Another branch of projective geometry concerns lines. There is a four-fold infinity of lines in space, of which we may form a subset. A subset containing a threefold infinity of lines is called a LINE COMPLEX. An example which is simple to define is the TETRAHEDRAL COMPLEX: given a tetrahedron, a general line in space cuts its four faces in four points:
These four points have a cross ratio which may be any real number. We may select the set of lines all of which intersect the tetrahedron
in the same cross ratio. Since there are infinitely many possible cross ratios we thus select a three-fold infinity of lines from the
four-fold infinity of all possible lines. The resulting line complex has a definite structure such that through any point of space
it possesses a set of lines forming a cone, while in any plane of space it possesses a set of lines enveloping a conic.
Just as we have polarity wrt (with respect to) conics and quadrics, so we may have polarity wrt a line complex. This means that if we choose any line u then the complex determines a line u' polar to u. This is accomplished by taking the axial pencil of planes in u, and for each such plane finding the point P polar to u wrt the conic of the complex in that plane:
The points P in all the planes of the pencil lie on a straight line u' which is the polar of u. ( If u happens to be a line of the
complex then it is self-polar).
We may then find the polar of u', which is a third line u", and so on. An interesting question then arises: what figure is formed by such a sequence of polar lines?
The answer turns out to be quite simple: it is a ruled quadric which is self-polar wrt the tetrahedron. This means self-polar in the sense that the faces of the tetrahedron and their opposite vertices are harmonic wrt the quadric. Although we started with a discrete set of lines u,u',u''... it turns out that if we take any line v on a self-polar quadric Q then its polar line v' wrt the complex also lies on Q.
Since we could have chosen any cross ratio to define the complex, and since a quadric Q is self-polar wrt the tetrahedron irrespective of that cross ratio, we see that the lines on Q form a self-polar set for all possible tetrahedral complexes sharing the same base tetrahedron (such complexes are known as COSINGULAR COMPLEXES). Of course a given line v of Q will have different lines of Q as its polar for different cosingular complexes.
I found this result myself and have not seen it anywhere in the literature. Has anyone seen it published elsewhere?
The proof is available from me (via email).