; this representation is unique up to the multiplication of all
1341:-dimensional projective space, a projective frame is a tuple of
4879:
1861:. In general, some collineations are not homographies, but the
897:, respectively. The complex projective line is also called the
3408:
Projective spaces admit an equivalent formulation in terms of
1355:
of them are independent; that is, they are not contained in a
55:. A projective space may thus be viewed as the extension of a
3278:). The full space and the empty space are always subspaces.
4875:
346:, which are in one to one correspondence with the points of
67:, in such a way that there is one point at infinity of each
4204:
is the space of flags, i.e., chains of linear subspaces of
3634:
2957:, in a unique way, into a projective variety by adding its
1954:
of the topology of a finite dimensional real vector space.
1672:
with only one nonzero entry, equal to 1), and the image by
1242:, the intersection of all projective subspaces containing
296:
are considered. Then, the central projection maps a point
27:, parallel (horizontal) lines in the plane intersect at a
3497:, to the effect that every projective space of dimension
3214:
There exist four points, no three of which are collinear.
2714:
where hats means that the corresponding term is missing.
1762:
for the basics of projective geometry in two dimensions.
904:
All these definitions extend naturally to the case where
3489:
Dimension at least 3: Two non-intersecting lines exist.
281:, in blue on the figure) is a plane not passing through
3714:
induce mappings of the corresponding projective spaces
2259:{\displaystyle \mathbf {P} (V)=\bigcup _{i=0}^{n}U_{i}}
1845:
of projective spaces, induced by an isomorphism of the
1717:
are permitted that transform the extra points (called "
16:
Completion of the usual space with "points at infinity"
4477:
4385:
3872:. Thus if one identifies the scalar multiples of the
3374:
in such a chain is said to have (geometric) dimension
1458:
on this basis are all nonzero. By rescaling the first
4615:
Projective geometry: from foundations to applications
4430:
Mauro
Biliotti, Vikram Jha, Norman L. Johnson (2001)
3541:
Further information on finite projective planes:
3293:
2772:
2530:
2304:
2207:
2106:
1975:
1030:
3437:
Dimension 0 (no lines): The space is a single point.
1269:
if its span is not the span of any proper subset of
1108:
is a projective space, which can be identified with
393:. As the intersection of two planes passing through
285:, which is often chosen to be the plane of equation
268:
Mathematically, the center of projection is a point
5145:
5081:
5019:
4973:
4913:
4234:(rather than only fields) yields, for example, the
4056:. The automorphisms of the complex projective line
3281:The geometric dimension of the space is said to be
2751:, that is of a projective line, there are only two
4706:
4647:
3353:
2942:is the set of points in a projective space, whose
2791:
2706:
2516:
2258:
2182:
2039:A (finite dimensional) projective space is compact
2004:
1059:
463:Affine space § Vector spaces as affine spaces
113:(that is, vector subspaces of dimension one) in a
1849:from which the projective spaces derive. It is a
235:". Such statements are suggested by the study of
4715:Ergebnisse der Mathematik und ihrer Grenzgebiete
4613:Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
4410:and other planes that exhibit atypical behavior.
4274:Another generalization of projective spaces are
3543:Projective plane § Finite projective planes
2762:, which can each be identified to a copy of the
3588:, whose order (that is, number of elements) is
883:is one or two, a projective space of dimension
4675:, Toronto, Ont.: University of Toronto Press,
4079:When the construction above is applied to the
3260:, such that any line containing two points of
3218:To avoid the non-Desarguesian planes, include
2721:Manifold structure of the real projective line
410:defines a projective line which is called the
317:, in green on the figure) that passes through
4891:
4797:, Cambridge Studies in Advanced Mathematics,
3855:they differ by a scalar multiple, that is if
3188:that states which points lie on which lines.
3070:(the set of lines), satisfying these axioms:
3014:can be built by gluing together open sets of
158:, and a projective space of dimension 2 is a
8:
4504:Aufbau der Geometrie aus dem Spiegelsbegriff
2177:
2120:
4244:Patching projective spaces together yields
3773:; in this case the meaning of the class of
3490:
3388:and so on. If the full space has dimension
3006:, by gluing together smaller pieces called
2969:with respect to the homogenizing variable.
2923:that allows computing the genus of a plane
2903:asserts that the intersection of two plane
778:, since all projective spaces of dimension
4898:
4884:
4876:
3666:, with 15 points, 35 lines and 15 planes.
3116:are distinct points and the lines through
1863:fundamental theorem of projective geometry
1705:. This means that, compared to elementary
1662:of the elements of the canonical basis of
333:It follows that the lines passing through
3632:1, 1, 1, 1, 0, 1, 1, 4, 0, ... (sequence
3336:
3320:
3304:
3292:
3058:(the set of points), together with a set
2867:was the study of common zeros of sets of
2779:
2771:
2688:
2678:
2672:
2650:
2640:
2633:
2632:
2615:
2605:
2599:
2577:
2561:
2540:
2535:
2529:
2498:
2473:
2448:
2429:
2406:
2384:
2378:
2377:
2362:
2345:
2328:
2315:
2310:
2305:
2303:
2250:
2240:
2229:
2208:
2206:
2165:
2149:
2130:
2111:
2105:
2005:{\displaystyle \pi :S\to \mathbf {P} (V)}
1988:
1974:
1246:. This projective subspace is called the
1043:
1029:
4795:An Introduction to Invariants and Moduli
4780:, esp. chapters I.2, I.7, II.5, and II.7
4278:; these are themselves special cases of
3522:-dimensional projective space over some
2737:, it results that projective spaces are
1721:") to Euclidean points, and vice versa.
1277:is a spanning set of a projective space
4423:
4342:
3294:
3207:
3030:Algebraic geometry of projective spaces
2792:{\displaystyle x\mapsto {\frac {1}{x}}}
2071:As soon as a basis has been chosen for
1462:vectors, any frame can be rewritten as
1297:, such an independent spanning set has
1075:to its equivalence class, which is the
831:has been chosen, and, in particular if
445:and the (projective) line at infinity.
4744:Euclidean and non-Euclidean geometries
4578:Linear Algebra and Projective Geometry
4465:
4453:
4159:, not just the locally free ones. See
3380:. Subspaces of dimension 0 are called
3052:can be defined axiomatically as a set
2183:{\displaystyle U_{i}=\{,x_{i}\neq 0\}}
1060:{\displaystyle p:V\to \mathbf {P} (V)}
456:may be identified with its associated
4541:
4166:, Chap. II, par. 4 for more details.
3652:The smallest projective plane is the
3143:Any line has at least 3 points on it.
