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

§4. PARALLELISM AND INFINITY

1. Points at Infinity as “Ideal Points”

An examination of the previous section will disclose that some of our arguments fail if certain lines in the constructions, supposed to be produced until they intersect, are in fact parallel. For example, in the construction above the fourth harmonic point D fails to exist if the line IF is parallel to AB. Geometrical reasoning seems to be hampered at every step by the fact that two parallel lines do not intersect, so that in any discussion involving the intersection of lines the exceptional case of parallel lines has to be considered, and formulated separately. Likewise, projection from a center O has to be distinguished from parallel projection, which requires separate treatment. If we really had to go into a detailed discussion of every such exceptional case, projective geometry would become very complicated. We are therefore led to try an alternative—namely, to find extensions of our basic concepts that will eliminate the exceptions.

Here geometrical intuition points the way: if a straight line that intersects another is rotated slowly towards a parallel position, then the point of intersection of the two lines will recede to infinity. We might naively say that the two lines intersect at a “point at infinity.” The essential thing is then to give this vague statement a precise meaning, so that points at infinity, or, as they are sometimes called, ideal points, can be dealt with exactly as though they were ordinary points in the plane or in space. In other words, we want all rules concerning the behavior of points, lines, planes, etc. to persist, even when these geometric elements are ideal. To achieve this goal we can proceed either intuitively or formally, just as we did in extending the number system, where one approach was from the intuitive idea of measuring, and another from the formal rules of arithmetical operations.

First, let us realize that in synthetic geometry even the basic concepts of “ordinary” point and line are not mathematically defined. The so-called definitions of these concepts which are frequently found in textbooks on elementary geometry are only suggestive descriptions. In the case of ordinary geometrical elements our intuition makes us feel at ease as far as their “existence” is concerned. But all we really need in geometry, considered as a mathematical system, is the validity of certain rules by means of which we can operate with these concepts, as in joining points, finding the intersection of lines, etc. Logically considered, a “point” is not a “thing in itself,” but is completely described by the totality of statements by which it is related to other objects. The mathematical existence of “points at infinity” will be assured as soon as we have stated in a clear and consistent manner the mathematical properties of these new entities, i.e. their relations to “ordinary” points and to each other. The ordinary axioms of geometry (e.g. Euclid’s) are abstractions from the physical world of pencil and chalk marks, stretched strings, light rays, rigid rods, etc. The properties which these axioms attribute to mathematical points and lines are highly simplified and idealized descriptions of the behavior of their physical counterparts. Through any two actual pencil dots not one but many pencil lines can be drawn. If the dots become smaller and smaller in diameter then all these lines will have approximately the same appearance. This is what we have in mind when we state as an axiom of geometry that “through any two points one and only one straight line may be drawn”; we are referring not to physical points and lines but to the abstract and conceptual points and lines of geometry. Geometrical points and lines have essentially simpler properties than do any physical objects, and this simplification provides the essential condition for the development of geometry as a deductive science.

As we have noticed, the ordinary geometry of points and lines is greatly complicated by the fact that a pair of parallel lines do not intersect in a point. We are therefore led to make a further simplification in the structure of geometry by enlarging the concept of geometrical point in order to remove this exception, just as we enlarged the concept of number in order to remove the restrictions on subtraction and division. Here also we shall be guided throughout by the desire to preserve in the extended domain the laws which governed the original domain.

We shall therefore agree to add to the ordinary points on each line a single “ideal” point. This point shall be considered to belong to all the lines parallel to the given line and to no other lines. As a consequence of this convention every pair of lines in the plane will now intersect in a single point; if the lines are not parallel they will intersect in an ordinary point, while if the lines are parallel they will intersect in the ideal point common to the two lines. For intuitive reasons the ideal point on a line is called the point at infinity on the line.

The intuitive concept of a point on a line receding to infinity might suggest that we add two ideal points to each line, one for each direction along the line. The reason for adding only one, as we have done, is that we wish to preserve the law that through any two points one and only oneline may be drawn. If a line contained two points at infinity in common with every parallel line then through these two “points” infinitely many parallel lines would pass.

We shall also agree to add to the ordinary lines in a plane a single “ideal” line (also called the line at infinity in the plane), containing all the ideal points in the plane and no other points. Precisely this convention is forced upon us if we wish to preserve the original law that through every two points one line may be drawn, and the newly gained law that every two lines intersect in a point. To see this, let us choose any two ideal points. Then the unique line which is required to pass through these points cannot be an ordinary line, since by our agreement any ordinary line contains but one ideal point. Moreover, this line cannot contain any ordinary points, since an ordinary point and one ideal point determine an ordinary line. Finally, this line must contain all the ideal points, since we wish it to have a point in common with every ordinary line. Hence this line must have precisely the properties which we have assigned to the ideal line in the plane.

