27:
525:, but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse. Similarly, a parabola is a concept in affine geometry but not in projective geometry, where a parabola is simply a kind of conic. The geometry that is overwhelmingly used in algebraic geometry is projective geometry, with affine geometry finding significant but far less use.
536:), so the notion of "general position with respect to circles", namely "no four points lie on a circle" makes sense. In projective geometry, by contrast, circles are not distinct from conics, and five points determine a conic, so there is no projective notion of "general position with respect to circles".
411:
The basic condition for general position is that points do not fall on subvarieties of lower degree than necessary; in the plane two points should not be coincident, three points should not fall on a line, six points should not fall on a conic, ten points should not fall on a cubic, and likewise for
566:
of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative
Kodaira dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces
464:) of cubics, whose equations are the projective linear combinations of the equations for the two cubics. Thus such sets of points impose one fewer condition on cubics containing them than expected, and accordingly satisfy an additional constraint, namely the
649:
is said to be in general position only if no four of them lie on the same circle and no three of them are collinear. The usual lifting transform that relates the
Delaunay triangulation to the bottom half of a convex hull (i.e., giving each point
415:
This is not sufficient, however. While nine points determine a cubic, there are configurations of nine points that are special with respect to cubics, namely the intersection of two cubics. The intersection of two cubics, which is
150:
For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a
494:
requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with
550:
General position is a property of configurations of points, or more generally other subvarieties (lines in general position, so no three concurrent, and the like). General position is an
446:
591:
554:
notion, which depends on an embedding as a subvariety. Informally, subvarieties are in general position if they cannot be described more simply than others. An
168:
This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general
408:
polynomial at that point, this formalizes the notion that points in general position impose independent linear conditions on varieties passing through them.
509:
distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the
393:, but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.
374:
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g.
658:|) shows the connection to the planar view: Four points lie on a circle or three of them are collinear exactly when their lifted counterparts are
400:
maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points. Significantly, the
490:). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of
562:, and corresponds to a variety which cannot be described by simpler polynomial equations than others. This is formalized by the notion of
471:
For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a
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Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any
91:
521:
Different geometries allow different notions of geometric constraints. For example, a circle is a concept that makes sense in
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that any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree.
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situation, as opposed to some more special or coincidental cases that are possible, which is referred to as
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or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.
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162:
419:
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37:
247:. These conditions contain considerable redundancy since, if the condition holds for some value
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meaning with multiplicity 1, rather than being tangent or other, higher order intersections.
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341:-dimensional vector space are linearly independent if and only if the points they define in
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of the triangle they define), but four points in general do not (they do so only for
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Thus, in
Euclidean geometry three non-collinear points determine a circle (as the
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is biregular; as points under the
Veronese map corresponds to evaluating a degree
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points – i.e. the points do not satisfy any more linear relations than they must.
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cubic, while if they are contained in two cubics they in fact are contained in a
366:, as long as the points are in general linear position (no three are collinear).
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this kind of condition is frequently encountered, in that points should impose
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on the configuration space, or equivalently that points chosen at random will
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161:; if three points are collinear (even stronger, if two coincide), this is a
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452:), is special in that nine points in general position are contained in a
285:-dimensional affine space to be in general position, it suffices that no
593:), and similar classifications occur in higher dimensions, notably the
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397:
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If a set of points is not in general linear position, it is called a
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26:
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of a configuration space, or equivalently on a
Zariski-open set.
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for a set of points, or other geometric objects. It means the
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General position for
Delaunay triangulations in the plane
682:; in this context one means properties that hold on the
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or giving precise statements thereof, and when writing
157:. Similarly, three generic points in the plane are not
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422:
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points in general linear position is also said to be
147:. Its precise meaning differs in different settings.
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of vectors, or more precisely of maximal rank), and
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51:. Unsourced material may be challenged and removed.
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440:
478:, and is measured by the vanishing of the higher
362:A fundamental application is that, in the plane,
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701:(with probability 1) be in general position.