2808:Real projective spaces have a simple
2277:has at least one nonzero coordinate.
1892:by the addition of new points called
1422:is a projective frame if and only if
1097:is a union of lines. It follows that
165:Projective spaces are widely used in
47:originated from the visual effect of
7:
4489:
3966:sheaves generated by global sections
3254:of the projective space is a subset
1779:Italian school of algebraic geometry
473:So, a projective space of dimension
51:, where parallel lines seem to meet
4654:, New York: John Wiley & Sons,
3537:Finite projective spaces and planes
3128:meet, then so do the lines through
2840:-cell with the quotient projection
1815:of the projective transformations).
806:The elements of a projective space
4478:Beutelspacher & Rosenbaum 1998
4386:Beutelspacher & Rosenbaum 1998
3470:(those that are isomorphic with a
3384:, those of dimension 1 are called
3272:(that is, completely contained in
1713:, for a given dimension, and that
105:, a projective space of dimension
14:
4783:Hilbert, D. and Cohn-Vossen, S.;
4432:Foundations of Translation Planes
1378:is the canonical projection from
261:, and the image is formed on the
4964:
4871:Projective Planes of Small Order
4456:, chapter 4.4. Projective bases.
3025:projective space as a manifold.
2209:
1989:
1825:This section is an excerpt from
1809:projective differential geometry
1685:This section is an excerpt from
1044:
300:to the intersection of the line
4820:Projective geometry. Vols. 1, 2
4669:Coxeter, Harold Scott MacDonald
4644:Coxeter, Harold Scott MacDonald
3876:with the underlying field
3394:then any subspace of dimension
3242:The three diagonal points of a
2974:morphism of algebraic varieties
1374:-dimensional vector space, and
1020:is a vector space over a field
565:under the equivalence relation
423:. By identifying each point of
239:, which may be considered as a
2881:fundamental theorem of algebra
2776:
2583:
2554:
2504:
2422:
2419:
2412:
2355:
2338:
2219:
2213:
2155:
2123:
1999:
1993:
1985:
1908:and allows using the tools of
1789:resulting in the study of the
1532:with a common nonzero factor.
1083:with the zero vector removed.
1054:
1048:
1040:
762:projective space of dimension
579:if there is a nonzero element
373:. This suggests to define the
308:does not belong to the plane (
1:
4818:; Young, John Wesley (1965),
4717:, Band 44, Berlin, New York:
2946:are common zeros of a set of
2871:. These common zeros, called
2068:can be provided, as follows.
1016:be a projective space, where
915:Quaternionic projective space
477:can be defined as the set of
109:is defined as the set of the
96:axioms of projective geometry
4786:Geometry and the imagination
4522:Algebraic Theory of Lattices
4230:Generalizing to associative
4220:parametrize objects such as
3830:induce the same map between
3562:is a projective space where
2744:For example, in the case of
1969:, and consider the function
1281:, then there is a subset of
1162:are two different points of
4565:Encyclopedia of Mathematics
4309:Projective line over a ring
4236:projective line over a ring
3765:is not injective, it has a
2086:may be identified with its
1307:Contrarily to the cases of
1071:that maps a nonzero vector
663:is finite dimensional, the
491:This set can be the set of
452:with a distinguished point
200:, and more specifically in
5211:
4799:Cambridge University Press
4619:Cambridge University Press
4276:weighted projective spaces
3696:between two vector spaces
3540:
3027:
2953:Any affine variety can be
2883:asserts that a univariate
1942:
1824:
1703:projective transformations
1684:
1322:
1258:is a spanning set for it.
790:vector space of dimension
397:is a line passing through
127:. Equivalently, it is the
5166:
4962:
4831:(Reprint of 1910 edition)
4789:, 2nd ed. Chelsea (1999).
4746:, 2nd ed. Freeman (1980).
4558:Afanas'ev, V.V. (2001) ,
4331:projective representation
4326:projective transformation
3748:is a non-zero element of
3708:over the same field
3491:Veblen & Young (1965)
3074:Each two distinct points
1965:in a normed vector space
1820:Projective transformation
1752:can be said to meet in a
1715:geometric transformations
685:In the common case where
511:(in a space of dimension
507:in a sphere of dimension
355:, and those contained in
59:, or, more generally, an
4650:Introduction to Geometry
4576:Baer, Reinhold (2005) ,
4376:also referred to as the
4246:projective space bundles
4054:projective linear groups
3086:are in exactly one line.
2933:complex projective plane
2869:multivariate polynomials
1946:A projective space is a
1928:Pappus's hexagon theorem
1769:, the coordinates used (
1767:complex projective space
1642:of the projective space
1446:and the coefficients of
1267:projectively independent
875:complex projective space
635:. This is the case when
603:topological vector space
206:non-orientable manifolds
4793:Mukai, Shigeru (2003),
4594:Berger, Marcel (2009),
4253:Severi–Brauer varieties
3964:). Using the notion of
3560:finite projective space
3484:non-Desarguesian planes
3414:subdirectly irreducible
3201:non-Desarguesian planes
2948:homogeneous polynomials
2944:homogeneous coordinates
2088:homogeneous coordinates
1813:differential invariants
1771:homogeneous coordinates
1611:are the coordinates of
1541:homogeneous coordinates
695:, the projective space
503:as the set of pairs of
245:three dimensional space
4705:Dembowski, P. (1968),
4380:and mistakenly as the
4069:Möbius transformations
3974:. The latter form the
3613:is usually written as
3555:
3355:
3000:Alexander Grothendieck
2907:of respective degrees
2885:square-free polynomial
2855:as the attaching map.
2793:
2722:
2708:
2518:
2260:
2245:
2184:
2006:
1950:, as endowed with the
1906:projective coordinates
1734:affine transformations
1658:consists of images by
1537:projective coordinates
1061:
930:is sometimes used for
843:projective coordinates
224:
36:
4502:Bachmann, F. (1959),
4304:Grassmannian manifold
4216:Even more generally,
4197:sequence of subspaces
4186:Grassmannian manifold
4147:quasi-coherent sheaf
4075:Dual projective space
3550:
3504:is isomorphic with a
3356:
2823:can be obtained from
2794:
2725:These charts form an
2720:
2709:
2519:
2266:since every point of
2261:
2225:
2190:is an open subset of
2185:
2056:to a neighborhood of
2052:, the restriction of
2012:that maps a point of
2007:
1857:to lines, and thus a
1726:transformation matrix
1348:points such that any
1062:
871:real projective space
786:to it (because every
605:, the quotient space
294:Cartesian coordinates
220:Projective plane and
219:
25:graphical perspective
22:
5082:Dimensions by number
4757:, Berlin, New York:
3574:Wedderburn's theorem
3495:Veblen–Young theorem
3291:
3246:are never collinear.
