We define a new geometry based on a Klein quadratic set.

Let *C* and *C** be the two equivalence classes of planes of a Klein quadratic set *Q* in a 5-dimensional projective space **P**. We define the geometry **S** as follows:

- the
*points*of**S**are the planes of*C*. - the
*lines*of**S**are the points of*Q*. - the
*planes*of**S**are the planes of*C**. - the incidence between a line of
**S**and either a point or plane of**S**is induced by the incidence in**P**. - a point
_{1}of**S**is incident with a plane_{2}of**S**iff the planes_{1}and_{2}of**P**are not disjoint (then by 4.5.5 they intersect each other in a line of**P**).

**Lemma 4.6.1**: Let *Q* be a Klein quadratic set.

- Each
*Q*-line is on exactly one plane of each equivalence class. - If
**P**is finite of order q, then each point of*Q*is on exactly q+1 planes of each equivalence class.

**Theorem 4.6.2**: The geometry **S** is a 3-dimensional projective space; more precisely, **S** is isomorphic to a 3-dimensional subspace of **P**.

- q(av) = a
^{2}q(v) for all v in V and a in F. - the map B: V × V -> F defined by
B(v,w) = q(v+w) - q(v) - q(w) is a symmetric bilinear form.

*Examples*:

Consider the 3 dimensional vector space V over the reals.

= 2xx' + 2yy' + 2zz'.

Let v' = (a,b,c), then

= 2xx' + 2yy' + 2zz' + 2ax' + 2by' + 2cz' = B(v,w) + B(v',w).

B(dv,w) = 2dxx' + 2dyy' + 2dzz' = d B(v,w).

Consider the 3 dimensional vector space V over the reals.

= xy' + x'y - 2zz'.

Let v' = (a,b,c), then

= xy' + x'y -2zz' + ay' + x'b – 2cz' = B(v,w) + B(v',w).

B(dv,w) = dxy' + x'dy – 2dzz' = d B(v,w).

**Lemma 4.7.1**: Let {v_{1},...,v_{n}} be a basis of the vector space V.

- If a
_{ij}in F, then a quadratic form is defined by the following rule:q( b _{i}v_{i}) = a_{ij}b_{i}b_{j}. - Conversely, for any quadratic form q there are elements a
_{ij}in F such that for all v in V we have the above form.

**Def**: A quadratic form is * nondegenerate* if q(v) = 0 and B(x,v) = 0 for all x in V implies that v = O.

**Def**: Let q be a quadratic form of the vector space V. The * quadric* of the projective space P(V) corresponding to q is the set of all points <v> of P(V) with q(v) = 0.

*Examples*:

By Theorem 2.4.4 the points on a regulus in a 3-dimensional projective space can be given coordinates that satisfy,

The quadratic form x_{0}^{2} – x_{1}^{2} - x_{2}^{2} yields a quadric in the plane which is an oval. The quadratic form x_{0}^{2} – x_{1}^{2} + x_{2}^{2} yields a quadric in the plane which is the union of two lines.

In a 3-dimensional projective space,
the quadric given by x_{0}^{2} - x_{1}^{2} – x_{2}^{2} – x_{3}^{2} is an ovoid,
while the quadric given by x_{0}^{2} + x_{1}^{2} - x_{2}^{2} - x_{3}^{2} is a hyperboloid.

**Lemma 4.7.2**: Let q be a quadratic form of a vector space V, and let *Q* be the corresponding quadric in P(V). Then, if a line g contains three points of *Q*, each point of g lies in *Q*.

**Def**: Let q be a quadratic form of the vector space V. For a non-zero vector v in V we define

**Lemma 4.7.3**: Let q be a quadratic form of the vector space V, and let *Q* be the corresponding quadric of P(V). Then for each non-zero vector v in V we have:

- <v>
^{perp}is a subspace of V, hence also a subspace of P(V). - <v>
^{perp}is either a hyperplane or equal to V. - Suppose that <v> in
*Q*. Then each line of <v>^{perp}through <v> is a tangent line of*Q*. - Suppose that <v> in
*Q*. Then each line through <v> that is not contained in <v>^{perp}intersects*Q*in precisely one further point.

**Theorem 4.7.4**: Each quadric is a quadratic set.

A fundamental theorem of Buekenhout states that any nondegenerate quadratic set is either a quadric or an ovoid. We have already proved this for hyperbolic sets in 3-dimensional projective spaces (by showing that these non-ovoids are quadrics.) In the next section we prove an analogous statement for Klein quadratic sets.

In the case d = 2 the corresponding theorem was first proved by B. Segre. In the case d = 3 this was first done by Barlotti and Panella. Segre's theorem is particularly remarkable.

**Theorem 4.7.5**: Any oval in a finite Desgarguesian projective plane of odd order is a conic (a nonempty, nondegenerate quadric in a projective plane).