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5 Calculating with Points on Lines 5 Calculating with points on lines Newsgroups: sci.math Subject: Re: I'm looking for axioms and proof in math texts Date: 6 Aug 2003 02:53:12 -0700 From: [email protected] (prometheus666) [...] and being passing familiar with the number line -- I'm sure if I don't say passing familiar somebody here will say, "You have to know vector tensor shmelaculus in 15 triad synergies to really understand the number line. It's not even called that, it's called the real torticular space." or something to that effect -- [...] Found on the WWW In the last section we have seen that cross ratios form a universal link from projective geometry to the underlying coordinate field. In this chapter we want to elaborate more on this topic. Thereby we will see that one can recover the structure of the real numbers from the purely geometric setup of the real projective plane. A line has (seemingly) by far less geometric structure than the plane. In the plane there can be collinear and non-collinear points, and we have seen that being collinear is a crucial property that stays invariant under projective transformations. On a line all triples of points are automatically collinear, so we cannot hope for this being a crucial characterizing property of projective transformations. The simplest non-trivial property that stays invariant under a projective transformation on a line is the specific value of the cross ratio of a quadruple of points. It will be the ultimate goal of this chapter to show that a map that fixes a certain value for the cross ratio of four points automati- cally fixes all cross ratios and will automatically be a projective transforma- tion. From there we will prove Theorem 3.3 that claimed that any bijective 2 transformation in RP that preserves collinearity is automatically a projective transformation. 86 5 Calculating with points on lines 5.1 Harmonic points We will now focus on the geometric properties of quadruples of points on a line that have cross ratio of (a, b; c, d) = 1. It is clear that if we fix points a, b, c on a line then this condition uniquely−determines the position of the last point d. So, from a projective viewpoint there is essentially only one point set (up to projective transformations) that satisfies this condition. However, under the aspect of coordinates there are many interesting cases that can be considered. Definition 5.1. Two pairs of points (a, b) and (c, d) are in harmonic position with respect to each other if the cross ratio (a, b; c, d) is 1. − First we observe that this definition is well defined, this means that it really is invariant under transposition of the points in the pairs. Lemma 5.1. If (a, b; c, d) = 1 we also have (b, a; c, d) = (a, b; d, c) = (c, d; a, b) = 1. − − Proof. This lemma is an immediate consequence of Theorem 4.2 and the fact that 1/( 1) = 1. − − #" In fact, if a = b and c = d and the cross ratio (a, b; c, d) stays invariant under interchange $ of the last$ two letters the points must be in harmonic position. This is the case since we then have 1 1 (a, b; c, d) = = (a, b; d, c) (a, b; c, d) The only values for the cross ratio that satisfy this equality are 1 and 1. Since a = b and c = d this excludes the first case. − If a, $b, c on a lin$ e ! are given the fourth harmonic point can be constructed in the following way: Start with an auxiliary point o, • connect this point to a, b, and c, • on join(o, c) choose another auxiliary point p, • construct d by • d = meet(!, join(meet(join(o, a), join(p, b)), meet(join(o, b), join(p, s))). Figure 5.1 shows the construction. The construction is not feasible when a and b coincide. Lemma 5.2. Independent of the choice of the auxiliary points o and p the construction presented above will end up with the same point d. This point satisfies (a, b; c, d) = 1. − 5.1 Harmonic points 87 o PSfrag replacements a! c! b! p a c b d Fig. 5.1. Construction of harmonic points. Proof. For the labelling we refer to Figure 5.1 . The point o can be considered as the center of projection from ! to the line spanned by b! and a!. Thus we get (a, b; c, d) = (a!, b!; c!, d). Similarly, p could be considered as a center of projection which implies (a!, b!; c!, d) = (b, a; c, d). Taking these two relations together we obtain (a, b; c, d) = (b, a; c, d) which implies (a, b; c, d) = 1. This also implies the independence