3197:theorem of Desargues
3193:Veblen–Young theorem
3167:consisting of a set
2961:, which consists of
2921:genus–degree formula
2915:consists of exactly
2804:CW complex structure
2770:
2528:
2302:
2205:
2104:
2035:, it follows that:
1973:
1805:projective varieties
1740:, the concept of an
1183:linearly independent
1028:
912:; see, for example,
817:are commonly called
678:is the dimension of
259:center of projection
254:Pinhole camera model
140:equivalence relation
5195:Space (mathematics)
5190:Projective geometry
4673:Projective geometry
4598:, Springer-Verlag,
4320:Projective geometry
4314:Space (mathematics)
4257:algebraic varieties
4112:-dimensional, then
3797:Birational geometry
3647:Bruck–Ryser theorem
3480:Desargues's theorem
3468:Desarguesian planes
3450:is equivalent to a
3418:compactly generated
3244:complete quadrangle
3150:incidence structure
2980:(that is, it is an
2873:algebraic varieties
2548:
1932:Desargues's theorem
1835:projective geometry
1746:perspective drawing
1699:projective geometry
1687:Projective geometry
1680:Projective geometry
1185:. It follows that:
1124:projective subspace
877:, respectively. If
620:, endowed with the
556:equivalence classes
493:equivalence classes
321:and is parallel to
175:projective geometry
43:, the concept of a
5011:Degrees of freedom
4914:Dimensional spaces
4847:"Projective Space"
4844:Weisstein, Eric W.
4755:Algebraic Geometry
4560:"projective space"
4518:Robert P. Dilworth
4516:Peter Crawley and
4378:Veblen–Young axiom
4259:over a field
4213:other subvarieties
4184:is generalized to
3784:is problematic if
3556:
3351:
3181:incidence relation
3041:synthetic geometry
3035:Synthetic geometry
2959:points at infinity
2940:projective variety
2865:algebraic geometry
2859:Algebraic geometry
2789:
2739:analytic manifolds
2735:analytic functions
2723:
2704:
2531:
2514:
2512:
2256:
2180:
2002:
1918:synthetic geometry
1914:incidence geometry
1894:points at infinity
1878:Euclidean geometry
1795:synthetic geometry
1787:Erlangen programme
1738:Euclidean geometry
1719:points at infinity
1707:Euclidean geometry
1220:synthetic geometry
1057:
241:central projection
225:
222:central projection
84:synthetic geometry
65:points at infinity
37:
5177:
5176:
4986:Lebesgue covering
4951:Algebraic variety
4808:978-0-521-80906-1
4768:978-0-387-90244-9
4751:Hartshorne, Robin
4742:Greenberg, M.J.;
4709:Finite geometries
4628:978-0-521-48277-6
4605:978-3-540-11658-5
4587:978-0-486-44565-6
4530:978-0-13-022269-5
4524:. Prentice-Hall.
4292:Geometric algebra
3199:and are known as
3179:of lines, and an
3173:of points, a set
3022:Proj construction
2897:zeros at infinity
2787:
2694:
2660:
2656:
2621:
2393:
2031:. As spheres are
1952:quotient topology
1948:topological space
1754:point at infinity
974:is often denoted
797:is isomorphic to
626:subspace topology
622:quotient topology
618:topological space
379:projective points
5202:
4974:Other dimensions
4968:
4936:Projective space
4900:
4893:
4886:
4877:
4861:Projective Space
4857:
4856:
4830:
4811:
4779:
4739:
4712:
4701:
4664:
4653:
4639:
4608:
4590:
4572:
4545:
4539:
4533:
4514:
4508:
4507:
4499:
4493:
4487:
4481:
4475:
4469:
4463:
4457:
4451:
4445:
4428:
4411:
4404:
4398:
4395:
4389:
4374:
4368:
4365:
4359:
4356:
4350:
4347:
4270:
4264:
4224:of a given kind.
4209:
4193:
4183:
4158:
4152:
4139:
4133:
4122:
4111:
4105:
4099:
4093:
4087:
4066:
4051:
4047:
4035:
4020:
3985:
3973:
3963:
3957:
3932:
3913:
3902:
3887:
3881:
3871:
3864:
3851:
3840:
3829:
3814:
3808:
3803:Two linear maps
3789:
3783:
3772:
3764:
3753:
3747:
3738:
3732:
3713:
3707:
3701:
3695:
3664:
3659:
3637:
3624:
3612:
3600:
3593:
3587:
3567:
3531:
3521:
3515:
3503:
3477:
3465:
3452:projective plane
3449:
3426:modular lattices
3400:
3393:
3379:
3373:
3360:
3358:
3357:
3352:
3341:
3340:
3325:
3324:
3312:
3311:
3286:
3277:
3271:
3265:
3259:
3220:Pappus's theorem
3187:
3178:
3172:
3166:
3139:
3133:
3127:
3121:
3115:
3109:
3103:
3097:
3085:
3079:
3069:
3063:
3057:
3051:
3045:projective space
3019:
2998:, introduced by
2978:Zariski topology
2918:
2914:
2910:
2905:algebraic curves
2901:Bézout's theorem
2894:
2890:
2854:
2839:
2834:by attaching an
2833:
2822:
2798:
2796:
2795:
2790:
2788:
2780:
2761:
2750:
2713:
2711:
2710:
2705:
2700:
2696:
2695:
2693:
2692:
2683:
2682:
2673:
2662:
2661:
2655:
2654:
2645:
2644:
2635:
2634:
2622:
2620:
2619:
2610:
2609:
2600:
2590:
2586:
2582:
2581:
2566:
2565:
2547:
2539:
2523:
2521:
2520:
2515:
2513:
2503:
2502:
2484:
2483:
2459:
2458:
2434:
2433:
2411:
2410:
2395:
2394:
2389:
2388:
2379:
2367:
2366:
2350:
2349:
2333:
