Arcs and Ovals in Projective Geometry
Steven Di Lisio
MATH 4220
University of Colorado Denver
1 Introduction
An oval is familiar shape that looks like an elongated circle. When we delve into the
world of Projective Geometry an oval is a new, interesting, abstract idea. The origin of an oval
comes from the idea of k-arcs, which is a set of k points, no three collinear. A k-arc with the
maximum number of points is called an oval. Segre was the first to use the word oval to describe
a maximum set of points of a projective plane of order q with the property that no three are
collinear (Brown, 2000). An oval has many different properties, and has direct realtions to
conics.
2 Preliminaries
We must first define a projective plane and a few of its properties. The following
definitions and theorems are from the book Projective Geometry (Casse, 2006).
Definition 2.1. A projective space of dimension 2 is called a projective plane.
Theorem 2.1. Let π be a projective plane which has a finite number of points N2. Then
1. Every line of π has exactly N1 = q+1 of points, q is the order of the plane π.
2. The number of points N2 = q2 + q + 1.
Proof.
1. Let l, m any two distinct lines of π. Let P = l ∩ m. Let the points A and B be elements of l
distinct from P and C and D be elements of m also distinct from P. Define Q = AD ∩ BD,
thus the point Q is not on l or on m. If we take any point on l, say Pi, then the line QPi
will intersect the line m in a point. Therefore l has at least as many points as m, and m has
at least as many points as l. Therefore l and m have the same number of points. Let N1
represent the number of points. Let N1 = q + 1.
2.
Let n be any line in π, and P be a points not on n. The line n has q + 1 points, so there are
q + 1 lines connecting each point of n to P. Each of these lines will have q points, other
than P. Thus we have
N2 = q(q + 1) + 1
= q2 + q + 1. 
Notation: A field plane with order n and order q is denoted by PG(n,q)
To help understand a proof in Section 3 we will need to define a conic and few of its
properties (Hirschfeld, 1979).
Definition 2.2. The points of PG(2, q) that are the zeros of an irreducible quadratic form are the
points of a (q + 1)-arc called a conic.
Arcs and Ovals
Steven Di Lisio
Definition 2.3. The point of intersection of the tangents to a conic in PG(2, q), with q even, is the
nucleus.
Lemma 2.2. Every conic in PG(2,q) is a (q +1)-arc.
Lemma 2.3. In PG(2,q) for q even, the q +1 tangents to a conic are concurrent.
The ovals we will be looking into will be in the projective plane PG(2,q). For n > 2, there
are different names for sets of points where no three are collinear. Segre called the set of points,
no three collinear in PG(3,q) an ovaloid (Brown, 2000).
3 k-arcs
We can now look into where an oval comes from and how an oval is constructed. In this
section we will exclusively be in the projective plane PG(2,q). For convenience denote the plane
PG(2,q) as П. A k-arc is a set of points in which no three are collinear. We will define m(2, q) as
the maximum number of points any k-arc can have (Hirschfeld, 1979). We will define a few
terms to help identify all of the other lines in the plane Π.
Definition 3.1. A line that does not intersect the k-arc  is called a 0-secant. This line is also
referred to as an external line.
Definition 3.2. A line that intersects the k-arc  in exactly one point is called a tangent.
Definition 3.3. A line that intersects the k-arc  in exactly two points is called a secant.
Every line in the plane П will be one of these types of lines. Let P be a point in П that is
not on the k-arc  and let t(P) be the number of tangents through P. We define τt to be the
number of tangents, τs to be the number of secants and τ0 to be the number of external lines.
Lemma 3.1. t(P) = q + 2 – k and let this number be equal to t.
Proof.
 will have k – 1 secants through P and q + 1 lines, thus t(P) = (q + 1) – (k – 1) = t.
In Section 1 we showed that in a plane П, of order q, a line will have q + 1 points. If we
look at a k-arc with k = q + 1 we see that t = 1. This means for every point in the plane П there is
one tangent line that goes through it. Thus there are q + 1 tangents to the (q + 1)-arc. 
The order of the plane П can either be even or odd. The maximum number of points that
a k-arc can have is m(2,q), this value changes for q odd or even. To help show these differences
we need to define two terms for a k-arc  (Hirschfeld, 1979):
Definition 3.1. The number of secants through Q is called the index of Q with respect to. We
denote this quantity as σ2(Q).
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Steven Di Lisio
Definition 3.2. The number of tangents through Q is called the grade of Q with respect to . We
denote this quantity as σ1(Q).
Lemma 3.2. For any point Q in П\.
σ1(Q) + 2σ2(Q) = k.
This brings us to our last theorem about k-arcs and will bring us to the first few
properties of ovals. Theorem 3.3 was proved by Bose (1947) showing the different maximum
points for q even or q odd (Hirschfeld, 1979).
Theorem 3.3. The maximum number of points in  for q even is q + 2. The maximum number of
points in  for q odd is q + 1.
Proof.
From Lemma 3.1 we have t(P) = q + 2 – k  0 for any k-arc , thus k  q + 2. When q is even,
the tangents to a conic  are concurrent at the nucleus, Lemma 2.3. Thus   {} is a (q + 2)arc.