of d from the specific choice of o and p−. "# There are a few remarkable collections of relative positions of points that generate a harmonic position. The following collection of equations collects some of them. For sake of simplicity we identify points on a line with the corresponding real numbers on R (including the point for the point at infinity). ∞ Lemma 5.3. Let x R and y R be arbitrary numbers. We have ∈ ∈ (i) ( x, x; 0, ) = 1, (ii) (0−, 2x; x, ∞) = −1, x+y∞ − (iii) (x, y; 2 , ) = 1, (iv) ( 1, 1; x, 1∞/x) = −1, (v) (−x, x; 1, x2) = −1. − − Proof. We start with a proof of property (v). We can simply check it via a calculation. representing determinants as differences of numbers (as described in Section 4.4.1.) We can calculate: ( x 1)(x x2) x( x 1)(1 x) − − − = − − − = 1 ( x x2)(x 1) x( 1 x)(x 1) − − − − − − − Statement (iv) is a consequence of statement (v) and arises by dividing all entries by x (this is a projective transformation and does not change the cross ratio). Statement (i) can be considered as a limit case of (iv) if x 0. Statement (ii) arises from (i) by adding x to all entries (this is again→a projective transformation). Statement (iii) arises from (i) also by scaling and shifting. #" 88 5 Calculating with points on lines 5.2 Projective scales We now fix three distinct points on a line, and call them 0, 1 and . Every point x on the line can be associated with a unique cross ratio (0,∞ ; 1, x). This construction allows us to equip the line with a scale that behave∞s “as if” point 0 is the origin, 1 is a unit point, and point is infinitely far away. In particular we have: ∞ Lemma 5.4. The following equations hold (i) (0, ; 1, 0) = 0 (ii) (0, ∞; 1, 1) = 1 (iii) (0, ∞; 1, ) = ∞ ∞ ∞ Proof. The lemma is a direct application of the definition of the cross ratio. "# The next lemma shows that the cross ratios with respect to these three points can be used to reconstruct the coordinates of a point from its geometric location. For this we set 0, 1 and to the position 0, 1 and on a real number line. ∞ ∞ Lemma 5.5. Assume that we have the specific homogeneous coordinates 0 = (0, 1), 0 = (1, 1), = (1, 0), x = (x, 1) for x R, then we have (0, ; 1, x) = x. ∞ ∈ ∞ Proof. We can proof this fact by direct calculation. 0 1 1 x det det 1 1 0 1 (0, ; 1, x) = ! " ! " = x ∞ 0 x 1 1 det det 1 1 0 1 ! " ! " "# If we single out three distinct points on a line we can always consider them as a projective basis and refer all measurements to these three points. By this we can assign to every point on the line a real number. The resulting number is then invariant under projective transformations. In a way this is like reconstructing the original scenery from a perspectively distorted photograph if the original positions of three points are given. 5.3 From geometry to real numbers From 1850 to 1856 the mathematician Karl Georg Christian von Staudt (1798- 1867) published a series of books under the title “Beitra¨ge zur Geometrie der Lage”. In these books von Staudt develops a completely synthetic (this 5.3 From geometry to real numbers 89 means there are no calculations) setup for projective geometry. In his setup for projective geometry he works on a similar axiomatic level as Euclid did it for Euclidean Geometry. One of the major achievements of von Staudt’s work was to provide a method that starts with a purely projective setup and reconstruct an underlying algebraic structure. In particular, he was able to reconstruct the field structure of the underlying coordinate field from properties of geometric constructions. We will not follow exactly these lines here. However, we will demonstrate how intimately the concepts of real numbers, cross ratios and projective trans- formations are interwoven. In particular, we will provide the promised proof 2 of Theorem 3.3 that claimed, that whenever we have a bijective map in RP that maps collinear points to collinear points, then it must be a projective transformation. We will even work a little harder and provide the proof for a similar fact on the projective line from which Theorem 3.3 will easily follow. We will show: 1 1 Theorem 5.1. Let τ: RP RP be a bijective map with the property that harmonic quadruples of poin→ts are mapped to harmonic quadruples of points. Then τ is a projective transformation. We subdivide the proof of this rather strong theorem in several smaller parts.
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