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| ISBN-13: | 9780486154732 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/17/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 240 |
| File size: | 11 MB |
| Note: | This product may take a few minutes to download. |
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Lectures in Projective Geometry
By A. Seidenberg
Dover Publications, Inc.
Copyright © 2005 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-15473-2
CHAPTER 1
PROJECTIVE GEOMETRY AS AN EXTENSION OF HIGH SCHOOL GEOMETRY
1. Two approaches to projective geometry. There are two ways to study projective geometry: (1) as a continuation of Euclidean geometry as usually taught in high schools, and (2) as an independent discipline, with its own definitions, axioms, theorems, etc. Of these two ways, the first corresponds to the actual historical development of the subject. As Euclidean geometry developed, a large body of theorems having a common character was built up, and theorems having that character (which we will describe more explicitly below) were called projective theorems. But precisely because projective geometry arose from Euclidean geometry, attention began to be directed more and more to the axioms. It was for this reason that the so-called "axiomatic method," which nowadays is applied in all branches of mathematics, was first applied to projective geometry, from which subject it gradually spread to all the others.
Of the two ways of studying the subject, each has disadvantages. As to the first method, although Euclid may have been definite enough about his axioms, the high schools frequently are not, and as a consequence one is never quite sure what these axioms were. This is also partly the fault of Euclid, and today we can be somewhat more exact than he was. Another disadvantage is that projective geometry consists of theorems of a certain special type, so that Euclid's axioms, even if made precise, are not entirely suitable.
The axiomatic method of introducing the subject also has disadvantages. For one thing, one has to enter somewhat fully into what is involved in an axiomatic system, and this takes us away from geometry and other fairly familiar things to things that at first sight may seem to be bizarre and to have no bearing on the subject. Besides, the content of an axiomatic system depends on what we want to study: for algebra we would have one system of axioms; for geometry, another; therefore if we really want to come to grips with geometric problems, we have to know, first, roughly what these problems are.
Of the two methods, we will start with the first. Since we do not say precisely what we are assuming, we shall never be able to tell precisely what we are proving. This kind of situation always gives rise to misunderstandings, especially if we insist that we are actually proving things. The first chapter will therefore not be directed toward establishing theorems with complete rigor, but toward supplying background material that will be helpful in the axiomatic method part of the book. At the same time, although we will not insist upon complete rigor, neither will we allow ourselves to make obvious mistakes in logic.
2. An initial question. The question toward which projective geometry first directed itself was, "How do we see things?" or "What is the relation between a thing itself, say a figure, and the thing seen?" More particularly, "Do we ever see a thing exactly as it is?" Take this rectangular table: we know it is rectangular. But a little thought shows us that we do not see it as rectangular. In fact, a little lack of thought will also show this. For suppose we try to draw the table. We draw a rectangle (Fig. 1.1). This is the top of the table. But now we try to put the legs on, and we see that we cannot. To do so, we have to draw the rectangle in a nonrectangular shape; then we can put the legs on (Fig. 1.2).
This argument will certainly convince us that angles are not always what they seem. Let us try, however, to make the argument more precise. For the following, we assume only one eye.
Let us suppose we are looking through the window onto the ground (which may be sloping), as illustrated in Fig. 1.3. On the ground is a figure, say a large letter S. The way we see this letter is by light passing from the points of S and entering the eye. We therefore have to consider lines joining the point E, representing the eye, to the various points of the figure S. These lines cut the windowpane in various points. If P is a point of S, let EP cut the windowpane in a point P', which let us call the image of P: the set of all the images of the points of the figure S will be called the image of S, or its projection (from E). We have two figures, then, the figure F, on the ground, and the figure F', on the windowpane. Since the rays of light emerging from F' and reaching E are the same as those emerging from F and reaching E, it is clear that as far as shape is concerned, we see the figure F' in precisely the same way we see the figure F.
Our original question becomes essentially the following: What is the relation between a figure F, given in a plane π, and its projection in another plane π'? (The projection is said to be from π to π' with center E.) Or "What properties of the figure F are also found in F'?" It is fairly clear that this is essentially the same question, because if we wanted to draw the figure F on π', we would draw F'.
