After studying points and lines in a plane, the next type of structure to look at are those of a quadratic nature (circles, ellipses, parabolas, etc.). The term quadratic refers to the fact that the coordinates of the points in these structures satisfy quadratic equations.

A *quadratic form***Q** on a vector space **V** defined over the field **F** is a map **Q**: **V** -> **F** which satisfies **Q**(k*v*) = k^{2}**Q**(*v*), for all vectors *v* in **V** and scalars k in **F** and **Q**(*v*+*w*) = **Q**(*v*) + **Q**(*w*) + b(*v*,*w*), where b(*v*,*w*) is a bilinear form. In terms of coordinates, if *v* = (x,y,z) then the most general quadratic form is:

**Examples:** Consider PG(2,5), and the quadratic form **Q**(x,y,z) = x^{2} + y^{2} + z^{2}. The quadric determined by this quadratic form consists of the points with coordinates (1,2,0), (1,3,0), (2,0,1), (0,2,1), (3,0,1), and (0,3,1).

In the same plane, the quadratic form **Q**(x,y,z) = x^{2} - z^{2} gives the quadric with points (0,1,0), (1,0,1),(1,1,1),(1,2,1),(1,3,1),(1,4,1),(4,0,1),(4,1,1),(4,2,1),(4,3,1), and (4,4,1). Note that these points lie on the two lines with homogeneous equations x + z = 0 and x - z = 0 (or with non-homogeneous equations x = 1 and x = 4).

Again in this plane, the quadratic form **Q**(x,y,z) = x^{2} - 3z^{2} gives the quadric whose only point is (0,1,0).

There are 5 possible types of point sets for a quadric in a projective plane. The first four types are known as ** degenerate** quadrics. A degenerate quadric may be a pair of lines, a single line (repeated twice), a point or the empty set (but this last only in infinite planes). A non-degenerate quadric is called a

It may be shown that no three of the points of a conic lie on a line and through each point of the conic there passes exactly one tangent line (a line which meets the conic in only one point). In general, these geometric conditions can be satisfied by sets of points which are not conics as well. Also, if the projective plane is finite, i.e., PG(2,q), then we have that a conic consists of q+1 points.

We will now examine what happens to a conic when we pass to the Desarguesian affine plane AG(2,F) obtained by removing the points of the line z = 0. First, consider the intersection of the line z = 0 with the general quadric Ax^{2} + Bxy + Cy^{2} + Dxz + Eyz + Fz^{2} = 0. We must determine the solutions of

**Example:** In our example of a conic in PG(2,5) we see that there are two points on the line z = 0, so the remaining 4 points form a hyperbola in AG(2,5). This can also be seen without determining all the points of the conic. From the quadratic form in that example we see that B = 0, A = C = 1. Thus, the discriminant is 0 -4(1)(1) = -4 = 1 in **Z**_{5} which is a square.

It is easy to see what the degenerate quadrics will be in an affine plane. If in the projective plane the degenerate quadric is a point, then in the affine plane we either get a point or the empty set (when the point is on the line z = 0). If the degenerate quadric is a repeated line, we get either a repeated line or the empty set (if the repeated line is z = 0). Finally, if the degenerate quadric is a pair of lines we get either a pair of lines, a pair of parallel lines (when the point of intersection is on z = 0) or a single line (when one of the pair is the line z = 0).