2332:
2320:
2319:
2314:
2291:is associated a
2290:
2276:
2265:
2263:
2262:
2257:
2255:
2254:
2244:
2239:
2212:
2200:
2189:
2187:
2186:
2181:
2170:
2169:
2154:
2153:
2135:
2134:
2116:
2115:
2099:
2085:
2074:
2059:
2055:
2051:
2047:
2044:For every point
2029:antipodal points
2026:
2015:
2011:
2009:
2008:
2003:
1992:
1968:
1960:
1791:classical groups
1775:invariant theory
1759:Projective plane
1675:
1667:
1661:
1657:
1634:
1614:
1610:
1582:
1553:
1531:
1519:
1488:
1461:
1457:
1445:
1441:
1421:
1392:
1381:
1377:
1373:
1365:
1354:
1347:
1340:
1335:projective basis
1331:projective frame
1325:Projective frame
1303:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1264:
1257:
1253:
1245:
1241:
1237:
1213:
1199:
1180:
1176:
1172:
1161:
1150:
1136:
1118:
1107:
1096:
1092:
1082:
1074:
1066:
1064:
1063:
1058:
1047:
1023:
1019:
1015:
996:Related concepts
985:
973:
957:
949:
945:
929:
907:
895:projective plane
888:
882:
868:
861:
857:
840:
830:
816:
802:
796:
789:
781:
775:
769:
765:
758:is often called
757:
741:
730:
721:
705:
694:
681:
677:
662:
654:
644:
638:
634:
615:
600:
596:
586:
582:
578:
568:
564:
553:
542:projective space
539:
532:
517:
510:
505:antipodal points
502:
487:
476:
468:projectivization
455:
444:
431:
422:
412:line at infinity
409:
400:
396:
392:
384:
372:
363:
354:
345:
336:
329:
320:
316:
307:
303:
299:
291:
284:
280:
271:
263:projection plane
160:projective plane
152:antipodal points
149:
137:
126:
119:
108:
45:projective space
5210:
5209:
5205:
5204:
5203:
5201:
5200:
5199:
5180:
5179:
5178:
5173:
5162:
5141:
5077:
5015:
4969:
4960:
4926:Euclidean space
4909:
4904:
4842:
4841:
4838:
4814:
4809:
4792:
4769:
4759:Springer-Verlag
4749:
4729:
4719:Springer-Verlag
4704:
4683:
4667:
4662:
4642:
4629:
4612:
4606:
4593:
4588:
4575:
4557:
4554:
4549:
4548:
4540:
4536:
4515:
4511:
4501:
4500:
4496:
4488:
4484:
4476:
4472:
4464:
4460:
4452:
4448:
4434:, p. 506,
4429:
4425:
4420:
4415:
4414:
4405:
4401:
4396:
4392:
4375:
4371:
4366:
4362:
4357:
4353:
4348:
4344:
4339:
4298:Generalizations
4288:
4280:toric varieties
4266:
4260:
4222:elliptic curves
4205:
4200:More generally
4189:
4179:
4172:
4170:Generalizations
4165:
4154:
4148:
4135:
4128:
4113:
4107:
4101:
4100:. That is, if
4095:
4089:
4083:
4077:
4057:
4049:
4041:
4029:
4022:
3990:
3979:
3969:
3959:
3940:
3915:
3904:
3893:
3883:
3877:
3866:
3856:
3842:
3831:
3816:
3810:
3804:
3785:
3774:
3770:
3760:
3749:
3743:
3740:
3736:
3715:
3709:
3703:
3697:
3678:
3672:
3662:
3657:
3643:
3633:
3614:
3602:
3595:
3589:
3581:
3563:
3545:
3539:
3527:
3517:
3505:
3498:
3471:
3455:
3444:
3434:
3395:
3389:
3375:
3372:
3364:
3332:
3316:
3300:
3289:
3288:
3282:
3273:
3267:
3266:is a subset of
3261:
3255:
3183:
3174:
3168:
3152:
3135:
3129:
3123:
3117:
3111:
3105:
3099:
3093:
3081:
3075:
3065:
3059:
3053:
3047:
3037:
3032:
3015:
3010:, similarly as
2993:
2925:algebraic curve
2916:
2912:
2908:
2899:. For example,
2892:
2888:
2861:
2841:
2835:
2824:
2813:
2806:
2768:
2767:
2760:
2752:
2745:
2731:transition maps
2684:
2674:
2646:
2636:
2611:
2601:
2598:
2594:
2573:
2557:
2553:
2549:
2526:
2525:
2511:
2510:
2494:
2469:
2444:
2425:
2415:
2402:
2380:
2358:
2352:
2351:
2341:
2334:
2324:
2309:
2300:
2299:
2295:, which is the
2289:
2281:
2267:
2246:
2203:
2202:
2191:
2161:
2145:
2126:
2107:
2102:
2101:
2091:
2076:
2072:
2057:
2053:
2049:
2045:
2042:
2027:consist of two
2017:
2013:
1971:
1970:
1966:
1958:
1944:
1941:
1936:
1935:
1926:. Equivalently
1880:, and the term
1830:
1822:
1817:
1816:
1799:finite geometry
1711:Euclidean space
1690:
1682:
1673:
1663:
1659:
1651:
1643:
1640:canonical frame
1632:
1623:
1616:
1612:
1609:
1600:
1593:
1584:
1580:
1566:
1555:
1544:
1530:
1521:
1518:
1508:
1500:
1490:
1486:
1475:
1463:
1459:
1456:
1447:
1443:
1439:
1430:
1423:
1419:
1405:
1394:
1383:
1379:
1375:
1367:
1363:
1349:
1342:
1338:
1327:
1321:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1262:
1255:
1251:
1248:projective span
1243:
1239:
1235:
1228:
1204:
1190:
1178:
1174:
1163:
1152:
1141:
1127:
1109:
1098:
1094:
1090:
1088:linear subspace
1080:
1072:
1026:
1025:
1021:
1017:
1006:
1003:
998:
975:
967:
959:
955:
947:
939:
931:
919:
918:. The notation
905:
891:projective line
884:
878:
866:
859:
855:
832:
828:
807:
798:
791:
787:
779:
773:
772:the projective
767:
763:
751:
743:
732:
723:
715:
707:
696:
686:
679:
668:
660:
657:complex numbers
650:
640:
636:
629:
606:
598:
588:
584:
580:
570:
566:
559:
544:
537:
530:
524:
512:
508:
500:
482:
474:
453:
443:
437:
430:
424:
421:
415:
408:
402:
398:
394:
390:
387:projective line
382:
371:
365:
362:
356:
