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FUNDAMENTAL THEOREM
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FUNDAMENTAL THEOREM .—The See also:cross-ratio of four points in a See also:line is equal to the cross-ratio of their projections on any other line which lies in the same See also:plane with it. § 14. Before we draw conclusions from this result, we must investigate the meaning of a cross-ratio somewhat more fully. If four points A, B, C, D are given, and we wish to See also:form their cross-ratio, we have first to See also:divide them into two See also:groups of two, the points in each See also:group being taken in a definite See also:order. Thus, let A, B be the first, C, D the second pair, A and C being the first points in each pair. The cross-ratio is then the ratio AC:CB divided by AD: DB. This will be denoted by (AB, CD), so that (AB, CD)= CB:ACADDB This is easily remembered. In order to write it out, make first the two lines for the fractions, and put above and below these the letters A and B in their places, thus, P'B : PIB; and then fill up, crosswise, the first by C and the other by D. § 15. If we take the points in a different order, the value of the cross-ratio will See also:change. We can do this in twenty-four different ways by forming all permutations of the letters. But of these twenty-four cross-ratios groups of four are equal, so that there are really only six different ones, and these six are reciprocals in pairs. We have the following rules: I. If in a cross-ratio the two groups be interchanged, its value remains unaltered, i.e. (AB, CD) = (CD, AB) re (BA, DC) = (DC, BA). II. If in a cross-ratio the two points belonging to one of the two groups be interchanged, the cross-ratio changes into its reciprocal, i.e. (AB, CD) = I/(AB, DC) = i/(BA, CD) = I/(CD, BA) = 'ADC, AB). From I. and II. we see that eight cross-ratios are associated with (AB, CD). [§ i6. If X.= (AB, CD), A= (AC, DB), v= (AD, BC), then A, At, v and their reciprocals i/A, I/µ, qv are the values of the See also:total number of twenty-four cross-ratios. Moreover, A, ti, v are connected by the relations P' A-i-Iiµ=,s+I/v=v+IIA= —Aµv= 1; this proposition may be Droved by substituting for A, is, v andreducing to a See also:common origin. There are therefore four equations between three unknowns; hence if one cross-ratio be given, the remaining twenty-three are determinate. Moreover, two of the quantities A, µ, v are See also:positive, and the remaining one negative. The following See also:scheme shows the twenty-four cross-ratios expressed in terms of A, is, v.] (AB, CD) = (AB, DC) = (BA, CD). For four See also:harmonic points the six cross-ratios become equal two and two: A=—I,I—A=2,AI=2, =—I,IIA,Ar1=2. Hence if we get four points whose cross-ratio is 2 or , then they are harmonic, but not arranged so that conjugates are paired. If this is the See also:case the cross-ratio = — I. § 19. If we equate any two of the above six values of the cross-ratios, we get either X= I, o, oo, or X= -1, 2, 1, or else A becomes a See also:root of the See also:equation A2—A+I =o, that is, an imaginary See also:cube root of —I. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no See also:interest, whilst harmonic points are all-important. § 20. From the See also:definition of 'harmonic points, and by aid of § I I, the following properties are easily deduced. If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are "separated " by C and D. For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B. If A and B are different points, and if C coincides with A or B, D does. But if A and B coincide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, " if of four harmonic conjugates two -coincide, then a third coincides with them, and the See also:fourth may be any point in the line." If C is the See also:middle point between A and B, then D is the point at infinity; for AC: CB=+1, hence AD:DB must be equal to —i. The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB. This important See also:property gives a first example how metric properties are connected with projective ones. [§ 21. Harmonic properties of the See also:complete See also:quadrilateral and quadrangle. (AB, CD) ' (AC, DB)' (CD DC) - A t—µ I/(I—v) (CB CA I( —X — 1)/v A,' BD) (DC, BA) , (DB, AC)_ (AB, DC) ' (AD, BC)1 (BA, CD) . IjA IJ(I —ti) C, AB) Bv (~ —I)/A t+ l(µ—I) A) I(BC,AD) , D- (CB, DA) (DA, CB) J L (AC, BD) ' (AD, CB) (BD , AC) 1 —A µ v/(v—1) (BC, DD)~ ) Iµ--I)/i I/ V ( DB) (DA, BC) A/(A— ' (DB, .. § 17. If one of the points of which a cross-ratio is formed is the point at infinity in the line, the cross-ratio changes into a See also:simple ratio. It is convenient to let the point at infinity occupy the last See also:place in the symbolic expression for the cross-ratio. Thus if I is a point at infinity, we have (AB, CI) = —AC/CB, because AI : IB = —i. Every common ratio of three points in a line may thus be ex-pressed as a cross-ratio, by adding the point at infinity to the group of points. Additional information and CommentsThere are no comments yet for this article.
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