8.2
Null systems in 3-space
The matrix of a null system in projective 3-space is 4-by-4, invertible, and skew- symmetric. Note that there are six degrees of freedom in such a matrix, one for each entry above the diagonal. Since an overall scalar multiple doesn’t matter, we deduce that there are five degrees of freedom in the choice of a null system in 3-space. In this section, we study the geometry of such a null system. One good source for this material is Maxwell [28].
Recall from the previous section that all null systems in 3-space are equivalent, and thus have the same geometry, up to our choice of coordinate system. The five degrees of freedom determine only how the null system sits in 3-space.
There isn’t much to say about points or planes: The polar plane of any point P is a plane P∗ containing P; and the pole of any planeπ is a pointπ∗lying on π. The relationship between a line`and its polar line`∗ is more interesting.
Proposition 8.2-1 Given any null system in 3-space, the polar`∗ of a line`is ei- ther skew to`or coincides with`.
In the former case, we shall call the pair of lines{`, `∗}a skew-polar pair. In the latter case, where`=`∗, the line`is called self-polar.
Proof For any line`, whenever the point P lies on`, the plane P∗ must pass through the line`∗. Thus, as P slides along`, the polar plane P∗ rotates around
`∗, and does so in such a way that P∗ always passes through P. In typical cases,
the two lines`and`∗are skew, and hence form a skew-polar pair{`, `∗}.
Could it happen that a line `and its polar`∗ intersected without coinciding? No. As the point P slid along`in such a case, the polar plane P∗ would have to remain fixed at the unique plane spanned by`and`∗. But distinct points must have distinct polars.
We conclude that, whenever`and`∗intersect, they must actually coincide. In this case, as P slides along`, the plane P∗ rotates around that same line` = `∗. Thus, for any self-polar line `, the null system provides a projective correspon- dence between the range of points along`and the pencil of planes through`. tu By the way, the geometry of a polar system in 3-space is quite different from that of a null system [30]. In a polar system, just because a line`intersects its polar
`∗does not mean that the two must coincide. Indeed, a line intersects its polar just
when it is tangent to the quadric surface associated with the polar system, while it coincides with its polar only when it lies entirely inside — that is, is a generating line of — that quadric. But back to null systems.
Proposition 8.2-2 Given any null system in 3-space, let`be any line and let P be any point on`. The line`is self-polar just when`lies in the plane P∗.
106 CHAPTER 8. NULL SYSTEMS
Proof The line`∗always lies in the plane P∗. If the line`is self-polar, then` =
`∗, so the line`lies in P∗ also. Conversely, suppose that`does lie in P∗. Since
both`and`∗lie in P∗, they are not skew, so we know from Proposition 8.2-1 that they coincide — that is,`is self-polar. tu
If we fix a point P in space, we deduce that the lines through P that are self- polar are precisely those lying in the plane P∗. Thus, while there are 4 degrees of freedom in the choice of an arbitrary line in 3-space, there are 3 degrees of free- dom in the choice of a self-polar line. In fact, the set of self-polar lines forms a
general linear complex,2and knowing this linear complex is equivalent to know- ing the null system. Indeed, Maxwell talks more about linear complexes [31] than he does about null systems as such [32].
As a corollary of Proposition 8.2-2, we can show that there are no triangles with all three edges self-polar.
Proposition 8.2-3 If the points A, B, and C are distinct and all three of the lines
AB, AC, and BC are self-polar with respect to some fixed null system, then A, B and C are collinear.
Proof If A, B, and C were not collinear, the polar of each vertex of the triangle
ABC would have to be the plane ABC, since the two triangle edges that meet at
that vertex are self-polar. But distinct points must have distinct polars. tu Our next result gives us an easy way to prove that a line is self-polar.
Proposition 8.2-4 Given any null system in 3-space, if the lines{`, `∗}are a skew- polar pair, then any line m that meets both`and`∗ is self-polar.
Proof Let m be a line that meets both` and `∗, say at the points A and B re- spectively. Since m passes through A, the line m∗must lie in the plane A∗, which we know is the unique plane through`∗ that passes through A. Similarly, since m passes through B, the line m∗must lie in the plane B∗, which is the unique plane through` that passes through B. It follows that m∗ must coincide with the inter- section A∗∩B∗, which is precisely the line m = AB. Thus, m is self-polar. tu
So suppose that the two opposite edges` = AC and`∗ = B D of some tetra-
hedron ABC D form a skew-polar pair. It follows from Proposition 8.2-4 that each of the other four edges AB, BC, CD, and D A of the tetrahedron is self-polar. We close this section with a converse to that result.
Proposition 8.2-5 Given any null system in 3-space, let ABC D be a skew quadri- lateral all four of whose edges are self-polar lines. Then, the two diagonal lines {AC,B D} form a skew-polar pair.
2The word ‘general’ here means ‘nonspecial’, a special linear complex being the 3-parameter