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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 ...
an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
s. However, a conic is only defined in a
pappian plane In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of Collinearity, collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line (mathemati ...
, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but there are many examples, both infinite and finite, of ovals in pappian planes which are not conics. As mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
by differentiability conditions in the
real plane In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
. The higher dimensional analog of an oval is an
ovoid 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 ...
in a
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 ...
. A generalization of the oval concept is an abstract oval, which is a structure that is not necessarily embedded in a projective plane. Indeed, there exist abstract ovals which can not lie in any projective plane.


Definition of an oval

*In a
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 ...
a set of points is called an oval, if: # Any line meets in at most two points, and # For any point there exists exactly one tangent line through , i.e., . When the line is an ''exterior line'' (or ''passant''), if a ''tangent line'' and if the line is a ''secant line''. For ''finite'' planes (i.e. the set of points is finite) we have a more convenient characterization: * For a finite projective plane of ''order'' (i.e. any line contains points) a set of points is an oval if and only if and no three points 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 common line). A set of points in an ''affine'' plane satisfying the above definition is called an ''affine oval''. An affine oval is always a projective oval in the projective closure (adding a line at infinity) of the underlying affine plane. An oval can also be considered as a special ''
quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space). Definition of a qu ...
''.


Examples


Conic sections

In any pappian projective plane there exist nondegenerate projective conic sections and any nondegenerate projective conic section is an oval. This statement can be verified by a straightforward calculation for any of the conics (such as the
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
or
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
). Non degenerate conics are ovals with special properties: *''
Pascal's Theorem In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by ...
'' and its various degenerations are valid. * There are many projectivities which leave a conic invariant.


Ovals, which are not conics

;in the
real plane In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
# If one glues one half of a circle and a half of an ellipse smoothly together, one gets a non-conic oval. #If one takes the inhomogeneous representation of a conic oval as a parabola plus a point at infinity and replaces the expression by , one gets an oval which is not a conic. #If one takes the inhomogeneous representation of a conic oval as a hyperbola plus two points at infinity and replaces the expression by , one gets an oval which is not a conic. #The implicit curve is a non conic oval. ;in a finite plane of ''even'' order #In a finite pappian plane of even order a nondegenerate conic has a ''nucleus'' (a single point through which every tangent passes), which can be exchanged with any point of the conic to obtain an oval which is not a conic. #For the field with elements let ::\Omega=\ \; \cup \; \ ::For and and coprime, the set is an oval, which is not a conic. Further finite examples can be found here:


Criteria for an oval to be a conic

For an oval to be a conic the oval and/or the plane has to fulfill additional conditions. Here are some results: #An oval in an arbitrary projective plane, which fulfills the incidence condition of
Pascal's theorem In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by ...
or the 5-point degeneration of it, is a nondegenerate conic. #If is an oval in a ''pappian'' projective plane and the group of projectivities which leave invariant is 3-transitive, i.e. for 2 triples of points there exists a projectivity with . In the finite case ''2-transitive'' is sufficient. # An oval in a ''pappian'' projective plane of characteristic is a conic if and only if for any point of a tangent there is an involutory
perspectivity In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. Graphics The science of graphical perspective uses perspectivities to make realistic images in p ...
(symmetry) with center which leaves invariant. #If is an oval in a ''finite Desarguesian'' (pappian) projective plane of ''odd'' order, , then 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 implies that, after a possible change of coordinates, every oval of with odd has the parametrization : :: \\cup \. For topological ovals the following simple criteria holds: :5. Any ''closed'' oval of the complex projective plane is a conic.


