ANALYTIC REPRESENTATION - PROJECTIVE GEOMETRY. AXIOMATICS. NON-EUCLIDEAN GEOMETRIES - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NON-EUCLIDEAN GEOMETRIES

§6. ANALYTIC REPRESENTATION

1. Introductory Remarks

In the early development of projective geometry there was a strong tendency to build everything on a synthetic and “purely geometric” basis, avoiding the use of numbers and of algebraic methods. This program met with great difficulties, since there always remained places where some algebraic formulation seemed unavoidable. Complete success in building up a purely synthetic projective geometry was only attained toward the end of the nineteenth century, at a rather high cost in complication. In this respect the methods of analytic geometry have been much more successful. The general tendency in modern mathematics is to base everything on the number concept, and in geometry this tendency, which started with Fermat and Descartes, has had decisive triumphs. Analytic geometry has developed from the status of a mere tool in geometrical reasoning to a subject where the intuitive geometrical interpretation of the operations and results is no longer the ultimate and exclusive goal, but has rather the function of a guiding principle that aids in suggesting and understanding the analytical results. This change in the meaning of geometry is the product of a gradual historical growth that has greatly enlarged the scope of the classical geometry, and at the same time has brought about an almost organic union of geometry and analysis.

In analytic geometry the “coördinates” of a geometrical object are any set of numbers which characterize that object uniquely. Thus a point is defined by giving its rectangular coördinates x, y or its polar coördinates ρ, θ, while a triangle can be defined by giving the coördinates of its three vertices, which requires six coördinates in all. We know that a straight line in the x, y-plane is the geometrical locus of all points P (x, y) (see p. 75 for this notation) whose coördinates satisfy some linear equation

(1) ax + by + c = 0.

We may therefore call the three numbers a, b, c the “coördinates” of this line. For example, a = 0, b = 1, c = 0 define the line y = 0, which is the x-axis; a = 1, b = –1, c = 0 define the line x = y, which bisects the angle between the positive x-axis and the positive y-axis. In the same way, quadratic equations define “conic sections”:

x2 + y2 = r2

a circle, center at origin, radius r,

(x – a)2 + (yb)2 = r2

a circle, center at (a, b), radius r,

image

an ellipse,

etc.

The naive approach to analytic geometry is to start with purely “geometric” concepts—point, line, etc.—and then to translate these into the language of numbers. The modern viewpoint is the reverse. We start with the set of all pairs of numbers x, y and call each such pair a point, since we can, if we choose, interpret or visualize such a pair of numbers by the familiar notion of a geometrical point. Similarly, a linear equation between x and y is said to define a line. Such a shift of emphasis from the intuitive to the analytical aspect of geometry opens the way for a simple, yet rigorous, treatment of the points at infinity in projective geometry, and is indispensable for a deeper understanding of the whole subject. For those readers who possess a certain amount of preliminary training we shall give an account of this approach.

*2. Homogeneous Coördinates. The Algebraic Basis of Duality

In ordinary analytic geometry, the rectangular coördinates of a point in the plane are the signed distances of the point from a pair of perpendicular axes. This system breaks down for the points at infinity in the extended plane of projective geometry. Hence if we wish to apply analytic methods to projective geometry it is necessary to find a coördinate system which shall embrace the ideal as well as the ordinary points. The introduction of such a coördinate system is best described by supposing the given X, Y-plane π imbedded in three-dimensional space, where rectangular coördinates x, y, z (the signed distances of a point from the three coördinate planes determined by the x, y, and z axes) have been introduced. We place π parallel to the x, y coördinate plane and at a distance 1 above it, so that any point P of π will have the three-dimensional coördinates (X, Y, 1). Taking the origin O of the coördinate system as the center of projection, we note that each point P determines a unique line through O and conversely. (See p. 184. The lines through O and parallel to π correspond to the points at infinity of π.)

We shall now describe a system of “homogeneous coördinates” for the points of π. To find the homogeneous coördinates of any ordinary point P of π, we take the line through O and P and choose any point Q other than O on this line (see Fig. 93). Then the ordinary three-dimensional coördinates x, y, z of Q are said to be homogeneous coördinates of P. In particular, the coördinates (X, Y, 1) of P itself are a set of homogeneous coördinates for P. Moreover, any other set of numbers (tX, tY, t) with t ≠ 0 will also be a set of homogeneous coördinates for P, since the coördinates of all points on the line OP other than O will be of this form. (We have excluded the point (0, 0, 0) since it lies on all lines through O and does not serve to distinguish one from another.)

