Conics

The Conic as a Locus of Points

We have seen that two pencils of lines may be projectively related. Say we have two such pencils, one {a} lying on the point P and one {a'} lying on the point P. What will be the locus of intersections of corresponding lines?

We may get some insight if we look at a few examples in the affine plane:
Let the pencil of lines on P=(0,-1) be y=kx-1
Let the pencil of lines on P'=(0,+1) be y=1-2x/k
The pencils are projectively related because the slopes are related by an a fractional linear transformation: k -> -2/k

To find the locus of all possible intersections, we eliminate k from these equations. (y+1)/x=k=-2x/(y-1);
yy-1=-2xx;
2xx+yy=1.
This is the equation of an ellipse.

What if our two pencils are y=kx-1 and y=1+2x/k? Then we will have -2xx+yy=1, which is the equation of a hyperbola.

We might guess from this that the locus of these intresections will always be a conic section. Alternatively, we state this as a defintion:

The locus of the points of intersection of the corresponding lines in two projective pencils is a conic.

We will sometimes call this a point conic, for reasons that will become clear soon. Then we can prove that such a locus, when interpreted in the affine plane, is exactly a conic in the sense of analytic geometry.

Now let's go back to the projective plane and see how this definition works.

Once again we take a pencil {a} on P and a pencil {a'} on P'. The projective correspondence between these two pencils of lines will be determined once we have three pairs of lines in the correspondence. Alternatively, five points will determine this correspondence: say we are given P, P', Q, R, and S; we set up a correspondence between the pencil on P and the pencil on P' by PQ->P'Q, PR->P'R, and PS->P'S. We can construct other lines in the correspondence, and by intersecting corresponding pairs can find other points on the conic. Then all of the points we have named will lie on the conic. In particular, P will lie on the conic as the intersection of the line PP' (considered as part of the pencil at P') and whatever line in the pencil in P corresponds to it.

We can generalize this to the following theorem:

Given five points in the plane, there is exactly one conic which lies on all of them.