Suppose there exists a (q + 2)-arc  for q odd. Then from Lemma 3.1 tells us that t = 0 and thus
σ1(Q) = 0 for any Q in П\. Thus 2σ2(Q) = q + 2, but q is odd so this is a contradiction. So
m(2,q)  q + 1, but a conic is always a (q + 1)-arc. Therefore m(2,q) = q + 1 for q odd. 
4 Ovals and Hyperovals
Dembowski also defined an oval as a set of q + 1 points, no three collinear (Dembowski,
1968). When the plane is odd both Segre and Dembowski defined an oval to be the same. On the
other hand, Segre only defined planes as a maximum; Dembowski actually put a value on the
number of points in the oval. Segre’s oval for a plane having even order later was renamed
hyperoval. If the projective plane П has an even order we can extend the oval to include the
nucleus to make a hyperoval (Brown, 2000). We will define these terms as so:
Definition 4.1. An oval of PG(2,q) is a set of q + 1 points no three collinear.
Definition 4.2. A hyperoval of PG(2,q) is a set of q + 2 points no three collinear.
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Steven Di Lisio
Now that an oval is defined we will show that every oval in a projective plane of even
order has a nucleus. This means that in a projective plane of even order we can always extend an
oval , out to a unique hyperoval . This oval  is said to be complete to  (Brown, 2000).
Theorem 4.1. Le t  be an oval of PG(2,q), q even. The q + 1 tangents of  are concurrent,
intersecting at the nucleus.
Proof.
Let P be any point of PG(2,q) \ . The lines through P partition the points of .The quantity
q + 1 is odd, so P must be on at least one tangent to . Now let  be a secant to  with  ∩  =
{Q, R}. The tangents of  at the points  \ {Q, R} meet  in distinct points, thus any point not
on  that lies on a secant must lie on exactly one tangent. If we take the intersection of two
tangents to  this point lies on two tangents and so cannot lie on any secants. Consequently it
lies on all the tangents to . 
The following two theorems are fairly simple, but are just a small piece of this oval
shaped pie. The next two theorems each deal with q either being even or odd. A further
investigation of ovals past these few proofs goes into the classification of ovals. Classifying
ovals deals primarily with how each oval “behaves” for different orders of q (Brown, 2000).
Next we will see another proof for the projective plane П of even order. This is a small,
but interesting result (Hirschfeld, 1979).
Theorem 4.2. If two ovals have more than half their points in common when q is even, they
coincide.
Proof.
Suppose the ovals  and  1 have [(q + 2)/ 2] + n points in common, with n > 0. Take P  1\.
We know  has no tangents so any line through P and a point of  \ 1 meets  again in a point
of  \ 1. So  has 2[[(q + 2)/ 2] + n] = (q + 2) + 2n points, contradicting Theorem 3.3. Thus
there is no such P and 1 = . 
Theorem 4.2 describes an oval in a projective plane of even order only. The next theorem
is applicable to projective planes of odd order.
Theorem 4.3. Let  be an oval of PG(2,q), q odd. Every point off  is incident with either 0 or 2
tangents to .
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Steven Di Lisio
Proof.
Let  be a tangent to  at the point P. If Q is another point on , the lines through Q partition the
q + 1 points of . Since q is odd and Q is already incident with one tangent it must lie on at least
one other tangent. This is true for each of the q points of  \{P} and there are q tangents to  that
are not . This shows that each point of  \{P} must be on exactly one other tangent. Therefore if
a point off of the oval  is on one tangent, then it is on exactly two. 
There are many more theorems dealing with how ovals behave whether they are in a
plane of odd or even order. The classification of ovals is the next piece of the puzzle. Classifying
ovals studies how ovals behave in an odd or even ordered plane. A common classification in
PG(2,q) with an odd q is the conics. Although ovals in a projective plane of even order have a
number of infinite families of ovals and hyperovals, they have not yet been classified. (Brown,
2000).
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Steven Di Lisio
References
Barocci, B. S. (1971). Ovali ed altre curve nei piani di Galois di caratteristica. In Acta Arith (pp. 423-449).
Bose, R. (1947). Mathematical Theory of the Symmetrical Factorial Design. In Sankhya 8 (pp. 107-166).
Brown, M. (2000). (Hyper)ovals and ovoids in projective spaces. Ghent University.
Casse, R. (2006). Projective Geometry: An Introduction. New York: Oxford Press.
Dembowski, P. (1968). Finite Geometries. Berlin: Springer Verlag.
Hirschfeld, J. (1979). Projective Geometries Over Finite Fields. London: Oxford Press.
Qvist, B. (1952). Some Remarks Concerning Curves of the Second Degree in a Finite Plane. Ann. Acad.
Sci. Fennicae. Ser. A , no. 134, 27pp.
Segre, B. (1954). Ovals in Fininte Projective Plane. In J. Canad, Math (pp. 414-416).
Weisstein, E. W. (n.d.). Conic Section. Retrieved april 23, 2009, from MathWorld- A Wolfram Web
Resource: http://mathworld.wolfram.com/ConicSection.html.
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