353:
347:
344:
338:
334:
328:
322:
318:
315:
309:
305:
301:
297:
286:
282:
279:
273:
269:
214:
202:manifold theory
171:affine geometry
156:projective line
147:
132:
121:
117:
106:
57:Euclidean space
29:vanishing point
17:
12:
11:
5:
5208:
5206:
5198:
5197:
5192:
5182:
5181:
5175:
5174:
5167:
5164:
5163:
5161:
5160:
5155:
5149:
5147:
5143:
5142:
5140:
5139:
5131:
5126:
5121:
5116:
5111:
5106:
5101:
5096:
5091:
5085:
5083:
5079:
5078:
5076:
5075:
5070:
5065:
5063:Cross-polytope
5060:
5055:
5050:
5048:Hyperrectangle
5045:
5040:
5035:
5029:
5027:
5017:
5016:
5014:
5013:
5008:
5003:
4998:
4993:
4988:
4983:
4977:
4975:
4971:
4970:
4963:
4961:
4959:
4958:
4953:
4948:
4943:
4938:
4933:
4928:
4923:
4917:
4915:
4911:
4910:
4905:
4903:
4902:
4895:
4888:
4880:
4874:
4873:
4868:
4858:
4837:
4836:External links
4834:
4833:
4832:
4816:Veblen, Oswald
4812:
4807:
4790:
4781:
4767:
4747:
4740:
4727:
4702:
4681:
4665:
4660:
4640:
4627:
4610:
4604:
4591:
4586:
4573:
4553:
4550:
4547:
4546:
4544:, example 3.72
4534:
4509:
4494:
4482:
4480:, pp. 6–7
4470:
4458:
4446:
4422:
4421:
4419:
4416:
4413:
4412:
4399:
4390:
4382:axiom of Pasch
4369:
4360:
4351:
4341:
4340:
4338:
4335:
4334:
4333:
4328:
4322:
4321:
4317:
4316:
4311:
4306:
4300:
4299:
4295:
4294:
4287:
4284:
4250:
4249:
4242:
4239:
4228:
4225:
4214:
4211:
4198:
4195:
4176:
4171:
4168:
4163:
4153:over a scheme
4076:
4073:
4048:.) The groups
4026:quotient group
3989:
3853:if and only if
3735:
3671:
3668:
3631:
3538:
3535:
3534:
3533:
3487:
3441:
3438:
3433:
3432:Classification
3430:
3368:
3350:
3347:
3344:
3339:
3335:
3331:
3328:
3323:
3319:
3315:
3310:
3307:
3303:
3299:
3296:
3248:
3247:
3232:characteristic
3228:
3227:
3216:
3215:
3145:
3144:
3141:
3087:
3064:of subsets of
3036:
3033:
3008:affine schemes
2992:
2989:
2976:is closed for
2860:
2857:
2812:structure, as
2805:
2802:
2786:
2783:
2778:
2775:
2756:
2729:, and, as the
2703:
2699:
2691:
2687:
2681:
2677:
2671:
2668:
2665:
2659:
2653:
2649:
2643:
2639:
2631:
2628:
2625:
2618:
2614:
2608:
2604:
2597:
2593:
2589:
2585:
2580:
2576:
2572:
2569:
2564:
2560:
2556:
2552:
2546:
2543:
2538:
2534:
2509:
2506:
2501:
2497:
2493:
2490:
2487:
2482:
2479:
2476:
2472:
2468:
2465:
2462:
2457:
2454:
2451:
2447:
2443:
2440:
2437:
2432:
2428:
2424:
2421:
2418:
2416:
2414:
2409:
2405:
2401:
2398:
2392:
2387:
2383:
2376:
2373:
2370:
2365:
2361:
2357:
2354:
2353:
2348:
2344:
2340:
2337:
2335:
2331:
2327:
2323:
2318:
2313:
2308:
2307:
2297:homeomorphisms
2285:
2253:
2249:
2243:
2238:
2235:
2232:
2228:
2224:
2221:
2218:
2215:
2211:
2179:
2176:
2173:
2168:
2164:
2160:
2157:
2152:
2148:
2144:
2141:
2138:
2133:
2129:
2125:
2122:
2119:
2114:
2110:
2037:
2033:compact spaces
2001:
1998:
1995:
1991:
1987:
1984:
1981:
1978:
1940:
1937:
1910:linear algebra
1831:
1823:
1821:
1818:
1811:(the study of
1750:parallel lines
1691:
1683:
1681:
1678:
1676:of their sum.
1668:(that is, the
1647:
1628:
1621:
1605:
1598:
1588:
1575:
1564:
1526:
1514:
1506:
1495:
1481:
1473:
1451:
1442:is a basis of
1435:
1428:
1414:
1403:
1323:Main article:
1320:
1317:
1227:
1224:
1216:
1215:
1201:
1173:, the vectors
1056:
1053:
1050:
1046:
1042:
1039:
1036:
1033:
1002:
999:
997:
994:
963:
935:
899:Riemann sphere
747:
711:
554:is the set of
523:
520:
441:
434:disjoint union
428:
419:
406:
369:
360:
351:
342:
326:
313:
277:
229:coplanar lines
213:
210:
185:; one for the
179:conic sections
103:linear algebra
73:parallel lines
15:
13:
10:
9:
6:
4:
3:
2:
5207:
5196:
5193:
5191:
5188:
5187:
5185:
5172:
5171:
5165:
5159:
5156:
5154:
5151:
5150:
5148:
5144:
5138:
5136:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5086:
5084:
5080:
5074:
5071:
5069:
5066:
5064:
5061:
5059:
5056:
5054:
5053:Demihypercube
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5030:
5028:
5026:
5022:
5018:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4978:
4976:
4972:
4967:
4957:
4954:
4952:
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4918:
4916:
4912:
4908:
4901:
4896:
4894:
4889:
4887:
4882:
4881:
4878:
4872:
4869:
4866:
4862:
4859:
4854:
4853:
4848:
4845:
4840:
4839:
4835:
4829:
4825:
4821:
4817:
4813:
4810:
4804:
4800:
4796:
4791:
4788:
4787:
4782:
4778:
4774:
4770:
4764:
4760:
4756:
4752:
4748:
4745:
4741:
4738:
4734:
4730:
4728:3-540-61786-8
4724:
4720:
4716:
4711:
4710:
4703:
4700:
4696:
4692:
4688:
4684:
4682:0-8020-2104-2
4678:
4674:
4670:
4666:
4663:
4661:0-471-18283-4
4657:
4652:
4651:
4645:
4641:
4638:
4634:
4630:
4624:
4620:
4616:
4611:
4607:
4601:
4597:
4592:
4589:
4583:
4579:
4574:
4571:
4567:
4566:
4561:
4556:
4555:
4551:
4543:
4538:
4535:
