36:
167:, even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in
334:
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275:
in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. This rudimentary idea was refined and abstracted, first in a
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279:
context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.
984:
1406:
920:
301:
The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of
791:
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The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example:
1685:
290:
is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The
1153:. In the case of orthogonal projections, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well.
1214:
and can be considered a generalization of the projection map. The image of a retraction is called a retract of the original space. A retraction which is
1573:
1411:
226:
to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see
100:
53:
72:
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1547:
1520:
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includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the
79:
1695:
1557:
1530:
1503:
1352:
1549:
The
Relational Database Dictionary: A Comprehensive Glossary of Relational Terms and Concepts, with Illustrative Examples
86:
17:
1786:
119:
1605:"The Arabic version of Ptolemy's Planisphere or Flattening the Surface of the Sphere: Text, Translation, Commentary"
817:
68:
807:
57:
1735:
1581:
159:(or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The
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1365:
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to each factor. This projection will take many forms in different categories. The projection from the
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760:
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46:
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152:
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1157:
1146:
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is a projection. This type of projection naturally generalizes to any number of dimensions
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8:
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onto a plane in it, like the shadow example. The two main projections of this kind are:
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635:
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1340:
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314:
1241:
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1223:
811:
208:
172:
1309:
1244:, the above notion of Cartesian product of sets can be generalized to arbitrary
1339:. Graduate Texts in Mathematics. Vol. 218 (Second ed.). p. 606.
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803:
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is a set of attribute names. The result of such projection is defined as the
291:
1736:"Product of a family of objects in a category - Encyclopedia of Mathematics"
1711:
1660:
1201:
1161:
1049:, which otherwise has no projection on the plane. A common instance is the
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1288:
283:
141:
133:
793:, and the evaluation map is a projection map from the Cartesian product.
337:
The commutativity of this diagram is the universality of the projection
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688:
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230:
for a formalization of this terminology). The projection of the point
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with itself. A projection may also refer to a mapping which has a
1473:{\displaystyle \pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}}
1014:
1002:
156:
1204:
on its image. This satisfies a similar idempotency condition
927:
1021:
C diametrically opposite the point of tangency. Any point
333:
255:
is the intersection of the plane with the line parallel to
267:
for an accurate definition, generalized to any dimension.
1637:"Stereographic projection - Encyclopedia of Mathematics"
1103:
operator. For example, the mapping that takes a point
763:
1414:
1368:
940:
882:
820:
381:. Both notions are strongly related, as follows. Let
27:
Mapping equal to its square under mapping composition
60:. Unsourced material may be challenged and removed.
1472:
1408:are topological spaces. Show that each projection
1400:
1033:intersecting the plane at the projected point for
1001:, projection of a sphere upon a plane was used by
978:
914:
868:
785:
171:to denote the projection of the three-dimensional
1778:
373:, which means that a projection is equal to its
1061:is frequently projected onto a plane using the
163:to a subspace of a projection is also called a
1602:
1310:"Direct product - Encyclopedia of Mathematics"
1053:where the compactification corresponds to the
869:{\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)}
757:can be identified with the Cartesian product
1291:and even surjective, they do not have to be.
973:
941:
1491:
1287:, etc. Although these morphisms are often
1712:"Retraction - Encyclopedia of Mathematics"
1661:"Projection - Encyclopedia of Mathematics"
1690:. Springer Science & Business Media.
1525:. Springer Science & Business Media.
1498:. Springer Science & Business Media.
1492:Brown, Arlen; Pearcy, Carl (1994-12-16).
1076:that remains unchanged if applied twice:
120:Learn how and when to remove this message
1603:Sidoli, Nathan; Berggren, J. L. (2007).
1037:. The correspondence makes the sphere a
332:
979:{\displaystyle \{a_{1},\ldots ,a_{n}\}}
717:A mapping that takes an element to its
294:are also at the basis of the theory of
14:
1779:
1518:
1145:for the codomain of the mapping. See
1683:
488:has a right inverse). Conversely, if
1545:
732:The evaluation map sends a function
385:be an idempotent mapping from a set
369:) is a projection if the mapping is
181:projection from a point onto a plane
58:adding citations to reliable sources
29:
1401:{\displaystyle X_{1},\ldots ,X_{k}}
1332:
915:{\displaystyle a_{1},\ldots ,a_{n}}
239:projection parallel to a direction
203:onto a plane that does not contain
24:
1756:
1226:to refer to any split epimorphism.
822:
25:
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1119:in three dimensions to the point
195:, then the projection of a point
1336:Introduction to Smooth Manifolds
1275:(which is always surjective and
34:
1728:
1704:
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353:Generally, a mapping where the
45:needs additional citations for
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1629:
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1539:
1522:Relational Database Technology
1512:
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1218:to the identity is known as a
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265:Affine space ยง Projection
13:
1:
1295:
1222:. This term is also used in
1045:is included to correspond to
786:{\textstyle \prod _{i\in X}Y}
585:An operation typified by the
328:
69:"Projection" mathematics
1684:Roman, Steven (2007-09-20).
7:
1773:Historical Math Collection.
1519:Alagic, Suad (2012-12-06).
1495:An Introduction to Analysis
1151:Projection (linear algebra)
214:with the plane. The points
207:is the intersection of the
10:
1803:
1552:. "O'Reilly Media, Inc.".
