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    Projective Geometry
    Projective Geometry
    Projective Geometry
    Ebook184 pages2 hours

    Projective Geometry

    By T. Ewan Faulkner

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    About this ebook

    This text explores the methods of the projective geometry of the plane. Some knowledge of the elements of metrical and analytical geometry is assumed; a rigorous first chapter serves to prepare readers. Following an introduction to the methods of the symbolic notation, the text advances to a consideration of the theory of one-to-one correspondence. It derives the projective properties of the conic and discusses the representation of these properties by the general equation of the second degree. A study of the relationship between Euclidean and projective geometry concludes the presentation. Numerous illustrative examples appear throughout the text.
    LanguageEnglish
    PublisherDover Publications
    Release dateFeb 20, 2013
    ISBN9780486154893
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      Projective Geometry - T. Ewan Faulkner

      INDEX

      CHAPTER I

      INTRODUCTION : THE PROPOSITIONS OF INCIDENCE

      1. Historical note.—The study of geometry began well over two thousand years ago, and it is inevitable that in that long period there should have been several courses of development. Modern geometry is built on more than one foundation and cannot be fully appreciated without some knowledge of its history. There are three main lines of approach to the study of geometry—the metrical, the projective and the analytical—and it is important to understand the contribution which each has made to our present knowledge.

      The first method of study began with the Greek geometers, and is associated with the name of Euclid. Euclidean geometry is based upon the fundamental notion of distance, or length ; distance is never defined, but is regarded as an intuitive concept which underlies every geometrical theorem. Euclidean geometry is metrical, for it assumes that every segment or angle can be measured and expressed in terms of a standard distance or standard angle.

      However, in addition to theorems which were very obviously concerned with distance, geometers were interested in theorems involving the concurrency of lines or the collinearity of points. A typical example is Pappus’ theorem, proved by Pappus using the methods of metrical geometry about the year A.D. 300. These projective theorems, as they were called, were for many centuries merely added to the propositions of Euclid, and were not regarded as being of a different character. Development was slow, and it was not until the seventeenth century that Desargues, and to a lesser extent Pascal, established the main theorems of projective geometry. Both Desargues and Pascal made full use of the theorems of metrical geometry, and it was only after the publication of Geometrie der Lage by von Staudt in 1847 that projective geometry was established as a science built upon a different set of axioms from those of Euclid. It was shown that the theorems of projective geometry were independent of the concept of distance, and that distance itself could be expressed in terms of simpler projective elements. The theorems of metrical geometry were found to be special cases of the more general theorems of projective geometry, with Euclidean geometry as only part of the field covered by the science of projective geometry.

      The third method of geometrical study is that known as coordinate or analytical geometry. It was introduced by Descartes, who represented a point by a set of numbers, and thus applied the methods of algebra to the solution of geometrical problems. Descartes used the idea of distance, and his geometry is thus metrical ; his achievement was that, by expressing geometrical ideas in the language of algebra, he was able to provide simple proofs of many theorems difficult to deal with by the traditional methods of metrical geometry. However, the methods of analytical geometry have not been limited in their application to metrical problems only, and since the time of Descartes, geometers such as Poncelet and Cayley have applied these methods, with modification, to the whole field of projective geometry. The Cartesian coordinates of Descartes have been replaced by homogeneous coordinates, which, since they are independent of metrical concepts, are able to deal more conveniently with projective problems.

      Complex points.—The application of algebra to geometry had a very important consequence. Once the theory of complex numbers was established and it was agreed that every quadratic equation had two roots whether real or complex, it was a simple matter to deduce the existence of complex or imaginary points. Previously the problem of finding the points common to a line and a conic could not be solved satisfactorily, but, when it was known that the problem was identical with that of solving a quadratic equation, it became clear that a line and conic always had two points in common, but that these two points could be real, coincident or complex. The use of complex points opened up a very fruitful field of study, for it enabled geometers to elaborate general theorems which would not be true if the field of real points only were considered. In particular, the discovery of the circular points made it possible to generalise well-established theorems about circles and obtain theorems about conics through two fixed points.

      2. The projective method.—The raw material of projective geometry consists of a number of elements, points, lines and planes. We make no attempt to define these concepts, but regard them as undefined elements related to each other according to certain axioms which we call the propositions of incidence. These axioms are not the only set of axioms upon which a logical and self-consistent geometry could be built, but they are chosen partly for their intrinsic simplicity, and partly because they provide us with a generalised geometry out of which Euclidean geometry appears as a special development.

      The propositions of incidence.—In projective geometry we take a point as a completely undefined element and suppose a line to consist of an infinite set of points, and to be completely determined by any two distinct points of the set. Thus, if the line determined by the points A and B contains the points X and Y, it follows that the line determined by X and Y is the same line and contains A and B. Three or more points which belong to a line are said to be collinear or to lie on the line.