4531:
4527:
4523:
4519:
4513:
4510:
4505:
4498:
4495:
4491:
4486:
4483:
4479:
4474:
4471:
4467:
4462:
4459:
4455:
4450:
4447:
4444:
4443:0-8247-0609-9
4440:
4437:
4436:Marcel Dekker
4433:
4427:
4424:
4417:
4409:
4403:
4400:
4394:
4391:
4387:
4383:
4379:
4373:
4370:
4364:
4361:
4355:
4352:
4346:
4343:
4336:
4332:
4329:
4327:
4324:
4323:
4319:
4318:
4315:
4312:
4310:
4307:
4305:
4302:
4301:
4297:
4296:
4293:
4290:
4289:
4285:
4283:
4281:
4277:
4272:
4269:
4263:
4258:
4254:
4247:
4243:
4240:
4237:
4233:
4229:
4226:
4223:
4219:
4218:moduli spaces
4215:
4212:
4208:
4203:
4202:flag manifold
4199:
4196:
4192:
4187:
4182:
4177:
4174:
4173:
4169:
4167:
4162:
4157:
4151:
4146:
4141:
4138:
4131:
4126:
4120:
4116:
4110:
4104:
4098:
4092:
4086:
4082:
4074:
4072:
4070:
4064:
4060:
4055:
4045:
4039:
4033:
4027:
4018:
4014:
4010:
4006:
4002:
3998:
3994:
3988:
3986:
3983:
3977:
3972:
3967:
3962:
3955:
3951:
3947:
3943:
3939:
3938:automorphisms
3934:
3930:
3926:
3922:
3918:
3911:
3907:
3900:
3896:
3891:
3886:
3882:, the set of
3880:
3875:
3869:
3863:
3859:
3854:
3849:
3845:
3838:
3834:
3827:
3823:
3819:
3813:
3807:
3801:
3799:
3798:
3793:
3788:
3781:
3777:
3768:
3763:
3758:
3752:
3746:
3734:
3730:
3726:
3722:
3718:
3712:
3706:
3700:
3693:
3689:
3685:
3681:
3677:
3669:
3667:
3665:
3655:
3650:
3648:
3641:
3636:
3630:
3626:
3622:
3618:
3610:
3606:
3598:
3592:
3585:
3579:
3575:
3571:
3566:
3561:
3554:
3549:
3544:
3536:
3530:
3525:
3524:division ring
3520:
3513:
3509:
3501:
3496:
3492:
3488:
3485:
3481:
3475:
3469:
3463:
3459:
3453:
3447:
3442:
3439:
3436:
3435:
3431:
3429:
3427:
3423:
3419:
3415:
3411:
3406:
3404:
3398:
3392:
3387:
3383:
3378:
3371:
3367:
3361:
3348:
3345:
3342:
3337:
3333:
3329:
3326:
3321:
3317:
3313:
3308:
3305:
3301:
3297:
3285:
3279:
3276:
3270:
3264:
3258:
3253:
3245:
3241:
3240:
3239:
3237:
3233:
3225:
3224:
3223:
3222:as an axiom;
3221:
3213:
3212:
3211:
3209:
3208:Coxeter (1969
3204:
3202:
3198:
3194:
3189:
3186:
3182:
3177:
3171:
3164:
3160:
3156:
3151:
3142:
3138:
3132:
3126:
3120:
3114:
3108:
3102:
3096:
3092:'s axiom: If
3091:
3088:
3084:
3078:
3073:
3072:
3071:
3068:
3062:
3056:
3050:
3046:
3042:
3034:
3031:
3026:
3023:
3018:
3013:
3009:
3005:
3001:
2997:
2996:Scheme theory
2991:Scheme theory
2990:
2988:
2985:
2983:
2982:algebraic set
2979:
2975:
2970:
2968:
2964:
2960:
2956:
2951:
2949:
2945:
2941:
2936:
2934:
2930:
2929:singularities
2926:
2922:
2906:
2902:
2898:
2886:
2882:
2878:
2875:belong to an
2874:
2870:
2866:
2858:
2856:
2852:
2848:
2844:
2838:
2831:
2827:
2820:
2816:
2811:
2803:
2801:
2784:
2781:
2773:
2765:
2759:
2755:
2748:
2742:
2740:
2736:
2732:
2728:
2719:
2715:
2701:
2697:
2689:
2685:
2679:
2675:
2669:
2666:
2663:
2657:
2651:
2647:
2641:
2637:
2629:
2626:
2623:
2616:
2612:
2606:
2602:
2595:
2591:
2587:
2578:
2574:
2570:
2567:
2562:
2558:
2550:
2544:
2541:
2536:
2532:
2507:
2499:
2495:
2491:
2488:
2485:
2480:
2477:
2474:
2470:
2466:
2463:
2460:
2455:
2452:
2449:
2445:
2441:
2438:
2435:
2430:
2426:
2417:
2407:
2403:
2399:
2396:
2390:
2385:
2381:
2374:
2371:
2368:
2363:
2359:
2346:
2342:
2336:
2329:
2325:
2321:
2316:
2311:
2298:
2294:
2288:
2284:
2278:
2274:
2270:
2251:
2247:
2241:
2236:
2233:
2230:
2226:
2222:
2216:
2198:
2194:
2174:
2171:
2166:
2162:
2158:
2150:
2146:
2142:
2139:
2136:
2131:
2127:
2117:
2112:
2108:
2098:
2094:
2089:
2083:
2079:
2069:
2067:
2063:
2062:homeomorphism
2040:
2036:
2034:
2030:
2024:
2020:
1996:
1982:
1979:
1976:
1964:
1955:
1953:
1949:
1938:
1933:
1929:
1925:
1921:
1919:
1915:
1911:
1907:
1903:
1900:over a given
1899:
1895:
1891:
1890:affine spaces
1887:
1883:
1879:
1875:
1871:
1866:
1864:
1860:
1856:
1852:
1848:
1847:vector spaces
1844:
1840:
1836:
1828:
1819:
1814:
1810:
1806:
1802:
1800:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1768:
1763:
1761:
1760:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1722:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1688:
1679:
1677:
1671:
1666:
1655:
1650:
1646:
1641:
1636:
1631:
1627:
1620:
1615:on the basis
1608:
1604:
1597:
1591:
1587:
1578:
1574:
1570:
1563:
1559:
1551:
1547:
1542:
1538:
1533:
1529:
1524:
1517:
1512:
1504:
1498:
1493:
1484:
1479:
1471:
1467:
1454:
1450:
1438:
1434:
1427:
1417:
1413:
1409:
1402:
1398:
1390:
1386:
1371:
1360:
1358:
1352:
1345:
1336:
1332:
1326:
1318:
1316:
1314:
1313:affine spaces
1310:
1309:vector spaces
1305:
1301:
1268:
1265:of points is
1259:
1249:
1233:
1225:
1223:
1221:
1211:
1207:
1202:
1197:
1193:
1188:
1187:
1186:
1184:
1170:
1166:
1159:
1155:
1148:
1144:
1138:
1134:
1130:
1125:
1120:
1116:
1112:
1105:
1101:
1089:
1084:
