544:
388:
601:. Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
239:
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695:
or dual
Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes.
980:
304:
961:
17:
604:
If flats do not intersect, and no line from the first flat is parallel to a line from the second flat, then these are
256:
defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in
807:
813:
793:
The aforementioned facts do not depend on the structure being that of
Euclidean space (namely, involving
1045:
153:
30:"Euclidean subspace" redirects here. For a subspace that contains the zero vector or a fixed origin, see
822:
192:. For example, a line in two-dimensional space can be described by a single linear equation involving
189:
261:
1050:
1055:
122:
597:
If each line from one flat is parallel to some line from another flat, then these two flats are
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125:
are points, lines, planes, and the space itself. The definition of flat excludes non-straight
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692:
608:. It is possible only if sum of their dimensions is less than dimension of the ambient space.
206:
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778:
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8:
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114:
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69:
820:
There is the distance between two flats, equal to 0 if they intersect. (See for example
794:
539:{\displaystyle x=5+2t_{1}-3t_{2},\;\;\;\;y=-4+t_{1}+2t_{2}\;\;\;\;z=5t_{1}-3t_{2}.\,\!}
130:
118:
62:
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variables describes a hyperplane, and a system of linear equations describes the
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31:
1002:
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849:
149:
651:. If two flats intersect, then the dimension of the containing flat equals to
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that is itself an affine space. Particularly, in the case the parent space is
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and can build systematic coordinates for flats in any dimension, leading to
798:
50:
1021:
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on the left-hand side and a third parallel line on the right-hand side.
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there exists the minimal flat which contains them, of dimension at most
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1007:
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80:
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There is the distance between a flat and a point. (See for example
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of those hyperplanes. Assuming the equations are consistent and
383:{\displaystyle x=2+3t,\;\;\;\;y=-1+t\;\;\;\;z={\frac {3}{2}}-4t}
393:
while the description of a plane would require two parameters:
244:
In three-dimensional space, a single linear equation involving
836:
126:
581:
549:
In general, a parameterization of a flat of dimension
294:. A line can be described by equations involving one
402:
307:
209:
175:
to distinguish it from other manifolds or varieties.
576:
997:
290:A flat can also be described by a system of linear
92:; flats one dimension lower than the parent space,
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382:
233:
904:Gallier, J. (2011). "Basics of Affine Geometry".
839:between two flats, which belongs to the interval
535:
1037:
152:, as geometric realizations of solution sets of
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731:is a point not lying on the same plane, then
683:) make the set of all flats in the Euclidean
698:However, the lattice of all flats is not a
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667:minus the dimension of the intersection.
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272:equations describes a flat of dimension
903:
14:
1038:
947:
582:Intersecting, parallel, and skew flats
137:having different notions of distance:
1017:
998:
974:Primitives for Computational Geometry
788:
763:, both representing a line. But when
675:These two operations (referred to as
121:, and the plane itself; the flats in
918:An affine subspace is also called a
27:Affine subspace of a Euclidean space
24:
906:Geometric Methods and Applications
25:
1067:
991:
590:of flats is either a flat or the
577:Operations and relations on flats
808:Distance from a point to a plane
853:(between two planes). See also
814:Distance from a point to a line
183:
178:
897:
13:
1:
931:
285:
188:A flat can be described by a
61:which inherits the notion of
950:Oriented Projective Geometry
944:, Krieger, New York, page 7.
616:For two flats of dimensions
167:, and is sometimes called a
113:(two-dimensional space) are
7:
914:10.1007/978-1-4419-9961-0_2
863:
154:systems of linear equations
10:
1072:
978:DEC SRC Research Report 36
829:Skew lines § Distance
823:Distance between two lines
190:system of linear equations
29:
797:) and are correct in any
890:
826:(in the same plane) and
801:. In a Euclidean space:
671:Properties of operations
234:{\displaystyle 3x+5y=8.}
65:from its parent space.
611:
123:three-dimensional space
948:Stolfi, Jorge (1991),
908:. New York: Springer.