      A plane is assumed to consist of an infinite set of points and to be completely determined by any three distinct non-collinear points upon it. It is further assumed that a line defined by any two points of the plane lies completely in the plane. Since a plane contains an infinite number of points, not all collinear, and since any two points determine a line, it is clear that a plane also contains an infinite number of lines. It is assumed further that any two distinct lines lying in the plane have one point in common, or, expressed otherwise, a point is determined by any two of the infinite number of lines which pass through it. This last assumption forms the basis of the principle of duality in the plane which is discussed below. Lines through a point are said to be concurrent, and the point is called the point of intersection of the lines.

      Extending, we may suppose three-dimensional space to consist of an infinite number of points and to be completely determined by any four non-coplanar points within it. A plane, and a line not lying in the plane, are assumed to have one point in common.

      It follows that two planes have one line in common. For, if we take two lines in the first plane meeting the second plane in A and B, the points A and B clearly lie in both planes, and thus the line defined by A and B lies in both planes.

      Since two planes have a line in common, and a third plane, which does not contain this line, meets it in one point, it follows that three planes, supposed not to have a common line, have one point in common. This result forms the basis of the principle of duality in space.

      The principle of duality.—If we examine the propositions of incidence carefully we find a certain dual relationship between them. Thus, in a plane, a line is determined by two points upon it, while a point is determined by two lines which pass through it. Thus, if, by using the propositions of incidence, we are able to prove a theorem involving points and lines, then, by using similar reasoning, we should be able to prove a corresponding theorem involving lines and points. The dual theorem is obtained from the original theorem merely by the interchange of certain words, point and line , collinear and concurrent , lie on and intersect in , and so on. This is known as the principle of duality in the plane.

      In space there is another form of duality. From the propositions of incidence we see that a plane is determined by three of its points, and a point is determined by three of the planes which pass through it. A line is determined, either by two of its points, or by two planes which contain it. Thus, in space, points and planes are dual elements, while a line is a self-dual element. Hence to any theorem involving points, lines and planes there corresponds the dual theorem involving planes, lines and points.

      We shall have several opportunities later of illustrating the principle of duality in specific instances.

      3. Desargues’ theorem.—As an illustration of the propositions of incidence we prove Desargues’ theorem, a theorem of very great importance in the foundations of projective geometry.

      If two triangles ABC and A′B′C′, lying in the same or in different planes, are such that AA′, BB′, CC′ meet in a point O ; then BC meets B′C′ in L, CA meets C′A′ in M, and AB meets A′B′ in N, where L, M, N are collinear.

      We first consider the case when the triangles ABC and A′B′C′ are in different planes π and π′ respectively. Since the lines BB′ and CC′ intersect in 0, it follows that B, B′, C, C′ lie in a plane and BC meets B′C′ in a point L. Similarly CA meets C′A′ in M and AB meets A′B′ in N. The three points L, M, N evidently lie in each of the planes π and π′, and are thus collinear on the line of intersection of these planes. The two triangles ABC and A′B′C′ are said to be in perspective from O.

      We next consider the case when the triangles are in the same plane π. Let OPP′ be any line through O not lying in π ; then, since PP′ meets AA′ in 0, the four points P, P′, A, A′ are coplanar, and PA meets P′A′ in a point A″. Similarly PB meets P′B′ in B″ and PC meets P′C′ in C″. The four points B, B″, C, C″ are evidently coplanar and so BC meets B″C″. The lines BC, B′C′, B″C″ are the three lines of intersection of the three planes PBC, P′B′C′ and π taken in pairs, and so BC, B′C′, B″C″ meet in a point L. Similarly CA, C′A′, C″A″ meet in M and AB, A′B′, A″B″ meet in N. The two triangles ABC and A″B″C″ lie in different planes and AA″, BB″, CC″ meet in P, and thus L, M, N are collinear. This establishes the required result.

      FIG. 1

      It is important to note that in proving Desargues’ theorem for the case of coplanar triangles we made use of points and lines outside the plane of the triangles. It is an interesting result that if we make no assumptions other than the propositions of incidence for the plane no proof is possible, and in fact geometries have been constructed for which the propositions of incidence for the plane hold but for which Desargues’ theorem is not true.

      4. The analytical method.—In the analytical approach to the study of geometry a point is represented by a set of numbers. This enables us to express elementary geometrical concepts in the language of algebra, and to solve geometrical theorems by the application of algebraic laws. In this section we give an analytical representation of the propositions of incidence and in this way provide a tool which will be of considerable use in the further development of the subject.

      The symbolic notation.—We denote points by A, B, C, etc. and we let x, y, z etc. denote numbers or multipliers. We suppose that the point denoted by xA, where x is any number

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