3. Projective invariants. Let us apply these considerations to the corner of a room (Fig. 1.4). Let us draw this comer, as shown in Fig. 1.5. After drawing this picture, we realize that we do not always see a right angle as a right angle. In fact, there are three right angles at the corner, totaling 270°, but the angles have images totaling 360°. At least one of the angles must appear as more than 90°.
More generally, take two angles α, α'. Place α in plane π and α' in π'. Let l be the intersection of π and π', and place α and α' so that l is one of the sides of each angle (Fig. 1.6). On the other side of α take a point A; and similarly a point A' for α'. On AA' take a point E. Since, α, α' are of arbitrary size, we see that it is always possible to look at a given angle so that it appears as large as any other given angle.
If a property P of a figure F is also shared by the projection F' of F (for every possible projection), then P is called a projective invariant.
Using this terminology, we can say that size of angle is not a projective invariant. In particular, "right angle" is not a projective invariant. Since A, A' in Fig. 1.6 could be taken arbitrarily far from O, we see that distance is not a projective invariant.
One may now ask for an example of a property that is a projective invariant. Roughly speaking, a point is a projective invariant (or the property of a figure that it be a point is a projective invariant—but we will express ourselves more simply in the above way; we will also say that a point is a projectively invariantive concept). A line is a projective invariant, at least roughly speaking, because if we see a plane object as a line, then it is a line. (We say "roughly speaking," because there is a slight qualification that must be made, which we shall, come to shortly.)
4. Vanishing points. Let us take two planes π and π' and a projection from the center E. The center is understood not to be in π or π'. In the plane π take two lines, p, q, that intersect in a point A. Let p' and q' be the images of p and q, and let A' be the image of A. Since A is on p, A' will be on the image of p, i.e., on p '; thus A' is on p', and similarly A' is on q'. We can then say:
The intersection of the images of two lines is the image of the intersection of the two lines.
Let us call the line joining two points, the join of the points. Then we can say:
The join of the images of two points is the image of the join of the two points.
These two theorems will enable us to make certain simple and obvious conclusions about a figure F' that corresponds by projection to the figure F. We shall pass readily from F to F' and back to F again.
Of course, in the above situation it may well be that A has no image. This will be so if and only if EA is parallel to the plane π'. In the case that EA is parallel to π' we will say that A disappears (under the projection from π to π') or that it vanishes or is a vanishing point. If A vanishes, then one sees that p' and q' do not meet (for if they did, the intersection would be the image of A). Thus intersecting lines may project into parallel lines; and projecting back from π' to π, we see that parallel lines may project into intersecting lines. Parallelism is not a projective invariant. If A is a vanishing point, then the whole system of lines passing through A projects into a system of parallel lines in π'. Regarding this plane π' as the ground (as above) and π as the window-pane, we see why a system of parallel lines appears as a system of lines passing through a point.
5. Vanishing lines. Let A be a vanishing point. Then A has no image. Because of this we cannot say that a point is a projective invariant, because if a figure F consists of just one point A, then its image F' may consist of no points at all. This is a difficulty, but it is a difficulty we want to surmount. The Greeks called this saving the phenomenon. We want to save the idea that a point is a projective invariant. On the other hand, size of angle is not a projective invariant, and, moreover, we do not want to save the concept.
A line is not quite a projective invariant either. For one thing, consider all the vanishing points. These are found by taking the plane through E parallel to π' and seeing where it intersects π. It intersects it in a line, say v (Fig. 1.7). The vanishing points are just the points of v. Thus we see:
[ILLUSTRATION OMITTED]
The locus of the vanishing points is a straight line.
In particular, the line v has no image, and therefore the concept of a line is not quite a projective invariant; but also in the case of a line we want to save the phenomenon.
