CROSS-RATIO - 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

§3. CROSS-RATIO

1. Definition and Proof of Invariance

Just as the length of a line segment is the key to metric geometry, so there is one fundamental concept of projective geometry in terms of which all distinctively projective properties of figures can be expressed.

If three points A, B, C lie on a straight line, a projection will in general change not only the distances AB and BC but also the ratio AB/BC. In fact, any three points A, B, C on a straight line l can always be coördinated with any three points A′, B′, C′ on another line l′ by two successive projections. To do this, we may rotate the line l′ about the point C′ until it assumes a position l″ parallel to l (see Fig. 74). We then project l onto l″ by a projection parallel to the line joining C and C′, defining three points, A″, B″, and C″ (= C″). The lines joining A′, A″ and B′, B″ will intersect in a point O, which we choose as the center of a second projection. These two projections accomplish the desired result.

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Fig. 74.

As we have just seen, no quantity that involves only three points on a line can be invariant under projection. But—and this is the decisive discovery of projective geometry—if we have four points A, B, C, D on a straight line, and project these into A′, B′, C′, D′ on another line, then there is a certain quantity, called the cross-ratio of the four points, that retains its value under the projection. Here is a mathematical property of a set of four points on a line that is not destroyed by projection and that can be recognized in any image of the line. The cross-ratio is neither a length, nor the ratio of two lengths, but the ratio of two such ratios: if we consider the ratios CA/CB and DA/DB, then their ratio,

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is by definition the cross-ratio of the four points A, B, C, D, taken in that order.

We now show that the cross-ratio of four points is invariant under projection, i.e. that if A, B, C, D and A′, B′, C′, D′ are corresponding points on two lines related by a projection, then

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The proof follows by elementary means. We recall that the area of a triangle is equal to ½(base × altitude) and is also given by half the product of any two sides by the sine of the included angle. We then have, in Figure 75,

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It follows that

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Fig. 75. Invariance of cross-ratio under central projection.

Hence the cross-ratio of A, B, C, D depends only on the angles subtended at O by the segments joining A, B, C, D. Since these angles are the same for any four points A′, B′, C′, D′ into which A, B, C, D may be projected from O, it follows that the cross-ratio remains unchanged by projection.

That the cross-ratio of four points remains unchanged by a parallel projection follows from elementary properties of similar triangles. The proof is left to the reader as an exercise.

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Fig. 76. Invariance of cross-ratio under parallel projection.

So far we have understood the cross-ratio of four points A, B, C, D on a line l to be a ratio involving positive lengths. It is more convenient to modify this definition as follows. We choose one direction on l as positive, and agree that lengths measured in this direction shall be positive, while lengths measured in the opposite direction shall be negative. We then define the cross-ratio of A, B, C, D in that order as the quantity

(1) image

where the numbers CA, CB, DA, DB are understood to be taken with the proper sign. Since a reversal of the chosen positive direction on l will merely change the sign of every term of this ratio, the value of (ABCD) will not depend on the direction chosen. It is easily seen that (ABCD) will be negative or positive according as the pair of points A, B is or is not separated (i.e. interlocked) by the pair C, D. Since this separation property is invariant under projection, the signed cross-ratio (ABCD) is invariant also. If we select a fixed point O on l as

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Fig. 77. Sign of cross-ratio.

origin and choose as the coördinate x of each point on l its directed distance from O, so that the coördinates of A, B, C, D are x1, x2, x3, x4, respectively, then

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When (ABCD) = –1, so that CA/CB = –DA/DB, then C and D

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Fig. 78. Cross-ratio in terms of coördinates.

divide the segment AB internally and externally in the same ratio. In this case, C and D are said to divide the segment AB harmonically, and each of the points C, D is called the harmonic conjugate of the other with respect to the pair A, B. If (ABCD) = 1, then the points C and D (or A and B) coincide.

It should be kept in mind that the order in which A, B, C, D are taken is an essential part of the definition of the cross-ratio (ABCD). For example, if (ABCD) = λ, then the cross-ratio (BACD) is l/λ, while (ACBD) = 1 – λ, as the reader may easily verify. Four points A, B, C, D can be ordered in 4·3·2·1 = 24 different ways, each of which gives a certain value to their cross-ratio. Some of these permutations will yield the same value for the cross-ratio as the original arrangement A, B, C, D; e.g. (ABCD) = (BADC). It is left as an exercise for the reader to show that there are only six different values of the cross-ratio for these 24 different permutations of the points, namely

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These six quantities are in general distinct, but two of them may coincide—as in the case of harmonic division, when λ = –1.

