An (''simple'') arc in finite

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

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 -arcs. An important generalization of the -arc concept, also referred to as arcs in the literature, are the ()-arcs.

-arcs in a projective plane

In a finite

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 (namely, parallel lines) that do ...

(not necessarily

Desarguesian In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...

) a set of points such that no three points of are

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

(on a line) is called a . If the plane has order then , however the maximum value of can only be achieved if is even. In a plane of order , a -arc is called an

oval An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...

and, if is even, a -arc is called a

hyperoval In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of ...

. Every conic in the Desarguesian projective plane PG(2,), i.e., the set of zeros of an irreducible homogeneous quadratic equation, is an oval. A celebrated result of

Beniamino Segre Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry. Life and career He was born and studied in Turin ...

states that when is odd, every -arc in PG(2,) is a conic (

Segre's theorem In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: *Any oval in a ''finite pappian'' projective plane of ''odd'' order is a nondegenerate projective conic section. This statement was ...

). This is one of the pioneering results in

finite geometry Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

. If is even and is a -arc in , then it can be shown via combinatorial arguments that there must exist a unique point in (called the nucleus of ) such that the union of and this point is a ( + 2)-arc. Thus, every oval can be uniquely extended to a hyperoval in a finite projective plane of even order. A -arc which can not be extended to a larger arc is called a ''complete arc''. In the Desarguesian projective planes, PG(2,), no -arc is complete, so they may all be extended to ovals.

-arcs in a projective space

In the finite

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

PG() with , a set of points such that no points lie in a common

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

is called a (spatial) -arc. This definition generalizes the definition of a -arc in a plane (where ).

()-arcs in a projective plane

A ()-arc () in a finite

(not necessarily

) is a set, of points of such that each line intersects in at most points, and there is at least one line that does intersect in points. A ()-arc is a -arc and may be referred to as simply an arc if the size is not a concern. The number of points of a ()-arc in a projective plane of order is at most . When equality occurs, one calls a

maximal arc A Maximal arc in a finite 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 pai ...

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

Notes

References

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External links