Even if we take exception to the situation in which a figure completely vanishes, we shall still have difficulties. There is no difficulty for a point: the image of a point (if there is an image) is itself a point. But consider a line l, l ≠ v, the vanishing line. How do we see l? Let l cross v at A; then A vanishes. No point corresponds to A on l', the line containing the images of l (= intersection of plane (E, l) with π'). For the same reason, there will be a point B on l' that, in the projection π' to π, has no image on l —namely, if B is taken so that EB is parallel to π. But then B is not the image of any point of l. Thus the image of l is not l', but l' minus the point B. Thus a line is seen as a line minus a point. One might then try line minus a point as a projective invariant. But a line minus a point would be seen as a line minus two points; and a line minus two points as a line minus three points, etc. The best we can get out of these considerations is:
The projection of a figure F consisting of a finite number of lines minus a finite number of points is itself a figure F' consisting of a finite number of lines minus a finite number of points.
This saves the phenomenon, but we shall have a better way.
6. Some projective noninvariants. The kind of geometrical objects we would like to study in projective geometry are the projectively invariantive objects. Thus, distance is out, size of angle is out, and so is parallelism. Let us then consider some of the other familiar concepts, partly to clear the way. Take a circle. Does it necessarily appear as a circle? If we look at it on edge, it appears as a straight line segment. But in the situation in which we considered a projection from a plane π to a plane π', if the circle is in π, the above argument would require E to be in π, whereas we have stipulated that E should be neither in π nor in π'. Let us find another argument. We will prove that boundedness is not a projective invariant: by a bounded figure we mean one that can be enclosed within a circle. Thus a square is a bounded figure; so is a circle. A line is not bounded; neither is a parabola. We will now indicate how a bounded figure can be projected into an unbounded figure. Let us take a bounded figure, say a square for definiteness. Place this square in plane π SO that it cuts the vanishing line v, say at A (Fig. 1.8). Now project. The point A disappears. The points near to A have images, but far out. This argument could be made more definite, but it should be fairly clear that the image figure is not bounded—here is an example of an argument that is far from rigorous but that we may allow temporarily.
The argument shows not only that boundedness is not a projective invariant, but that any bounded figure can be projected into an unbounded figure. In particular, a circle can be so projected, so the concept of a circle is not a protectively invariantive concept.
Exercises
1. If A, B are two points on a circle and P varies on the circle, then angle APB remains constant (at least if P stays on one of the two arcs into which A and B divide the circle). Use this fact (instead of the notion of boundedness) to prove that a circle is not a projective invariant.
2. Let P' be the image of a point P under a projection. We shall make use of the fact that if Q approaches P, then Q' the image of Q, approaches P'. Let l, m be lines through P, l fixed, m variable; l', m' their images. Show that if m approaches l, then m' approaches l'.
3. With the notation of the preceding exercise, let C be a curve through P and let Q be a point on G approaching P. Then the limiting position of PQ as Q approaches P is the tangent to C at P. Two curves C, D through P are called tangent at P if they have the same tangent at P. Let C, D be tangent at P. Show that under projection the curves C, D go over into tangent curves at P'. In other words, tangency is a projective invariant.
7. Betweenness. Let A, B be two points. We can consider the line AB: this is usually understood to mean the line extended. We also speak of the segment AB. Any point P on this segment is said to be between A and B. To say that betweenness is invariantive is to say that if P is between A and B, and these points are projected (in any manner) into A', B', and P', then P' is between A' and B'; and to say that betweenness is not invariantive is to say that that conclusion is not necessarily so. In Fig. 1.9, we give a cross-section of the situation involving π, π', and E. In π we take three points A, B, P with P between A and B. First take E so that EP is parallel to plane π'. Let A', B' be the images of A, B. The point P vanishes. Now move E slightly off the line EP to a point E1. If E has been moved very little, the image A1 of A from the center E1 will be near A', the image of B1 of B near B' and the image of P will exist but be far out, hence certainly not between A1 and B1. Thus betweenness is not a projective invariant.
Exercise. A line divides the plane into two parts, which may be called the sides of the line. Show that the concept side of a line is not a protectively invariantive concept. (This could be done, for example, by showing that if two points lie in different sides of l in π, their images may well lie in the same side of l', the image of l.)