1078:
1070:
1069:canonical map
1051:
1037:
1034:
1031:
1013:
1009:
1000:
995:
993:
991:
990:
983:
979:
971:
966:
962:
953:
943:
938:
934:
927:
923:
917:
916:
911:
910:division ring
902:
900:
896:
892:
887:
881:
876:
872:
863:
853:
848:
844:
839:
835:
826:
822:
821:
814:
810:
804:
801:
794:
785:
777:
761:
755:
750:
746:
739:
735:
729:
726:
719:
714:
710:
703:
699:
693:
689:
683:
675:
671:
666:
658:
653:
649:or the field
648:
643:
639:is the field
632:
627:
623:
619:
613:
609:
604:
595:
591:
577:
573:
562:
557:
551:
547:
543:
536:
529:
521:
519:
515:
506:
497:
494:
489:
485:
480:
471:
469:
465:
464:
459:
451:
446:
440:
435:
427:
418:
413:
405:
388:
380:
377:(called here
376:
368:
359:
350:
341:
331:
325:
312:
295:
289:
276:
266:
264:
260:
256:
255:
250:
246:
242:
238:
234:
230:
223:
218:
211:
209:
207:
203:
199:
194:
192:
188:
184:
180:
176:
172:
168:
163:
161:
157:
153:
145:
141:
135:
130:
124:
120:of dimension
116:
112:
104:
99:
97:
93:
89:
85:
81:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
34:
30:
26:
21:
5168:
5134:
5073:Hyperpyramid
5038:Hypersurface
4935:
4931:Affine space
4921:Vector space
4850:
4819:
4794:
4784:
4754:
4743:
4708:
4672:
4649:
4614:
4595:
4577:
4563:
4537:
4521:
4512:
4503:
4497:
4492:, p. 71
4485:
4473:
4461:
4449:
4431:
4426:
4402:
4393:
4377:
4372:
4363:
4354:
4345:
4273:
4267:
4261:
4251:
4206:
4190:
4180:
4155:
4149:
4144:
4142:
4136:
4129:
4125:Grassmannian
4118:
4114:
4108:
4102:
4096:
4090:
4088:rather than
4084:
4078:
4062:
4058:
4043:
4031:
4023:
4016:
4012:
4008:
4004:
4000:
3996:
3992:
3981:
3970:
3960:
3953:
3949:
3945:
3941:
3935:
3928:
3924:
3920:
3916:
3909:
3905:
3898:
3894:
3884:
3878:
3874:identity map
3867:
3861:
3857:
3847:
3843:
3836:
3832:
3825:
3821:
3817:
3811:
3805:
3802:
3795:
3792:rational map
3786:
3779:
3775:
3769:larger than
3761:
3757:well-defined
3750:
3744:
3741:
3728:
3724:
3720:
3716:
3710:
3704:
3698:
3691:
3687:
3683:
3679:
3673:
3651:
3644:
3627:
3620:
3616:
3608:
3604:
3596:
3590:
3583:
3578:finite field
3569:
3564:
3559:
3557:
3528:
3518:
3511:
3507:
3499:
3473:
3461:
3457:
3445:
3422:complemented
3407:
3401:is called a
3396:
3390:
3385:
3381:
3376:
3369:
3365:
3362:
3283:
3280:
3274:
3268:
3262:
3256:
3251:
3249:
3236:Fano's axiom
3235:
3229:
3217:
3205:
3190:
3184:
3175:
3169:
3162:
3158:
3154:
3146:
3136:
3130:
3124:
3118:
3112:
3106:
3100:
3094:
3082:
3076:
3066:
3060:
3054:
3048:
3044:
3038:
3016:
2994:
2986:
2971:
2963:homogenizing
2954:
2952:
2937:
2932:
2896:
2891:has exactly
2877:affine space
2863:Originally,
2862:
2850:
2846:
2842:
2836:
2829:
2825:
2818:
2814:
2807:
2757:
2753:
2746:
2743:
2724:
2286:
2282:
2279:
2272:
2268:
2196:
2192:
2096:
2092:
2081:
2077:
2070:
2043:
2038:
2022:
2018:
1956:
1945:
1898:vector space
1881:
1867:
1859:collineation
1832:
1764:
1757:
1730:translations
1723:
1692:
1664:
1653:
1648:
1644:
1639:
1637:
1629:
1625:
1618:
1606:
1602:
1595:
1589:
1585:
1576:
1572:
1568:
1561:
1557:
1549:
1545:
1540:
1536:
1534:
1527:
1522:
1515:
1510:
1502:
1496:
1491:
1482:
1477:
1469:
1465:
1452:
1448:
1436:
1432:
1425:
1415:
1411:
1407:
1400:
1396:
1388:
1384:
1369:
1361:
1350:
1343:
1334:
1330:
1328:
1306:
1299:
1266:
1260:
1247:
1232:intersection
1229:
1217:
1209:
1205:
1203:A subset of
1195:
1191:
1168:
1164:
1157:
1153:
1146:
1142:
1139:
1132:
1128:
1123:
1121:
1114:
1110:
1103:
1099:
1085:
1068:
1011:
1007:
1004:
987:
981:
977:
969:
964:
960:
952:finite field
941:
936:
932:
925:
921:
913:
903:
889:is called a
885:
879:
864:
846:
837:
833:
818:
812:
808:
805:
799:
792:
771:
759:
753:
748:
744:
737:
733:
727:
724:
722:(as well as
717:
712:
708:
701:
697:
691:
687:
684:
673:
669:
664:
651:
647:real numbers
641:
630:
611:
607:
593:
589:
575:
571:
560:
549:
545:
541:
528:vector space
525:
513:
498:
490:
483:
479:vector lines
472:
461:
458:vector space
450:affine space
447:
438:
425:
416:
411:
403:
386:
378:
374:
366:
357:
348:
339:
332:
323:
310:
287:
274:
267:
262:
258:
252:
226:
195:
164:
133:
129:quotient set
122:
115:vector space
111:vector lines
100:
91:
87:
77:
61:affine space
52:
44:
38:
5158:Codimension
5137:-dimensions
5058:Hypersphere
4941:Free module
4468:, chapter 4
4466:Berger 2009
4454:Berger 2009
4227:other rings
4067:are called
4052:are called
3794:, see also
3676:linear maps
3493:proved the
3363:A subspace
2524:such that
1963:unit sphere
1916:, see also
1874:projections
1870:perspective
1843:isomorphism
1783:Felix Klein
1695:mathematics
1554:on a frame
1543:of a point
1285:that spans
1079:containing
1077:vector line
845:of a point
706:is denoted
682:minus one.