540:
384:
235:
49:, i.e. a subset of an
938:Heinrich Guggenheimer
841:[0, π/2]
693:Grassmann coordinates
541:
385:
236:
972:Ph.D. dissertation,
856:Angles between flats
700:distributive lattice
400:
305:
292:parametric equations
266:linearly independent
207:
942:Applicable Geometry
870:N-dimensional space
847:. (See for example
777:are parallel, this
100:-flats, are called
1046:Euclidean geometry
1019:Weisstein, Eric W.
1000:Weisstein, Eric W.
983:2021-10-17 at the
843:between 0 and the
795:Euclidean distance
789:Euclidean geometry
536:
380:
231:
73:-dimensional space
59:Euclidean subspace
18:Euclidean subspace
963:978-0-12-672025-9
557:parameters, e.g.
369:
165:algebraic variety
98: − 1)
16:(Redirected from
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922:by some authors.
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79:-flats of every
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985:Wayback Machine
976:, available as
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702:. If two lines
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169:linear manifold
148:Flats occur in
143:geodesic length
129:and non-planar
109:The flats in a
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47:affine subspace
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32:Linear subspace
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1056:Linear algebra
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992:External links
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968:From original
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954:Academic Press
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850:Dihedral angle
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779:distributivity
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173:linear variety
150:linear algebra
57:, a flat is a
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835:There is the
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1003:"Hyperplane"
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949:
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827:
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806:
799:affine space
792:
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742:
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680:
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588:intersection
585:
566:
559:
548:
392:
289:
278:
274:
262:intersection
243:
187:
184:By equations
179:Descriptions
172:
168:
159:A flat is a
158:
147:
133:, which are
108:
101:
95:
88:
75:, there are
67:
58:
51:affine space
42:
36:
880:Coplanarity
845:right angle
103:hyperplanes
1040:Categories
932:References
606:skew flats
286:Parametric
139:arc length
86:from 0 to
1027:MathWorld
1008:MathWorld
687:-space a
592:empty set
516:−
458:−
429:−
372:−
340:−
296:parameter
135:subspaces
81:dimension
55:Euclidean
981:Archived
970:Stanford
940:(1977),
885:Isometry
864:See also
599:parallel
161:manifold
131:surfaces
63:distance
39:geometry
875:Matroid
689:lattice
163:and an
1022:"Flat"
960:
753:) ∩ (ℓ
252:, and
127:curves
115:points
68:In an
45:is an
891:Notes
837:angle
565:, …,
119:lines
111:plane
958:ISBN
920:flat
811:and
770:and
745:= (ℓ
741:) +
709:and
681:join
679:and
677:meet
625:and
612:Join
196:and
141:and
43:flat
41:, a
910:doi
737:∩ ℓ
722:∩ ℓ
649:+ 1
586:An
171:or
37:In
1042::
1024:.
1005:.
956:,
952:,
859:.)
832:.)
817:.)
757:+
749:+
733:(ℓ
659:+
642:+
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573:.
298::
282:.
277:−
248:,
229:8.
200::
156:.
117:,
106:.
1030:.
1011:.
987:.
912::
783:p
774:2
772:ℓ
767:1
765:ℓ
761:)
759:p
755:2
751:p
747:1
743:p
739:2
735:1
729:p
724:2
720:1
718:ℓ
713:2
711:ℓ
706:1
704:ℓ
685:n
664:2
661:k
657:1
654:k
647:2
644:k
640:1
637:k
631:2
628:k
622:1
619:k
569:k
567:t
563:1
560:t
555:k
551:k
532:.
527:2
523:t
519:3
511:1
507:t
503:5
500:=
497:z
488:2
484:t
480:2
477:+
472:1
468:t
464:+
461:4
455:=
452:y
445:,
440:2
436:t
432:3
424:1
420:t
416:2
413:+
410:5
407:=
404:x
378:t
375:4
367:2
364:3
359:=
356:z
349:t
346:+
343:1
337:=
334:y
327:,
324:t
321:3
318:+
315:2
312:=
309:x
279:k
275:n
270:k
258:n
254:z
250:y
246:x
226:=
223:y
220:5
217:+
214:x
211:3
198:y
194:x
96:n
94:(
89:n
84:k
77:k
71:n
34:.
20:)
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