Arc (projective geometry)
Encyclopedia
In mathematics, a-arc (k, d > 1) in a finite projective plane
π (not necessarily Desarguesian) is a set of k points of
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. When d = 2 it is typical to refer to a (k, d)-arc as simply a k-arc or an arc if the size is not a concern.
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(q + 1, 2)-arcs are precisely the ovals
and (q + 2, 2)-arcs are precisely the hyperovals (which can only occur for even q). Notice that hyperovals are maximal arcs.
A k-arc which can not be extended to a larger arc is called a complete arc. Complete arcs need not be maximal arcs.
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
π (not necessarily Desarguesian) is a set of k points of
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. When d = 2 it is typical to refer to a (k, d)-arc as simply a k-arc or an arc if the size is not a concern.Special cases
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 arcMaximal arc
Maximal arcs are -arcs in a projective plane, where k is maximal with respect to the parameter d and the ambient space.-Definition:Let \pi be a projective plane of order q...
.
(q + 1, 2)-arcs are precisely the ovals
Oval (projective plane)
In mathematics, an oval in a projective plane is a set of points, no three collinear, such that there is a unique tangent line at each point...
and (q + 2, 2)-arcs are precisely the hyperovals (which can only occur for even q). Notice that hyperovals are maximal arcs.
A k-arc which can not be extended to a larger arc is called a complete arc. Complete arcs need not be maximal arcs.