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As an example, a point is a hyperplane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is
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In projective space, a hyperplane does not divide the space into two parts; rather, it takes two hyperplanes to separate points and divide up the space. The reason for this is that the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.
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is a set of points with the property that for any two points of the set, all the points on the line determined by the two points are contained in the set. Projective geometry can be viewed as
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In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the
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Several specific types of hyperplanes are defined with properties that are well suited for particular purposes. Some of these specializations are described here.
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A hyperplane H is called a "support" hyperplane of the polyhedron P if P is contained in one of the two closed half-spaces bounded by H and
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of codimension 2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
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of codimension 1, only possibly shifted from the origin by a vector, in which case it is referred to as a
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In other kinds of ambient spaces, some properties from
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The
Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas
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between two non-parallel hyperplanes of a
Euclidean space is the angle between the corresponding
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Two intersecting planes: Two-dimensional planes are the hyperplanes in three-dimensional space.
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of a vector hyperplane). A hyperplane in a
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The difference in dimension between a subspace and its ambient space is known as its
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in n-dimensional
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The datapoint and its predicted value via a linear model is a hyperplane.
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Any hyperplane of a
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is a vector space, one distinguishes "vector hyperplanes" (which are
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Affine hyperplanes are used to define decision boundaries in many
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that fixes the hyperplane and interchanges those two half spaces.
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1130:. The product of the transformations in the two hyperplanes is a
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506:{\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b.\ }
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Projective
Geometry: From Foundations to Applications
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Victor V. Prasolov & VM Tikhomirov (1997, 2001)
87:. Two lower-dimensional examples of hyperplanes are
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1228:"Excerpt from Convex Analysis, by R.T. Rockafellar"
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1002:algorithms such as linear-combination (oblique)
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1018:In a vector space, a vector hyperplane is a
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1289:. Cambridge University Press. p. 13.
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1099:, hyperplanes are a key tool to create
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104:Most commonly, the ambient space is
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83:is one less than that of the
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409:is an arbitrary constant):
259:is a subspace of dimension
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1203:Arrangement of hyperplanes
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189:through a point which are
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1322:Applicable Geometry
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1047:projective subspace
1043:projective geometry
172:-dimensional sphere
58:mathematical spaces
1718:Euclidean geometry
1539:Degrees of freedom
1442:Dimensional spaces
1387:Weisstein, Eric W.
1368:Weisstein, Eric W.
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1371:"Hyperplane"
1340:
1336:
1321:
1307:
1285:
1267:
1249:
1243:
1234:
1222:
1188:Hypersurface
1145:
1121:
1113:
1094:
1076:
1073:Applications
1067:
1062:
1058:
1038:
1037:
1017:
997:
741:
739:connected).
737:
630:
526:inequalities
515:
352:affine space
339:
337:
329:
303:
301:
285:vector space
281:affine space
272:
271:. The space
268:
260:
256:
251:
246:
240:
224:
222:
220:is defined.
202:affine space
199:
162:
158:intersection
115:
103:
73:hypersurface
45:
39:
29:
1686:Codimension
1665:-dimensions
1586:Hypersphere
1469:Free module
1310:, page 62,
1086:convex sets
1008:perceptrons
811:dot product
348:codimension
316:half spaces
312:translation
293:coordinates
265:codimension
226:codimension
140:preserving
126:half spaces
101:on a line.
18:Hyperplanes
1712:Categories
1681:Hyperspace
1561:Hyperplane
1214:References
933:such that
522:complement
320:reflection
247:hyperplane
197:geodesic.
183:space form
130:reflection
46:hyperplane
1571:Hypercube
1549:Polytopes
1529:Minkowski
1524:Hausdorff
1519:Inductive
1484:Spacetime
1435:Dimension
1395:MathWorld
1376:MathWorld
1314:, Boston.
1166:∅
1163:≠
1157:∩
971:~
962:−
953:⋅
947:^
903:∈
862:∈
856:~
827:^
759:^
750:±
688:⋯
584:⋯
466:⋯
275:may be a
187:geodesics
150:generated
81:dimension
64:. Like a
62:dimension
1698:Category
1674:See also
1474:Manifold
1337:Geometry
1283:(1980).
1182:See also
1136:subspace
1132:rotation
1083:disjoint
1059:infinite
1020:subspace
350:1 in an
243:geometry
218:subspace
142:distance
77:subspace
42:geometry
1596:Simplex
1534:Fractal
1320:(1977)
1306:(1968)
520:of the
342:is an
283:, or a
156:is the
122:"flats"
1553:shapes
1390:"Flat"
1351:
1328:
1293:
1256:
1081:, two
1006:, and
723:
616:
501:
249:of an
195:normal
134:motion
99:points
79:whose
1657:Eight
1652:Seven
1632:Three
1509:Krull
1231:(PDF)
1053:with
1045:. A
354:. In
287:or a
146:group
92:lines
1642:Five
1637:Four
1617:Zero
1551:and
1349:ISBN
1326:ISBN
1291:ISBN
1254:ISBN
1122:The
1107:and
1024:flat
714:>
631:and
610:<
245:, a
128:. A
75:, a
70:flat
44:, a
1647:Six
1627:Two
1622:One
1095:In
1077:In
1061:or
346:of
338:An
302:If
241:In
212:or
174:or
163:In
56:to
52:in
40:In
1714::
1392:.
1373:.
1343:,
1233:.
1111:.
1092:.
1030:.
1010:.
995:.
233:.
1663:n
1427:e
1420:t
1413:v
1398:.
1379:.
1355:.
1332:.
1299:.
1237:.
1160:P
1154:H
983:0
980:=
977:)
968:b
959:x
956:(
944:n
919:1
916:+
913:n
908:R
900:x
878:1
875:+
872:n
867:R
853:b
824:n
795:1
792:+
789:n
784:R
756:n
720:.
717:b
709:n
705:x
699:n
695:a
691:+
685:+
680:2
676:x
670:2
666:a
662:+
657:1
653:x
647:1
643:a
613:b
605:n
601:x
595:n
591:a
587:+
581:+
576:2
572:x
566:2
562:a
558:+
553:1
549:x
543:1
539:a
498:.
495:b
492:=
487:n
483:x
477:n
473:a
469:+
463:+
458:2
454:x
448:2
444:a
440:+
435:1
431:x
425:1
421:a
397:b
375:i
371:a
304:V
273:V
269:V
261:n
257:V
252:n
231:1
170:n
136:(
116:n
114:(
106:n
20:)
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