According to our conventions, a point at infinity is determined or is represented by any family of parallel lines, just as an irrational number is determined by a sequence of nested rational intervals. The statement that the intersection of two parallel lines is a point at infinity has no mysterious connotation, but is only a convenient way of stating that the lines are parallel. This way of expressing parallelism, in the language originally reserved for intuitively different objects, has the sole purpose of making the enumeration of exceptional cases superfluous; they are now automatically covered by the same kind of linguistic expressions or other symbols that are used for the “ordinary” cases.

To sum up: our conventions regarding points at infinity have been so chosen that the laws governing the incidence relation between ordinary points and lines continue to hold in the extended domain of points, while the operation of finding the point of intersection of two lines, previously possible only if the lines are not parallel, may now be performed without restriction. The considerations which led to this formal simplification in the properties of the incidence relation may seem somewhat abstract. But they are amply justified by the result, as the reader will see in the following pages.

2. Ideal Elements and Projection

The introduction of the points at infinity and the line at infinity in a plane enables us to treat the projection of one plane onto another in a much more satisfactory way. Let us consider the projection of a plane π onto a plane π′ from a center O (Fig. 83). This projection establishes a correspondence between the points and lines of π and those of π′. To every point A of π corresponds a unique point A′ of π′, with the following exceptions: if the projecting ray through O is parallel to the plane π′, then it intersects π in a point A to which no ordinary point of π′ corresponds. These exceptional points of π lie on a line l to which no ordinary line of π′ corresponds. But these exceptions are eliminated if we make the agreement that to A corresponds the point at infinity in π′ in the direction of the line OA, and that to l corresponds the line at infinity in π′. In the same way, we assign a point at infinity in π to any point B′ on the line m′ in π′ through which pass all the rays from O parallel to the plane π. To m′ itself will correspond the line at infinity in π. Thus, by the introduction of the points and line at infinity in a plane, a projection of one plane onto another establishes a correspondence between the points and lines of the two planes which is biunique without exception. (This disposes of the exceptions mentioned in the footnote on p. 169.) Moreover, it is easily seen to be a consequence of our agreement that a point lies on a line if and only if the projection of the point lies on the projection of the line. Hence all statements about collinear points, concurrent lines, etc. that involve only points, lines, and the incidence relation, are seen to be invariant under projection in the extended sense. This enables us to operate with the points at infinity in a plane π simply by operating with the corresponding ordinary points in a plane π′ coördinated with π by a projection.

Fig. 83. Projection into elements at infinity.

* The interpretation of the points at infinity of a plane π by means of projection from an external point O onto ordinary points in another plane π′ may be used to give a concrete Euclidean “model” of the extended plane. To this end we merely disregard the plane π′ and fix our attention onπ and the lines through O. To each ordinary point of π corresponds a line through O not parallel to π; to each point at infinity of π corresponds a line through O parallel to π. Hence to the totality of all points, ordinary and ideal, of π corresponds the totality of all lines through the point O, and this correspondence is biunique without exception. The points on a line of π will correspond to the lines in a plane through O. A point and a line of π will be incident if and only if the corresponding line and plane through O are incident. Hence the geometry of incidence of points and lines in the extended plane is entirely equivalent to the geometry of incidence of the ordinary lines and planes through a fixed point of space.

*In three dimensions the situation is similar, although we can no longer make matters intuitively clear by projection. Again we introduce a point at infinity associated with every family of parallel lines. In each plane we have a line at infinity. Next we have to introduce a new element, theplane at infinity, consisting of all points at infinity of the space and containing all lines at infinity. Each ordinary plane intersects the plane at infinity in its line at infinity.

3. Cross-Ratio with Elements at Infinity

A remark must be made about cross-ratios involving elements at infinity. Let us denote the point at infinity on a straight line l by the symbol ∞. If A, B, C are three ordinary points on l, then we may assign a value to the symbol (ABC ∞) in the following way: choose a point P on l; then (ABC ∞) should be the limit approached by (ABCP) as P recedes to infinity along l. But

Fig. 84. Cross-ratio with a point at infinity.

and as P recedes to infinity, PA/PB approaches 1. Hence we define

(ABC ∞) = CA/CB.

In particular, if (ABC ∞) = –1, then C is the midpoint of the segment AB: the midpoint and the point at infinity in the direction of a segment divide the segment harmonically.

Exercises: What is the cross-ratio of four lines l₁, l₂, l₃, l₄ if they are parallel? What is the cross-ratio if l₄ is the line at infinity?