Further results on ovals in finite planes

An oval in a finite projective plane of order is a ()- arc, in other words, a set of points, no three collinear. Ovals in the
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 ...
(pappian) projective plane for odd are just the nonsingular conics. However, ovals in for even have not yet been classified. In an arbitrary finite projective plane of odd order , no sets with more points than , no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to the statistical design of experiments. Furthermore, by
Qvist's theorem In projective geometry, Qvist's theorem, named after the Finnish mathematician , is a statement on ovals in ''finite'' projective planes. Standard examples of ovals are non-degenerate (projective) conic sections. The theorem gives an answer to the ...
, through any point not on an oval there pass either zero or two tangent lines of that oval. When ''q'' is even, the situation is completely different. In this case, sets of points, no three of which collinear, may exist in a finite projective plane of order and they are called hyperovals; these are
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 ...
s of degree 2. Given an oval there is a unique tangent through each point, and if is even
Qvist's theorem In projective geometry, Qvist's theorem, named after the Finnish mathematician , is a statement on ovals in ''finite'' projective planes. Standard examples of ovals are non-degenerate (projective) conic sections. The theorem gives an answer to the ...
, () shows that all these tangents are concurrent in a point outside the oval. Adding this point (called the ''nucleus'' of the oval or sometimes the ''knot'') to the oval gives a hyperoval. Conversely, removing ''any'' one point from a hyperoval immediately gives an oval. As all ovals in the even order case are contained in hyperovals, a description of the (known) hyperovals implicitly gives all (known) ovals. The ovals obtained by removing a point from a hyperoval are projectively equivalent if and only if the removed points are in the same orbit of the automorphism group of the hyperoval. There are only three small examples (in the Desarguesian planes) where the automorphism group of the hyperoval is transitive on its points (see ) so, in general, there are different types of ovals contained in a single hyperoval.


Desarguesian Case: PG(2,2h)

This is the most studied case and so the most is known about these hyperovals. Every nonsingular conic in the projective plane, together with its nucleus, forms a hyperoval. These may be called ''hyperconics'', but the more traditional term is ''regular hyperovals''. For each of these sets, there is a system of coordinates such that the set is: : \\cup \\cup\. However, many other types of hyperovals of PG(2, ''q'') can be found if ''q'' > 8. Hyperovals of PG(2, ''q'') for ''q'' even have only been classified for ''q'' < 64 to date. In PG(2,2h), h > 0, a hyperoval contains at least four points no three of which are collinear. Thus, by the ''Fundamental Theorem of Projective Geometry'' we can always assume that the points with projective coordinates (1,0,0), (0,1,0), (0,0,1) and (1,1,1) are contained in any hyperoval. The remaining points of the hyperoval (when h > 1) will have the form (t, f(t),1) where t ranges through the values of the finite field GF(2h) and ''f'' is a function on that field which represents a permutation and can be uniquely expressed as a polynomial of degree at most 2h - 2, i.e. it is a
permutation polynomial In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x \mapsto g(x) is a bijection. In case the ring is a finite field, the Dickson polynomials, whic ...
. Notice that f(0) = 0 and f(1) = 1 are forced by the assumption concerning the inclusion of the specified points. Other restrictions on ''f'' are forced by the no three points collinear condition. An ''f'' which produces a hyperoval in this way is called an ''o-polynomial''. The following table lists all the known hyperovals (as of 2011) of PG(2,2h) by giving the o-polynomial and any restrictions on the value of ''h'' that are necessary for the displayed function to be an o-polynomial. Note that all exponents are to be taken mod(2h - 1).


Known Hyperovals in PG(2,2h)

a) The Subiaco o-polynomial is given by: f(t) = + t^, whenever tr(1/d) = 1 \hbox d \not\in GF(4) \hbox h \equiv 2 (\bmod 4), where ''tr'' is the absolute trace function of GF(2h). This o-polynomial gives rise to a unique hyperoval if h \not\equiv 2 (\bmod 4) and to two inequivalent hyperovals if h \equiv 2 (\bmod 4), h > 2. b) To describe the Adelaide hyperovals, we will start in a slightly more general setting. Let F = ''GF(q)'' and K = ''GF(q2)''. Let b \in K be an element of norm 1, different from 1, i.e. bq+1 = 1, b \neq 1. Consider the polynomial, for t \in F,
f(t) = (''tr''(b))−1''tr''(bm)(t + 1) + (''tr''(b))−1''tr''((bt + bq)m)(t + ''tr''(b)t½+ 1)1−m + t½,
where ''tr''(x) = ''tr''K/F(x) = x + xq. When ''q'' = 2h, with ''h'' even and m = ±(q - 1)/3, the above f(t) is an o-polynomial for the Adelaide hyperoval. c) The Penttila-O'Keefe o-polynomial is given by:
f(t) = t4 + t16 + t28 + η11(t6 + t10 + t14 + t18 + t22 + t26) + η20(t8 + t20) + η6(t12 + t24),
where η is a primitive root of GF(32) satisfying η5 = η2 + 1.