This method of introducing coördinates in the plane requires three numbers instead of two to specify the position of a point, and has the further disadvantage that the coördinates of a point are not determined uniquely but only up to an arbitrary factor t. But it has the great advantage that the points at infinity in π are now included in the coördinate representation. A point P at infinity in π is determined by a line through O parallel to π. Any point Q on this line will have coördinates of the form (x, y, 0). Hence the homogeneous coördinates of a point at infinity in π are of the form (x, y, 0).

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Fig. 93. Homogeneous coördinates.

The equation in homogeneous coördinates of a straight line in π is readily found by observing that the lines joining O to the points of this line lie in a plane through O. It is proved in analytic geometry that the equation of such a plane is of the form

ax + by + cz = 0.

Hence this is the equation in homogeneous coördinates of a straight line in π.

Now that the geometrical model of the points of π as lines through O has served its purpose, we may lay it aside and give the following purely analytic definition of the extended plane:

A point is an ordered triple of real numbers (x, y, z), not all of which are zero. Two such triples, (x1, y1, z1) and (x2, y2, z2), define the same point if for some t ≠ 0,

x2 = tx1,

y2 = ty1,

z2 = tz1

In other words, the coördinates of any point may be multiplied by any non-zero factor without changing the point. (It is for this reason that they are called homogeneous coördinates.) A point (x, y, z) is an ordinary point if z ≠ 0; if z = 0, it is a point at infinity.

A straight line in π consists of all points (x, y, z) which satisfy a linear equation of the form

(1′) ax + by + cz = 0,

where a, b, c are any three constants, not all zero. In particular, the points at infinity in π all satisfy the linear equation

(2) z = 0.

This is by definition a line, and is called the line at infinity in π. Since a line is defined by an equation of the form (1′), we call the triple of numbers (a, b, c) the homogeneous coördinates of the line (1′). It follows that (ta, tb, tc), for any t ≠ 0, are also coördinates of the line (1′), since the equation

(3) (ta)x + (tb)y + (tc)z = 0

is satisfied by the same coördinate-triples (x, y, z) as (1′).

In these definitions we observe the perfect symmetry between point and line: each is specified by three homogeneous coördinates (u, v, w). The condition that the point (x, y, z) lie on the line (a, b, c) is that

ax + by + cz = 0,

and this is likewise the condition that the point whose coördinates are (a, b, c) lie on the line whose coördinates are (x, y, z). For example, the arithmetical identity

2 · 3 + 1 · 4 – 5 · 2 = 0

may be interpreted equally well as meaning that the point (3, 4, 2) lies on the line (2, 1, –5) or that the point (2, 1, –5) lies on the line (3, 4, 2). This symmetry is the basis of the duality in projective geometry between point and line, for any relationship between points and lines becomes a relationship between lines and points when the coördinates are properly re-interpreted. In the new interpretation the previous coördinates of points and lines are now thought of as representing lines and points respectively. All the algebraic operations and results remain the same, but their interpretation gives the dual counterpart of the original theorem. It is to be noted that this duality does not hok1 in the ordinary plane of two coördinates X, Y, since the equation of a straight line in ordinary coördinates

aX + bY + c = 0

is not symmetrical in X, Y and a, b, c. Only by including the points and the line at infinity is the principle of duality perfectly established.

To pass from the homogeneous coördinates x, y, z of an ordinary point P in the plane π to ordinary rectangular coördinates, we simply set X = x/z, Yy/z. Then X, Y represent the distances from the point P to two perpendicular axes in π, parallel to the x- and y-axes, as shown in Figure 93. We know that an equation of the form

aX + bY + c = 0

will represent a straight line in π. On substituting X = x/z, Y = y/z and multiplying through by z we find that the equation of the same line in homogeneous coördinates is, as stated on page 195,

ax + by + cz = 0.

Thus the equation of the line 2x – 3y + z = 0 in ordinary rectangular coördinates X, Y is 2X – 3Y + 1 = 0. Of course, the latter equation fails for the point at infinity on this line, one set of whose homogeneous coördinates is (3, 2, 0).

One thing remains to be said. We have succeeded in giving a purely analytic definition of point and line, but what of the equally important concept of projective transformation? It may be proved that a projective transformation of one plane onto another as defined on page 169 is given analytically by a set of linear equations,

    xa1x + b1y + c1z,

(4)    ya2x + b2y + c2z,

    za3x + b2y + c3z,

connecting the homogeneous coördinates x′, y′, z′ of the points in the plane π′ with the homogeneous coördinates x, y, z of the points in the plane π. From our present point of view we may now define a projective transformation as one given by any set of linear equations of the form (4). The theorems of projective geometry then become theorems on the behavior of number triples (x, y, z) under such transformations. For example, the proof that the cross-ratio of four points on a line is unchanged by such transformations becomes simply an exercise in the algebra of linear transformations. We cannot go further into the details of this analytic procedure. Instead we shall return to the more intuitive aspects of projective geometry.