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	~~[[File:Hyperoval in Fano plane.svg\|thumb\|A 4-arc (red points) in the projective plane of order 2 (Fano plane).]]~~		The author's title is Christy. Distributing production is how he makes a living. To perform domino is some thing I really appreciate doing. For a while I've been in Mississippi but now I'm contemplating other options.<br><br>Here is my site ... [http://formalarmour.com/index.php?do=/profile-26947/info/ best psychic]

	A (''simple'') '''arc''' in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of ''curved'' figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called '''''k'''''-'''arcs'''. An important generalization of the ''k''-arc concept, also referred to as arcs in the literature, are the (''k'', ''d'')-arcs.

	~~==''k''-arcs in a projective plane==~~

	In a finite [[projective plane]] ''π'' (not necessarily [[Desarguesian]]) a set ''A'' of ''k'' (''k'' ≥ 3) points such that no three points of ''A'' are [[Collinear points\|collinear]] (on a line) is called a '''''k'''''-'''arc'''. If the plane ''π'' has order ''q'' then ''k'' ≤ ''q'' + 2, however the maximum value of ''k'' can only be achieved if ''q'' is ~~even~~.~~<ref>{{harvnb\|Hirschfeld\|1979\|loc=pg. 164, Theorem 8.1.3}}</ref> In a plane of order ''q'', a (''q'' + 1)-arc is called an '''[[Oval (projective plane)\|oval]]''' and, if ''q''~~ is ~~even,~~ a ~~(''q'' + 2)-arc is called a '''[[Oval (projective plane)\|hyperoval]]'''.~~

	~~A ''k''-arc which can not be extended to a larger arc is called a '''''complete arc'''''~~. ~~In the Desarguesian projective planes, PG(2,''q''), no ''q''-arc is complete, so they may all be extended to ovals.<ref>{{harvnb\|Dembowski\|1968\|loc=pg. 150, result 28}}</ref>~~

	~~==''k''-arcs in a projective space==~~

	~~In the finite [[projective space]] PG(''n'', ''q'') with ''n'' ≥ 3, a set ''A'' of ''k'' ≥ ''n'' + 1 points such that no ''n'' + 1 points lie in a common [[Hyperplane (geometry)\|hyperplane]]~~ is ~~called a (spacial) '''''k'''''-'''arc'''~~. ~~This definition generalizes the definition of~~ a '~~'k''-arc~~ in ~~a plane (where~~ '~~'n'' = 2)~~.

	~~==(''k'', ''d'')-arcs in a projective plane==~~

	~~A (''k'', ''d'')-'''arc''' (''k'', ''d'' > 1) in a finite [[projective plane]] ''π'' (not necessarily [[Desarguesian]]) is a set, ''A'' of ''k'' points of~~ <~~math~~>~~\pi~~<~~/math~~> such that each line intersects ''A'' in at most ''d'' points, and there is at least one line that does intersect ''A'' in ''d'' points. A (''k'', 2)-arc is a '''''k''-arc''' and may be referred to as simply an '''arc''' if the size is not a concern.

	~~The number of points ''k'' of a (''k'', ''d'')-arc ''A'' in a projective plane of order ''q''~~ is ~~at most ''qd'' + ''d'' − ''q''~~. ~~When equality occurs, one calls ''A'' a '''[[maximal arc]]'''~~.

	~~Hyperovals are maximal arcs~~. ~~Complete arcs need not be maximal arcs.~~

	~~==See also==~~

	[~~[Oval (projective plane)]]~~

	~~[[Normal rational curve]]~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{Citation \| last1=Dembowski \| first1=Peter \| title=Finite geometries \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 \| mr=0233275 \| year=1968 \| isbn=3-540-61786-8}}

	* {{citation\|last=Hirschfeld\|first=J.~~W.P~~.~~\|title~~=~~Projective Geometries over Finite Fields\|year=1979\|publisher=Oxford University Press\|location=New York\|isbn=0~~-~~19-853526-0}}~~

	~~==External links==~~

	*{{springer\|id=Arc_(projective_geometry)&oldid=25358\|title=Arc\|author=C.M. O'Keefe}}

	~~[[Category:Projective geometry]~~]

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[[File:Hyperoval in Fano plane.svg|thumb|A 4-arc (red points) in the projective plane of order 2 (Fano plane).]]

A (''simple'') '''arc''' in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of ''curved'' figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called '''''k'''''-'''arcs'''. An important generalization of the ''k''-arc concept, also referred to as arcs in the literature, are the (''k'', ''d'')-arcs.

==''k''-arcs in a projective plane==

In a finite [[projective plane]] ''π'' (not necessarily [[Desarguesian]]) a set ''A'' of ''k'' (''k'' ≥ 3) points such that no three points of ''A'' are [[Collinear points|collinear]] (on a line) is called a '''''k'''''-'''arc'''. If the plane ''π'' has order ''q'' then ''k'' ≤ ''q'' + 2, however the maximum value of ''k'' can only be achieved if ''q'' is even.<ref>{{harvnb|Hirschfeld|1979|loc=pg. 164, Theorem 8.1.3}}</ref> In a plane of order ''q'', a (''q'' + 1)-arc is called an '''[[Oval (projective plane)|oval]]''' and, if ''q'' is even, a (''q'' + 2)-arc is called a '''[[Oval (projective plane)|hyperoval]]'''.

A ''k''-arc which can not be extended to a larger arc is called a '''''complete arc'''''. In the Desarguesian projective planes, PG(2,''q''), no ''q''-arc is complete, so they may all be extended to ovals.<ref>{{harvnb|Dembowski|1968|loc=pg. 150, result 28}}</ref>

==''k''-arcs in a projective space==

In the finite [[projective space]] PG(''n'', ''q'') with ''n'' ≥ 3, a set ''A'' of ''k'' ≥ ''n'' + 1 points such that no ''n'' + 1 points lie in a common [[Hyperplane (geometry)|hyperplane]] is called a (spacial) '''''k'''''-'''arc'''. This definition generalizes the definition of a ''k''-arc in a plane (where ''n'' = 2).

==(''k'', ''d'')-arcs in a projective plane==

A (''k'', ''d'')-'''arc''' (''k'', ''d'' > 1) in a finite [[projective plane]] ''π'' (not necessarily [[Desarguesian]]) is a set, ''A'' of ''k'' points of <math>\pi</math> such that each line intersects ''A'' in at most ''d'' points, and there is at least one line that does intersect ''A'' in ''d'' points. A (''k'', 2)-arc is a '''''k''-arc''' and may be referred to as simply an '''arc''' if the size is not a concern.

The number of points ''k'' of a (''k'', ''d'')-arc ''A'' in a projective plane of order ''q'' is at most ''qd'' + ''d'' − ''q''. When equality occurs, one calls ''A'' a '''[[maximal arc]]'''.

Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.

==See also==

[[Oval (projective plane)]]

[[Normal rational curve]]

==Notes==
{{reflist}}

==References==
* {{Citation | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8}}

* {{citation|last=Hirschfeld|first=J.W.P.|title=Projective Geometries over Finite Fields|year=1979|publisher=Oxford University Press|location=New York|isbn=0-19-853526-0}}

==External links==

*{{springer|id=Arc_(projective_geometry)&oldid=25358|title=Arc|author=C.M. O'Keefe}}

[[Category:Projective geometry]]

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70.197.11.190 at 17:08, 1 June 2014

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