318:points in general linear position in affine
621:is used: subvarieties in general intersect
386:conditions on curves passing through them.
185:
111:Learn how and when to remove this message
516:
745:
274:. Thus, for a set containing at least
733:
717:
613:, both in algebraic geometry and in
396:General position is preserved under
49:adding citations to reliable sources
20:
13:
352:) are in general linear position.
14:
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500:Riemann–Roch theorem for surfaces
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666:Abstractly: configuration spaces
25:
689:This notion coincides with the
595:Enriques–Kodaira classification
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256:then it also must hold for all
36:needs additional citations for
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654:an extra coordinate equal to |
558:analog of general position is
307:(this is the affine analog of
1:
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391:five points determine a conic
364:five points determine a conic
18:Concept in algebraic geometry
662:in general linear position.
7:
693:notion of generic, meaning
441:{\displaystyle 3\times 3=9}
209:is a common example) is in
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617:, the analogous notion of
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482:groups of the associated
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670:In very abstract terms,
466:Cayley–Bacharach theorem
641:in the plane, a set of
639:Delaunay triangulations
586:{\displaystyle g\geq 2}
211:general linear position
186:General linear position
734:Yale, Paul B. (1968),
587:
498:, as discussed in the
496:superabundant divisors
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129:computational geometry
736:Geometry and Symmetry
635:Voronoi tessellations
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544:Further information:
534:cyclic quadrilaterals
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328:affine transformation
190:A set of points in a
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517:Different geometries
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305:affinely independent
45:improve this article
680:configuration space
674:is a discussion of
611:intersection theory
492:intersection number
309:linear independence
289:contains more than
753:Algebraic geometry
676:generic properties
615:geometric topology
599:algebraic surfaces
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523:Euclidean geometry
438:
380:algebraic geometry
179:generic complexity
125:algebraic geometry
60:"General position"
695:almost everywhere
691:measure theoretic
564:Kodaira dimension
296:A set of at most
221:of them lie in a
174:computer programs
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672:general position
633:When discussing
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488:invertible sheaf
480:sheaf cohomology
450:BĂ©zout's theorem
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412:higher degree.
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343:projective space
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215:general position
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145:special position
133:general position
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619:transversality
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605:Other contexts
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376:conic sections
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370:More generally
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345:(of dimension
337:vectors in an
322:-space are an
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699:almost surely
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462:linear system
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241:= 2, 3, ...,
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205:-dimensional
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62: –
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56:Find sources:
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40:
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34:This article
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23:
22:
16:
738:, Holden-Day
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560:general type
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546:General type
540:General type
530:circumcircle
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402:Veronese map
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141:general case
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43:Please help
38:verification
35:
15:
511:cross ratio
486:(formally,
484:line bundle
448:points (by
384:independent
333:Similarly,
231:dimensional
196:dimensional
728:References
330:for more.
287:hyperplane
281:points in
227:− 2)
137:genericity
71:newspapers
718:Yale 1968
578:≥
556:intrinsic
552:extrinsic
427:×
398:biregular
350:− 1
213:(or just
159:collinear
747:Category
720:, p. 164
473:regular
217:) if no
170:theorems
101:May 2014
645:in the
475:divisor
85:scholar
643:points
458:pencil
454:unique
378:). In
326:. See
87:
80:
73:
66:
58:
705:Notes
678:of a
647:plane
260:with
176:(see
92:JSTOR
78:books
637:and
262:2 ≤
236:for
234:flat
127:and
64:news
660:not
609:In
597:of
316:+ 1
301:+ 1
279:+ 1
245:+ 1
182:).
123:In
47:by
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601:.
513:.
502:.
266:≤
165:.
131:,
656:p
652:p
581:2
575:g
567:(
507:k
436:9
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430:3
424:3
406:d
348:n
339:n
335:n
320:d
314:d
299:d
291:d
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271:0
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229:-
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223:(
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201:(
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108:(
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89:·
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75:·
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