We may also define the cross-ratio of four coplanar (i.e. lying in a common plane) and concurrent straight lines 1, 2, 3, 4 as the cross-ratio of the four points of intersection of these lines with another straight line lying in the same plane. The position of this fifth line is immaterial because of the invariance of the cross-ratio under projection. Equivalent to this is the definition

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taken with a plus or minus sign according as one pair of lines does not or does separate the other. (In this formula, (1, 3), for example, means the angle between the lines 1 and 3.) Finally, we may define the cross-ratio of four coaxial planes (four planes in space intersecting in a line l, their axis). If a straight line intersects the planes in four points, these points will always have the same cross-ratio, whatever the position of the line may be. (The proof of this fact is left as an exercise.) Hence we may assign this value as the cross-ratio of the four planes. Equivalently, we may define the cross-ratio of four coaxial planes as the cross-ratio of the four lines in which they are intersected by any fifth plane (see Fig. 79).

The concept of the cross-ratio of four planes leads naturally to the question of whether a projective transformation of three-dimensional space into itself can be defined. The definition by central projection cannot immediately be generalized from two to three dimensions. But it can be proved that every continuous transformation of a plane into itself that correlates in a biunique manner points with points and lines with lines is a projective transformation. This theorem suggests the following definition for three dimensions: A projective transformation of space is a continuous biunique transformation that preserves straight lines. It can be shown that these transformations leave the cross-ratio invariant.

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Fig. 79. Cross-ratio of coaxial planes.

The preceding statements may be supplemented by a few remarks. Suppose we have three distinct points, A, B, C, on a line, with coördinates x1, x2, x3. Required, to find a fourth point D so that the cross-ratio (ABCD) = λ, where λ is prescribed. (The special case λ = –1, for which the problem amounts to the construction of the fourth harmonic point, will be taken up in more detail in the next article.) In general, the problem has one and only one solution; for, if x is the coördinate of the desired point D, then the equation

(2) image

has exactly one solution x. If x1, x2, x3 are given, and if we abbreviate equation (2) by setting (x3x1)/(x3x2) = k, we find on solving this equation that x = (kx2 – λx1)/(k– λ). For example, if the three points A, B, C are equidistant, with coördinates x1 = 0, x2 = d, x3 = 2d respectively; then k = (2d – 0)/(2dd) = 2, and x = 2d/(2 – λ).

If we project the same line l onto two different lines l′, l″ from two different centers O′ and O″, we obtain a correspondence PP′ between the points of l and l′, and a correspondence PP″ between those of l and l″. This sets up a correspondence P′ ↔ P″ between the points of l′ and those of l″ which has the property that every set of four points A′, B′, C′, D′ on l′ has the same cross-ratio as the corresponding set A″, B″, C″, D″ on l″. Any biunique correspondence between the points on two lines which has this property is called a projective correspondence, irrespective of how the correspondence is defined.

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Fig. 80. Projective correspondence between the points on two lines.

Exercises: 1) Prove that, given two lines together with a projective correspondence between their points, one can shift one line by a parallel displacement into such a position that the given correspondence is obtained by a simple projection. (Hint: Bring a pair of corresponding points of the two lines into coincidence.)

2) On the basis of the preceding result, show that if the points of two lines l and l′ are coördinated by any finite succession of projections onto various intermediate lines, using arbitrary centers of projection, the same result can be obtained by only two projections.

2. Application to the Complete Quadrilateral

As an interesting application of the invariance of the cross-ratio we shall establish a simple but important theorem of projective geometry. It concerns the complete quadrilateral, a figure consisting of any four straight lines, no three of which are concurrent, and of the six points where they intersect. In Figure 81 the four lines are AE, BE, BI, AF. The lines through AB, EG, and IF are the diagonals of the quadrilateral. Take any diagonal, say AB, and mark on it the points C and D where it meets the other two diagonals. We then have the theorem: (ABCD) = –1; in words, the points of intersection of one diagonal with the other two separate the vertices on that diagonal harmonically. To prove this we simply observe that

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Fig. 81. Complete quadrilateral.

x = (ABCD) = (IFHD) by projection from E,
(IFHD) = (BACD)  by projection from G.

But we know that (BACD) = l/(ABCD); so that x = 1/x, x2 = 1, x = ±1. Since C, D separate A, B, the cross-ratio x is negative and must therefore be –1, which was to be proved.

This remarkable property of the complete quadrilateral enables us to find with the straightedge alone the harmonic conjugate with respect to A, B of any third collinear point C. We need only choose a point E off the line, draw EA, EB, EC, mark a point G on EC, draw AG and BGintersecting EB and EA at F and I respectively, and draw IF, which intersects the line of A, B, C in the required fourth harmonic point D.

Problem: Given a segment AB in the plane and a region R, as shown in Figure 82. It is desired to continue the line AB to the right of R. How may this be done with straightedge alone so that the straightedge never crosses R during the construction? (Hint: Choose two arbitrary points C, C′ on the segment AB, then locate their harmonic conjugates D, D′ respectively by means of four quadrilaterals having A, B as vertices.)

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Fig. 82. Producing a line beyond an obstacle.