Hyperovals in PG(2, q), q even, q ≤ 64

As the hyperovals in the Desarguesian planes of orders 2, 4 and 8 are all hyperconics we shall only examine the planes of orders 16, 32 and 64.


PG(2,16)

In the details of a computer search for complete arcs in small order planes carried out at the suggestion of B. Segre are given. In PG(2,16) they found a number of hyperovals which were not hyperconics. In 1975, M. Hall Jr. showed, also with considerable aid from a computer, that there were only two classes of projectively inequivalent hyperovals in this plane, the hyperconics and the hyperovals found by Lunelli and Sce. Out of the 2040 o-polynomials which give the ''Lunelli-Sce hyperoval'', we display only one:
f(x) = x12 + x10 + η11x8 + x6 + η2x4 + η9x2,
where η is a primitive element of ''GF(16)'' satisfying η4 = η + 1. In his 1975 paper Hall described a number of collineations of the plane which stabilized the Lunelli-Sce hyperoval, but did not show that they generated the full automorphism group of this hyperoval. using properties of a related
generalized quadrangle In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 ...
, showed that the automorphism group could be no larger than the group given by Hall. independently gave a constructive proof of this result and also showed that in Desarguesian planes, the Lunelli-Sce hyperoval is the unique irregular hyperoval (non-hyperconic) admitting a transitive automorphism group (and that the only hyperconics admitting such a group are those of orders 2 and 4). reproved Hall's classification result without the use of a computer. Their argument consists of finding an upper bound on the number of o-polynomials defined over ''GF(16)'' and then, by examining the possible automorphism groups of hyperovals in this plane, showing that if a hyperoval other than the known ones existed in this plane then the upper bound would be exceeded. provides a group-theoretic construction of the Lunelli-Sce hyperoval as the union of orbits of the group generated by the elations of PGU(3,4) considered as a subgroup of PGL(3,16). Also included in this paper is a discussion of some remarkable properties concerning the intersections of Lunelli-Sce hyperovals and hyperconics. In it is shown that the Lunelli-Sce hyperoval is the first non-trivial member of theSubiaco family (see also ). In it is shown to be the first non-trivial member of the Adelaide family.


PG(2,32)

Since ''h'' = 5 is odd, a number of the known families have a representative here, but due to the small size of the plane there are some spurious equivalences, in fact, each of the Glynn type hyperovals is projectively equivalent to a translation hyperoval, and the Payne hyperoval is projectively equivalent to the Subiaco hyperoval (this does not occur in larger planes). Specifically, there are three classes of (monomial type) hyperovals, the hyperconics (f(t) = t2), proper translation hyperovals (f(t) = t4) and the Segre hyperovals (f(t) = t6).In smaller order planes these hyperovals are not distinct from hyperconics. The proof of their existence given in utilizes
linearized polynomial In mathematics, a linearised polynomial (or ''q''-polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of ''q'' and the coefficients come from some extension field of the finite field of order ''q''. We w ...
s.
There are also classes corresponding to the Payne hyperovals and the Cherowitzo hyperovals (for more details see . In the collineation groups stabilizing each of these hyperovals have been determined. Note that in the original determination of the collineation group for the Payne hyperovals the case of ''q'' = 32 had to be treated separately and relied heavily on computer results. In an alternative version of the proof is given which does not depend on computer computations. In 1991, O'Keefe and Penttila discovered a new hyperoval in this plane by means of a detailed investigation of the divisibility properties of the orders of automorphism groups of hypothetical hyperovals . One of its o-polynomials is given by:
f(x) = x4 + x16 + x28 + η11(x6 + x10 + x14 + x18 + x22 + x26) + η20(x8 + x20) + η6(x12 + x24),
where η is a primitive root of ''GF(32)'' satisfying η5 = η2 + 1. The full automorphism group of this hyperoval has order 3. cleverly structured an exhaustive computer search for all hyperovals in this plane. The result was that the above listing is complete, there are just six classes of hyperovals in PG(2,32).