8. Division of a segment in a ratio. Let A, B be two points, C a point on AB, not necessarily between A, B (Fig. 1.10). The point C is said to divide A B on the ratio CA/CB. Here C = B is not allowed, although C = A is. One may fix a direction (or orientation) on the line, either to the right or to the left; say to the right. We then count XY positive if Y is to the right of X, negative if Y is to the left of X, and, of course, X Y = 0 if X = Y. With the stated orientation, in Fig. 1.10, PQ, PR, QR are positive, RP, RQ, QP negative. If the orientation were fixed in the opposite way, all these segments would have the opposite sign. Hence if C is between A and B, then CA/CB is negative, whereas if C is not between A and B, then CA/CB is positive. From this it becomes clear that division into a given ratio is not a projective invariant, because by projection we could change the ratio from a negative quantity to a positive. In particular, mid-point is not a projective invariant.
(Continues...)
Excerpted from Lectures in Projective Geometry by A. Seidenberg. Copyright © 2005 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
I. Projective Geometry as an Extension of High School Geometry1. Two approaches to projective geometry
2. An initial question
3. Projective invariants
4. Vanishing points
5. Vanishing lines
6. Some projective noninvariants
7. Betweenness
8. Division of a segment in a ratio
9. Desargues' Theorem
10. Perspectivity; projectivity
11. Harmonic tetrads; fourth harmonic
12. Further theorems on harmonic tetrads
13. The cross-ratio
14. Fundamental Theorem of Projective Geometry
15. Further remarks on the cross-ratio
16. Construction of the projective plane
17. Previous results in the constructed plane
18. Analytic construction of the projective plane
19. Elements of linear equations
II. The Axiomatic Foundation
1. Unproved propositions and undefined terms
2. Requirements on the axioms and undefined terms
3. Undefined terms and axioms for a projective plane
4. Initial development of the system; the Principle of Duality
5. Consistency of the axioms
6. Other models
7. Independence of the axioms
8. Isomorphism
9. Further axioms
10. Consequences of Desargues' Theorem
11. Free planes
III. Establishing Coordinates in a Plane
1. Definition of a field
2. Consistency of the field axioms
3. The analytic model
4. Geometric description of the operations plus and times
5. Setting up coordinates in the projective plane
6. The noncommutative case
IV. Relations between the Basic Theorems
V. Axiomatic Introduction of Higher-Dimensional Space
1. Higher-dimensional, especially 3-dimensional projective space
2. Desarguesian planes and higher-dimensional space
VI. Conics
1. Study of the conic on the basis of high school geometry
2. The conic, axiomatically treated
3. The polar
4. The polar, axiomatically treated
5. Polarities
VII. Higher-Dimensional Spaces Resumed
1. Theory of dependence
2. Application of the dependency theory to geometry
3. Hyperplanes
4. The dual space
5. The analytic case
VIII. Coordinate Systems and Linear Transformations
1. Coordinate systems
2. Determinants
3. Coordinate systems resumed
4. Coordinate changes, alias linear transformations
5. A generalization from n = 2 to n = 1
6. Linear transformations on a line and from one line to another
7. Cross-ratio
8. Coordinate systems and linear transformations in higher-dimensional spaces
9. Coordinates in affine space
IX. Coordinate Systems Abstractly Considered
1. Definition of a coordinate system
2. Definition of a geometric object
3. Algebraic curves
4. A short cut to PNK
5. A result for the field of real numbers
X. Conic Sections Analytically Treated
1. Derivation of equation of conic
2. Uniqueness of the equation
3. Projective equivalence of conics
4. Poles and polars
5. Polarities and conics
Appendix to Chapter X
A1. Factorization of linear transformation into polarities
XI. Coordinates on a conic
1. Coordinates on a conic
2. Projectivities on a conic
XII. Pairs of Conics
1. Pencils of conics
2. Intersection multiplicities
XIII. Quadric Surfaces
1. Projectivities between pencils of planes
2. Reguli and quadric surfaces
3. Quadric surfaces over the complex field
4. Some properties of the sphere
XIV. The Jordan Canonical Form
Bibliographical Note
Index