Polariy and Quadric Surfaces

The basic algebra for handling projective geometry is introduced in the Basics page. Important is the duality it so clearly expresses which enables the concept of polarity to be handled conveniently. The equation of a quadric surface is a general homogeneous equation of the second degree in the homogeneous coordinates (x, y, z, w):

In matrix form this is

which may be expressed as x'Qx = 0 where x is a column vector, x' is the corresponding row vector and Q is the symmetrical 4x4 matrix representing the quadric surface. Now consider the equation y'Qx = 0. We may regard Qx as the coordinates u of a plane so that y'u = 0 simply expresses the fact that Y lies in U (using Y to denote the geometric point represented by y, etc.). Now Y need not lie on Q, so the expression requires it to lie in the plane U, and any value of x that ensures that satisfies the equation. As X varies in accordance with this we obtain all the planes U in Y. Looking at the situation the other way round, y'Q is also a plane, moreover a fixed plane v since Y is fixed, and as vx = 0, all the points X must lie in V. Thus for every fixed point Y the quadric Q determines a plane of points which is the polar plane of Y. Conversely given a plane V, v = y'Q for a unique point Y, since y' = Q"v determines Y uniquely, where Q" is the inverse matrix of Q. Y is the pole of V, and this polar relationship is one-to-one provided the quadric is not singular. If Y lies on Q then one solution for x is x = y as then y'Qy = 0 by definition of the quadric, so the plane Qy is the tangent plane at Y since no other point X can satisfy y'Qx = 0 and lie on the surface. If Y lies outside the quadric then planes U exist which touch Q in which case, by what we have just seen, X lies on Q (recalling that X is given by u = Qx). This corresponds to the diagram in the Basics page where the polar of Y intersects Q when Y lies outside Q, and conversely if Y lies inside Q then no real plane U can be a tangent plane so V does not intersect Q.

A quadric may also be expressed in terms of plane coordinates as follows. In x'Qx = 0, u' = x'Q touching at x, and u = Qx also touching at x. Thus x'Qx = x'QQ"Qx = u'Q"u = 0, giving the class quadric as the envelope of its tangent planes U, which is the same as the surface described by X. This connection is only valid for non-singular quadrics, but of course any quadric also has a class equation, singular or not (e.g. a cone possesses tangent planes). A slight economy is possible as it is not necessary to divide by the determinant of Q when deriving Q" since we are using homogeneous coordinates, so in the literature we usually find the class quadric expressed as u'(Q)u = 0 where (Q) is the adjoint matrix of Q.

Projective Classification of Quadrics

There are three distinct types of quadric in purely projective geometry, distinct because no real projective transformation can transform a member of one type into a member of another. By suitable change of coordinates it is possible to reduce the equation of the quadric to canonical form (c.f. e.g. Reference 8, 9 or 14) where only the terms in the leading diagonal of Q are non-zero. This gives an equation

which is singular if any of a b c or d is zero (cones if one is zero, plane pairs if two are zero, two coincident planes for three zero).

Three possibilities exist for the relative signs of a b c and d:

1. one of opposite sign to the other three,

2. two positive and two negative,

3. all of the same sign.

In the first case, taking d to be negative and reverting to Cartesian coordinates by dividing by w², we have ax² + by² + cz² = d which is the equation of an ellipsoid. A similar result is obtained if instead a b or c is negative, recalling that infinity is not invariant so all central quadrics are projectively equivalent.

In the second case, setting a=A² etc. such that A B C D are all positive, we have for example when b and d are negative the equation (Ax + By)(Ax - By) = (Cz + Dw)(Cz - DW). This is satisfied by any line which is the intersection of the two planes Ax+By-Cz-Dw = 0 and Ax-By-Cz+Dw = 0, and also by the plane pairs Ax+By-Cz+Dw = 0, AX-By-Cz-Dw = 0. It is thus a ruled quadric, the two alternatives yielding the two complementary sets of generators.

In the third case the quadric contains no real points and is accordingly purely imaginary.

Cayley's Metric Quadric

We will now briefly outline Cayley's derivation of metric from projective geometry (which followed an initial insight of Laguerre). The problem in projective geometry is that the only numerical invariant is the cross-ratio (of four points, lines or planes), so this is all that is available for use in defining a quantity that is to be thought of as distance or length. Metric geometry is so-called precisely because its legitimate transformations leave lengths and angles invariant, and also areas and volumes. This is untrue in projective geometry. Cayley proposed restricting the allowable projective transformations to those leaving a quadric surface invariant, which is known as the absolute quadric G. Then given two points P and Q, the line PQ intersects G in two points I and J say. The cross-ratio (PQ,IJ) is now available for the definition of length, as when we make a transformation P and Q move to P' and Q', say, and I and J move to I' and J' such that I' and J' lie on G since it is invariant (as a whole, note, not pointwise) and the cross-ratio (P'Q',I'J') = (PQ,IJ). Cayley chose the following expression for length:

s = log(P Q, I J)/(2i)

so that s is indeed invariant. s is imaginary for real G , and in addition I and J need not be real, so that gives a non-Euclidean geometry. If however G is an imaginary quadric then I and J are always imaginary, so log(PQ,IJ) is complex and if it is purely imaginary then s is real. If we select the singular imaginary disk quadric at infinity given by x² + y² + z² = 0 = w then we recover the familiar Pythagorean expression for length, which is of course why the above expression was selected by Cayley. This is rather messy and limiting arguments must be used. The clearest exposition is given in Reference 15 for two dimensions. The result is readily generalised to three dimensions giving for Euclidean geometry the length s between x and y as:

which reduces to Pythagoras' Theorem for Cartesian coordinates with x₃ = y₃ = 1.

For the angle between two planes U and V Cayley took the two planes I and J in the line (U,V) which are tangential to G, and then used

cos(theta) = log(U V, I J)/(2i)

For Euclidean G the angle between u and v in terms of their plane coordinates is then

which is the familiar normalised inner product for the cosine.

Thus choosing an imaginary circle in the plane at infinity gives the Euclidean metric. A circle may be regarded as a singular class quadric known as a disk quadric, in the sense that there are an infinite number of axial pencils in its tangents which may be thought of as its tangent planes. Those planes are of course imaginary in the present case.

Counterspace Metric

We may dualise a disc quadric as follows: we have the dual of the plane of the circle as a point O, the duals of its tangents as lines in O forming a cone, and the duals of its "tangent planes" (axial pencils in the tangents) as the points of the lines in O i.e. we simply have a cone of points, which is of course more readily felt to be a quadric! For an imaginary circle O is still real (as polar of the real plane at infinity), but its tangents and "tangent planes" are imaginary, so the cone is imaginary apart from its real vertex. It is however a wrap of tangent planes, and therefore a class quadric, since it is dual to G which is treated as being composed of imaginary points. George Adams (Reference 5) suggested using this quadric, say H, as the absolute quadric defining the metric for a quite different kind of space. It is different from the usual notion of non-Euclidean geometry in two main ways: first of all it is based on a class quadric, and secondly that implies the fundamental metric relates planes rather than points. In Relativity the metric tensor g determines how infinitesimal coordinate displacements may be related to the corresponding infinitesimal distance displacements. This is necessary for general coordinates e.g. even for ordinary spherical polar coordinates. Now g is a symmetrical matrix and thus may also be regarded as a quadric, which is exactly the connection between Cayley's work and the metric tensor. Indeed a grasp of Cayley's work gives an immediate intuitive feel for the metric tensor. For a curved space the components of g are functions of the coordinates, which obey special conditions to ensure the matrix is also a tensor, and thus we can visualise the absolute quadric varying from point to point in a curved space, which is all that is meant by the forbidding formalism of the metric tensor. An important point is that the so-called signature of the quadric cannot change. This simply means that no real transformation can change its type e.g. from any of an ellipsoid, hyperboloid, paraboloid, ruled quadric or imaginary quadric to another of them.

Adam's H fully expresses the metric of a new kind of space, but noting that g is always assumed to define the distance between points whereas H defines a new kind of displacement between planes that is not an angle. We discover this by dualising the above expression for distance in ordinary space, giving the displacement between two planes u and v as:

We will refer to tau as the turn between U and V. That it is not an angle is clear from the fact that it may become infinite if u₃ or v₃ is zero. It is fully analogous to distance in that sense, but refers to planes. Adams studied it by means of projective measures.

The geometry characterised in this way is technically polar-Euclidean geometry as is clear from its derivation, and it is usually referred to as counterspace when thought of as the geometry of another kind of space.

So far we have not said anything about the location of O. It acts as infinity for counterspace, being the dual of the plane at infinity, but the process of dualising does not otherwise locate it. Indeed we are free to locate this real point anywhere in our ordinary space, and in so doing we establish a linkage between the two spaces. Lacking linkages the the two spaces are quite disjoint. We refer to such a linkage as a CSI (counterspace infinity). The turn tau becomes infinite if either of the planes contains O i.e. is "at infinity".

We can also dualise the above expression for the angle between two planes in space to give the separation between two points in counterspace, which we will call shift:

Thus sigma behaves just like an angle, which is to be expected from the dualising process. In other words points are separated in counterspace by a quantity which is never infinite, a notion that takes some getting used to. We may have parallel points in counterspace, but not parallel planes. Thus if the vectors x and y are parallel but the points are distinct then sigma is zero, the dual of two distinct parallel planes. A useful "crutch" is to see that the numerical value of sigma equals that of the angle between the lines XO and YO in space, regarding x and y as position vectors with respect to O. However, x and y are shift coordinates, not distance coordinates, and such a visualisation is only valid if the points are linked.

Thus far we have treated counterspace as a "flat" space since its metric H does not vary with position.