PG(2,64)

By extending the ideas in to PG(2,64), were able to search for hyperovals whose automorphism group admitted a collineation of order 5. They found two and showed that no other hyperoval exists in this plane that has such an automorphism. This settled affirmatively a long open question of B. Segre who wanted to know if there were any hyperovals in this plane besides the hyperconics. The hyperovals are:
f(x) = x8 + x12 + x20 + x22 + x42 + x52 + η21(x4+x10+x14+x16+x30+x38+x44+x48+x54+x56+x58+x60+x62) + η42(x2 + x6 + x26 + x28 + x32 + x36 + x40),
which has an automorphism group of order 15, and
f(x) = x24 + x30 + x62 + η21(x4 +x8+x10+x14+x16+x34+x38 +x40 +x44+x46+x52+x54+x58+x60) + η42(x6+ x12+ x18+ x20+ x26+ x32 + x36+ x42+ x48+x50),
which has an automorphism group of order 60, where η is a primitive element of GF(64) satisfying η6 = η + 1. In it is shown that these are Subiaco hyperovals. By refining the computer search program, extended the search to hyperovals admitting an automorphism of order 3, and found the hyperoval:
f(x) = x4 + x8 + x14 + x34 + x42 + x48 + x62 + η21(x6+x16 +x26+x28+x30+x32+x40+x58) + η42(x10 + x18 + x24 + x36 + x44 + x50 + x52+ x60),
which has an automorphism group of order 12 (η is a primitive element of ''GF(64)'' as above). This hyperoval is the first distinct Adelaide hyperoval. Penttila and Royle have shown that any other hyperoval in this plane would have to have a trivial automorphism group. This would mean that there would be many projectively equivalent copies of such a hyperoval, but general searches to date have found none, giving credence to the conjecture that there are no others in this plane.


Abstract ovals

Following ( Bue1966), an ''abstract oval'', also called a ''B-oval'', of order n is a pair (F,) where F is a set of n+1 elements, called points, and is a set of involutions acting on F in a sharply quasi 2-transitive way, that is, for any two (a_1,a_2),(b_1,b_2)\in F with a_i\neq b_j for i,j\in\, there exists exactly one \sigma\in with \sigma(a_1)=a_2 and \sigma(b_1)=b_2. Any oval embedded in a projective plane of order q might be endowed with a structure of an abstract oval of the same order. The converse is, in general, not true for n\geq 8; indeed, for n=8 there are two abstract ovals which may not be embedded in a projective plane, see ( Fa1984). When n is even, a similar construction yields ''abstract hyperovals'', see ( Po1997): an abstract hyperoval of order n is a pair (F,) where F is a set of n+2 elements and is a set of fixed-point free involutions acting on F such that for any set of four distinct elements a,b,c,d\in F there is exactly one \sigma\in with \sigma(a)=b, \sigma(c)=d.


See also

* Ovoid (projective geometry)


Notes


References

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *{{Citation , last1= Segre , first1 = B. , last2 = Bartocci , first2 = U. , title= Ovali ed altre curve nei piani di Galois di caratteristica due , language= Italian , journal=
Acta Arithmetica ''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sc ...
, volume=18 , year =1971 , pages= 423–449 , mr=0295201 , doi=10.4064/aa-18-1-423-449, doi-access= free


External links


Bill Cherowitzo's Hyperoval Page
Projective geometry Incidence geometry