1546:Date, C. J. (2006-08-28).
1039:one-point compactification
934:are restricted to the set
926:that is obtained when all
751:. The space of functions
1767:A Treatise on Projections
1345:10.1007/978-1-4419-9982-5
307:projective transformation
1787:Mathematical terminology
1011:stereographic projection
602:, that takes an element
1687:Advanced Linear Algebra
1200:which restricts to the
1009:. The method is called
251:: The image of a point
191:is a point, called the
1771:University of Michigan
1740:encyclopediaofmath.org
1716:encyclopediaofmath.org
1665:encyclopediaofmath.org
1641:encyclopediaofmath.org
1474:
1402:
1314:encyclopediaofmath.org
1252:of some objects has a
1220:deformation retraction
1025:on the sphere besides
980:
916:
870:
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367:mathematical structure
350:
234:itself is not defined.
153:mathematical structure
1475:
1403:
1333:Lee, John M. (2012).
1233:(or resolute) of one
1158:differential topology
1147:Orthogonal projection
1099:. In other words, an
1074:linear transformation
1041:for the plane when a
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917:
871:
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691:and, when each space
430:viewed as a map from
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1578:www.cs.rochester.edu
1574:"Relational Algebra"
1412:
1366:
1254:canonical projection
938:
880:
818:
800:relational databases
761:
727:canonical projection
723:equivalence relation
492:has a right inverse
193:center of projection
54:improve this article
1135:for the domain and
1063:gnomonic projection
1057:. Alternatively, a
706:, this map is also
687:This map is always
303:projective geometry
248:parallel projection
228:Projective geometry
218:such that the line
1584:on 30 January 2004
1470:
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1273:topological spaces
1029:determines a line
1017:to a sphere and a
999:spherical geometry
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866:
783:
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422:. If we denote by
389:into itself (thus
351:
185:central projection
169:Euclidean geometry
18:Central projection
1697:978-0-387-72831-5
1559:978-1-4493-9115-7
1532:978-1-4612-4922-1
1505:978-0-387-94369-5
1354:978-1-4419-9982-5
1261:Cartesian product
1231:scalar projection
1168:and is therefore
1043:point at infinity
1013:and uses a plane
992:database-relation
764:
719:equivalence class
636:Cartesian product
325:of this article.
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16:(Redirected from
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468:), then we have
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1757:Further reading
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1172:and surjective.
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566:is idempotent.
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1360:Exercise A.32.
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1281:direct product
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1185:continuous map
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1055:Riemann sphere
1007:Planisphaerium
1005:(~150) in his
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288:map projection
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243:, onto a plane
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1051:complex plane
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736:to the value
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361:are the same
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305:. However, a
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71: โ
70:
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65:Find sources:
59:
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49:
48:
43:This article
41:
37:
32:
31:
19:
1763:Thomas Craig
1743:. Retrieved
1739:
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1719:. Retrieved
1715:
1706:
1686:
1679:
1668:. Retrieved
1664:
1655:
1644:. Retrieved
1640:
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1619:. Retrieved
1615:
1611:
1598:
1586:. Retrieved
1582:the original
1577:
1568:
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1521:
1514:
1494:
1487:
1359:
1358:
1335:
1328:
1317:. Retrieved
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1289:epimorphisms
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1210:
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1202:identity map
1196:
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1162:fiber bundle
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747:for a fixed
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90:
83:
76:
64:
52:Please help
47:verification
44:
814:written as
552:; that is,
375:composition
323:projections
296:perspective
284:cartography
161:restriction
134:mathematics
110:August 2021
1745:2021-08-11
1721:2021-08-11
1670:2021-08-11
1646:2021-08-11
1319:2021-08-11
1296:References
1246:categories
1181:retraction
1101:idempotent
1059:hemisphere
808:projection
708:continuous
689:surjective
593:, written
580:set theory
371:idempotent
329:Definition
273:projection
165:projection
155:) into a
151:(or other
142:idempotent
138:projection
80:newspapers
1621:11 August
1588:29 August
1458:→
1445:×
1442:⋯
1439:×
1417:π
1383:…
1216:homotopic
958:…
897:…
840:…
823:Π
773:∈
766:∏
484:(so that
454:(so that
444:injection
311:bijection
277:geometric
1781:Category
1362:Suppose
1257:morphism
1177:topology
704:topology
426:the map
359:codomain
345:and set
224:parallel
1765:(1882)
1612:Sciamvs
1250:product
1248:. The
1015:tangent
1003:Ptolemy
634:of the
623:, ...,
614:, ...,
496:, then
438:and by
145:mapping
94:scholar
1694:
1556:
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1351:
1285:groups
1267:, the
1235:vector
1160:, any
928:tuples
876:where
810:is a
806:, the
702:has a
654:ร โฏ ร
645:ร โฏ ร
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355:domain
263:. See
157:subset
140:is an
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1183:is a
990:is a
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309:is a
187:: If
147:of a
101:JSTOR
87:books
1692:ISBN
1623:2021
1590:2021
1554:ISBN
1527:ISBN
1500:ISBN
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1277:open
1265:sets
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1019:pole
802:and
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712:open
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666:proj
595:proj
533:โ Id
505:= Id
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442:the
365:(or
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237:The
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73:news
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