Projective Geometry may also be studied by means of algebra. Linear transformations are expressed as matrices, and the transformation of a
point or plane is accomplished by multiplying the vector representing it by the transformation matrix.
The Cartesian coordinates of a point may be expressed as (x,y,z) with respect to the three orthogonal axes. The problem encountered in using them, however, is that ideal points at infinity cannot be handled because x,y or z (or all three) become infinite. If a point moves towards infinity in a fixed direction then the ratios x : y : z remain constant. We may introduce a fourth number w and re-express the coordinates as x/w : y/w : z/w, noting that the ratios are unaffected. If we multiply all coordinates by a constant k the ratios are still unaffected. We now re-express the point as (x,y,z,w) as if we were working in four dimensions i.e. we regard w as a fourth coordinate. If w becomes zero then we see that x/w, y/w and z/w each become infinite to give us a point at infinity, but instead of retaining these improper ratios we instead express that fact as (x,y,z,0). This formulation retains intact the ratios of x : y : z of the point before it reached infinity, and we use w=0 to indicate we have gone to infinity. Thus for each direction in space (x,y,z,0) is unique, the twofold infinity of ratios x : y : z representing that direction and (x,y,z,0) its ideal point. Two aspects should be noted:
1. (x,y,z,1) returns us to the Cartesian coordinates when w=1 is discarded;
2. (kx,ky,kz,kw) is the same point as (x,y,z,w) as we are now only interested in ratios.
These coordinates are known as homogeneous coordinates because they still refer to three dimensions despite the use of four coordinates, and the coordinates are homogeneous in the sense that they are are not absolute but enter into equations fully symmetrically, just as a homogeneous equation contains all products of its variables to a fixed overall power.
(x,0,0,0) is the point at infinity on the x-axis, and similarly for the y and z axes.
Since we may divide throughout by k=x this simplifies to (1,0,0,0).
(0,0,0,1) is the origin.
Once we switch to homogeneous coordinates the axes need not remain orthogonal, and we end up with a tetrahedron of reference with vertices (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1). All connection with Cartesian coordinates is then lost as distances can no longer be associated with x,y,z and w. This expresses the non-metric nature of projective geometry. The infinite plane is no longer defined as the plane w=0 but can be any face of the tetrahedron, consistent with the fact that an infinite plane is not defined for projective geometry, only for affine and metric geometry.
If we take y=0 then we have all the points in the XZW plane. If we take x+y=0 then we have all the points in the plane for which x=-y. Generally a linear equation in x,y,z,w yields a plane i.e.
kx + ly + mz + nw = 0
for constant k,l,m,n is the equation of a plane. Now suppose we hold x,y,z,w constant and vary k,l,m,n while satisfying the equation. Clearly we obtain all possible quadruples (k,l,m,n) satisfying the equation for that fixed point (x,y,x,w) i.e. all possible planes containing (x,y,x,w), from which it is clear that (k,l,m,n) may be regarded as the coordinates of the planes. The duality of point and plane is beautifully expressed by the symmetry of the equation. The meaning of these coordinates may be appreciated if we think of the Cartesian special case with w=1. On the x-axis y=z=0 so x=-n/k, and similarly on the y- and z-axes, so the plane represented by the coordinates is as illustrated below.
A linear transformation (x',y',z',w') = f(x,y,z,w) is such that x y z w enter the function f homogeneously to the first power. This means we cannot have terms such as x2, xy, yw or xyz for example. Thus x' = ax+by+cz+dw for some constants a b c d, and similarly for y', z' and w'. The most convenient way of collecting together these linear equations for x' y' z' and w' is to express them in matrix form:
recalling that the inner product of a row of the square matrix with the right hand column vector gives the corresponding term in the left
hand column vector. We may denote this also as x'=Ax where emphasised capitals denote matrices and lower case letters represent
The same transformation may be applied to a plane (s,t,u,v) :
It has long been known that projective geometry may be expressed in terms of linear transformations such as these. However the actual
geometry is easily lost sight of if we are not careful !
Generally the point x' is distinct from x, but we may ask if there are any points that correspond to themselves. If so then such a point p is such that p=Tp. Because we are concerned with ratios rather than absolute values it is more accurate to set kp=Tp for some constant k. To solve this for p we need to multiply the left hand side by the unit matrix I (which has 1 in the leading diagonal and 0 elsewhere e.g. in the above square matrix that would mean a=f=l=r=1 and b=c=d= ... =q=0). Then we have the equation (T-kI)p=0, and as we do not want p=0 then T-kI=0. This really consists of four simultaneous equations in k which give rise to the characteristic equation which is a fourth order equation in k. The four roots are known as the characteristic values or eigenvalues and from them it is possible to derive four vectors p which transform into themselves i.e. in geometric parlance there are four invariant points. There can be no more than four provided the roots are distinct, and furthermore they need not be real. Applying the same idea to find the invariant planes u gives (T-kI)u=0 and hence the very same equation T-kI=0. The four planes must evidently each contain three of the invariant points as such a triple determines an invariant plane. The nett result is an invariant tetrahedron with four invariant vertices, four invariant planes and six invariant edges. It is non-degenerate if the four characteristic roots are real and distinct as then the invariant points are also real and distinct. If however some of the roots are complex the tetrahedron possesses imaginary elements. In particular the so-called semi-imaginary tetrahedron arises when two of the roots are complex conjugates, as then only two real invariant planes, points and lines arise. Pairs of equal real roots give rise to lines of invariant points (which are also the axes of axial pencils of invariant planes). This is all described for example in Reference 14.
Denoting a square transformation matrix such as that above by T, a path curve arises when it is repeatedly applied to an initial point. If that is a then a new point b=Ta arises. We now apply the same transformation to b to give c=Tb=TTa and so on. Continuing in this way the series of points a b c ... are found to lie on a curve. The nature of the curve depends upon the characteristic roots of the matrix T. If all four are real and distinct then there are four invariant points as we have just seen, and the curve passes from one to another of them. If two are conjugate imaginary then the egg and vortex spirals arise that are described on the Path Curve page.
Felix Klein discovered path curves and Sophus Lie gave the transformation for a continuous curve in place of the discrete recursive approach above which places points on that curve.
Instead we may start with an initial plane u and transform it to v=Tu and so on. This results in the polar of a locus which is called a developable, consisting of a single-parameter sequence of planes that are the osculating planes to the path curve, but the latter is now more strictly referred to as the cuspidal edge of the developable. An osculating plane has triple contact with the curve.