569:defined by
237:perspective
144:unit sphere
53:at infinity
49:perspective
41:mathematics
5184:Categories
5153:Hyperspace
5033:Hyperplane
4865:PlanetMath
4596:Geometry I
4552:References
4542:Mukai 2003
4408:Fano plane
4134:planes in
4081:dual space
3914:is simply
3767:null space
3674:Injective
3654:Fano plane
3553:Fano plane
3403:hyperplane
3028:See also:
2967:saturating
2887:of degree
2810:CW complex
2100:, the set
2095:= 0, ...,
1882:homography
1853:that maps
1839:homography
1827:Homography
1489:such that
1476:), ..., p(
1357:hyperplane
1304:elements.
958:elements,
784:isomorphic
587:such that
522:Definition
212:Motivation
183:hyperbolas
5043:Hypercube
5021:Polytopes
5001:Minkowski
4996:Hausdorff
4991:Inductive
4956:Spacetime
4907:Dimension
4852:MathWorld
4580:, Dover,
4570:EMS Press
4532:, p. 109.
4490:Baer 2005
4418:Citations
4175:dimension
3999:)) = Aut(
3890:morphisms
3865:for some
3670:Morphisms
3330:⋯
3327:⊂
3314:⊂
3306:−
3295:∅
3012:manifolds
2955:completed
2927:from its
2777:↦
2764:real line
2667:…
2658:^
2627:…
2571:⋯
2542:−
2533:φ
2489:⋯
2453:−
2439:⋯
2420:↦
2400:…
2391:^
2372:…
2339:→
2312:φ
2227:⋃
2172:≠
2140:⋯
1986:→
1977:π
1886:Euclidean
1851:bijection
1041:→
665:dimension
80:isotropic
69:direction
5170:Category
5146:See also
4946:Manifold
4753:(1977),
4671:(1969),
4646:(1974),
4520:, 1973.
4286:See also
4241:patching
3888:-linear
3663:PG(3, 2)
3658:PG(2, 2)
3478:satisfy
3252:subspace
3234:include
2280:To each
1939:Topology
1601:+ ... +
1567:), ...,
1509:+ ... +
1406:), ...,
1001:Subspace
526:Given a
233:parallel
198:topology
191:ellipses
187:parabola
167:geometry
31:(on the
5068:Simplex
5006:Fractal
4828:0179666
4777:0463157
4737:0233275
4691:0346652
4637:1629468
4123:is the
4015:=: PGL(
3638:in the
3635:A001231
3410:lattice
3020:. The
3004:schemes
2931:in the
2201:, and
1961:be the
1624:, ...,
1431:, ...,
1393:, then
1067:be the
1024:, and
989:PG(3,2)
823:. If a
655:of the
645:of the
624:of the
533:over a
292:, when
247:onto a
243:of the
150:in two
138:by the
33:horizon
5025:shapes
4826:
4805:
4775:
4765:
4735:
4725:
4699:977732
4697:
4689:
4679:
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4602:
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4528:
4441:
4038:center
3759:. (If
3742:where
3526:
3516:, the
3472:PG(2,
3466:. The
3382:points
3090:Veblen
2090:. For
1841:is an
1807:) and
1781:, and
1777:, the
1670:tuples
1366:is an
1261:A set
1254:, and
1230:Every
1086:Every
841:, the
820:points
776:-space
540:, the
448:As an
375:points
101:Using
5129:Eight
5124:Seven
5104:Three
4981:Krull
4337:Notes
4232:rings
4145:every
4007:= GL(
3976:group
3892:from
3733:via:
3607:, GF(
3570:order
3386:lines
2938:So a
2727:atlas
2293:chart
2066:atlas
2060:is a
1924:field
1902:field
1855:lines
1742:angle
1732:(the
1583:with
1319:Frame
1273:. If
1200:, and
986:(see
954:with
950:is a
946:. If
908:is a
893:or a
873:or a
852:up to
825:basis
770:, or
766:over
659:. If
633:\ {0}
616:is a
601:is a
597:. If
563:\ {0}
535:field
460:(see
251:(see
249:plane
136:\ {0}
88:point
63:with
5114:Five
5109:Four
5089:Zero
5023:and
4803:ISBN
4763:ISBN
4723:ISBN
4695:OCLC
4677:ISBN
4656:ISBN
4623:ISBN
4600:ISBN
4582:ISBN
4526:ISBN
4439:ISBN
4255:are
4042:Aut(
4024:the
4011:) /
4003:) /
3991:Aut(
3948:) →
3936:The
3841:and
3809:and
3723:) →
3702:and
3640:OEIS
3551:The
3134:and
3122:and
3080:and
3043:, a
2911:and
2733:are
1957:Let
1930:and
1888:and
1872:and
1837:, a
1728:and
1638:The
1535:The
1372:+ 1)
1311:and
1226:Span
1181:are
1177:and
1151:and
1005:Let
782:are
385:. A
92:line
90:and
5119:Six
5099:Two
5094:One
4863:at
4161:EGA
4132:− 1
4127:of
4106:is
4050:PGL
4040:of
4030:GL(
4028:of
3980:GL(
3903:to
3870:≠ 0
3815:in
3800:.)
3771:{0}
3615:PG(
3603:PG(
3599:+ 1
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3506:PG(
3502:≥ 3
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3448:= 2
3399:− 1
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2749:= 1
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1876:in
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1785:'s
1693:In
1539:or
1382:to
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1353:+ 1
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1093:of
992:).
976:PG(
920:PG(
865:If
858:in
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803:).
795:+ 1
760:the
731:or
667:of
628:of
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558:of
518:).
516:+ 1
486:+ 1
436:of
414:of
290:= 1
196:In
146:of
131:of
125:+ 1
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5186::
4849:.
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4773:MR
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4282:.
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3927:,
3862:λS
3860:=
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3737:→
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3649:.
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3611:))
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980:,
924:,
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836:=
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594:λy
592:=
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5135:n
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327:2
324:P
319:O
314:1
311:P
306:P
298:P
288:z
283:O
278:2
275:P
270:O
148:V
134:V
123:n
118:V
107:n
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