17 Euclidean Geometry Dieses scho¨ne Resultat [...] blieb aber lange unbeachtet, vermutlich, weil sich die Geometer an den Gedanken gew¨ohnt hatten, dass Metrik und projektive Geometrie in keiner Beziehung zueinander sta¨nden.’ Felix Klein about Laguerres formula, Vorlesungen u¨ber Nicht-Euklidische Geometrie, 1928 1 2 In this chapter we will merge two different worlds: CP and RP . Both can be considered as representing a real two dimensional plane. They have different algebraic structures and they both represent different compactifications of the 2 1 Euclidean plane: For RP we added a line at infinity. For CP we added a point at infinity. Both spaces have different weaknesses and strengths. In the first 2 two parts of the book we learned that RP is very well suited for dealing for instance with incidences of lines and points, with conics in their general form and with cross-ratios. We did not have a proper way to talk about 2 1 circles, angels and distances in RP . The last two chapters introduced CP . This space was very good for dealing with co-circularity and also for dealing with angles. Unfortunately this space was not suitable for properly dealing with lines. However, lines were purely supported by CP 1. They had to be considered as circles with infinite radius and they were not even projectively invariant objects. We will now introduce an algebraic system that is capable of merging advantages of both worlds. We will end up with a framework in which we express all Euclidean properties by projectively invariant expressions. 260 17 Euclidean Geometry 17.1 The points I and J The key to expressing Euclidean properties in projective geometry is as simple as it is powerful. We have to introduce two special points. All Euclidean properties will be expressed as projectively invariant expressions in which these two points play a special role. There are several possibilities for the choice of these points. However, there is a special choice for the two points under which all formulas become very simple and elegant. The points are: i i I = −1 and J = 1 . 0 0 2 Strictly speaking these points are not even members of RP , since they have 2 complex coordinates. We will consider them formally as members of CP . 2 All algebraic calculations will be carried out in CP (we use homogeneous coordinates with complex coordinate entries). However the start elements and 2 the results of our calculations will usually be in RP . All our considerations 2 will refer to the standard embedding of the Euclidean plane R2 into RP by x x y . y "→ % & 1 Thus as usual the line at infinity will have coordinates l = (0, 0, 1)T . We will collect a few useful properties of the points I and J. ∞ I and J span the line at infinity: The points I and J both are on l since they have a zero in their last entry. Since they are two different poin∞ts they even span l . ∞ i i 0 0 −1 1 = 0 = 2i 0 . 0 × 0 2i − 1 − 1 I and J can transfer finite points from RP to C: Consider a point p with 2 homogeneous coordinates (a, b, 1)T in RP . In the euclidean plane this point would represent the point (a, b)T . In the complex plane it would represent the point a + ib. Now consider the 3 3 determinant [p, I, l ]. We get: × ∞ a i 0 [p, I, l ] = b −1 0 = a + ib. ∞ ' ' ' 1 0 1 ' ' ' ' ' ' ' Similarly we get the conjugate a +'ib: ' a i 0 [p, J, l ] = b 1 0 = a ib. ∞ ' ' − ' 1 0 1 ' ' ' ' ' ' ' ' ' 17.2 Cocircularity 261 1 2 I and J can express determinants in CP as determinants in RP : We consider 2 also the standard embedding in CP : z z . "→ 1 % & 2 T Now consider two points p1 and p2 in RP represented by vectors (a1, b1, 1) T 1 and (a2, b2, 1) . In CP they would represent points p1 = (a1 + ib1, 1) and p2 = (a2 + ib2, 1). Considering the determinant [p1, p2, I] we get: ( a a i ( 1 2 − [p , p , I] = b b 1 1 2 ' 1 2 ' ' 1 1 0 ' ' ' = a' 2 ib1 'a1 + ib2 ' − − ' ' ' = (a + ib ) (a + ib ) 2 2 − 1 1 = [p2, p1]. And similarly we get the conjugate ( ( [p1, p2, J] = [p2, p1]. 1 Thus the use of I and J allows us to express determinants of CP (and their 2 ( ( conjugates) as determinants of RP and involving I and J. The fact that we rely on the standard embedding in both worlds will not harm later on since in both worlds we will only have to deal with projectively invariant conditions. The last property is crucial. It is the key to translate projective in variants 1 2 of CP to projective invariants of RP . 17.2 Cocircularity Let us start applying I and J to express Euclidean properties. The strategy here will be: 1 Express the property in CP as bracket identity. • 2 Translate the identity bracket by bracket to RP (using I and J). • 2 Consider the translated identity as a projective invariant in RP . • 1 One of the most fundamental properties of CP was cocircularity. Four 1 points of CP are cocircular if their cross ratio was real. Assume that we are 2 given four points A, B, C, D in RP (w.r.t standard embedding). We consider 1 their corresponding counterparts in CP . The complex points A, B, C, D. It is easy to express cocircularity in terms of A, B, C, D. The points are cocircular if ( ( ( ( ( ( ( ( 262 17 Euclidean Geometry [AC][BD] R. [AD][BC] ∈ ( ( ( ( Or equivalently ( ( ( ( [AC][BD] [AC][BD] = . [AD][BC] )[AD][BC] * ( ( ( ( ( ( ( ( Expressing the brackets in 2 we get (RP( ( ( ( ( ( ( [ACI][BDI] [ACJ][BDJ] = . [ADI][BCI] [ADJ][BCJ] We can also multiply by the denominators to obtain a projectively invariant polynomial equation: [ACI][BDI][ADJ][BCJ] = [ACJ][BDJ][ADI][BCI]. Summarizing we get the following characterization of cocircularity: 2 Theorem 17.1. The points A, B, C, D in RP are cocircular if [ACI][BDI][ADJ][BCJ] = [ACJ][BDJ][ADI][BCI]. Comparing the above bracket expression with the expression of derived in Section 10.2 that characterized when six points are on a conic we observe that the expression [ACI][BDI][ADJ][BCJ] = [ACJ][BDJ][ADI][BCI]. expresses the fact that A, B, C, D, I, J lie on a common conic (each point occurs quadratically on either side). Thus we can reformulate the last theorem and state 2 Theorem 17.2. The points A, B, C, D in RP are cocircular if A, B, C, D, I, J are on a common conic. Or in other words: Circles are conics through I and J! The last fact can also be derived in a different way. If we consider the (euclidean) equation of a circle with midpoint (mx, my) and radius r (x m )2 + (x m )2 = r2 − x − y we can translate this into a quadratic equation in homogeneous coordinates. We obtain x2 + y2 2m xz 2m yz + (m2 + m2 r2) z2 = 0 − x · − y · x y − · which is for suitably chosen parameters a, b, c the special conic: 17.3 Transformations 263 PSfrag replacements J A D A I B B C C D Fig. 17.1. Projective interpretation of cocircularity. x2 + y2 + a xz + b yz + c z2 = 0. · · · Inserting the coordinates of I in this equation we get ( i)2 + 12 + a 0 + b 0 + c 0 = 1 + 1 = 0. − · · · − Hence I lies on this arbitrarily chosen circle. A similar calculation also shows that J lies on any circle, as well. Figure 17.1 illustrates the projective interpretation of cocircularity. Pro- jectively cocircularity of four points has to be considered as coconicallity of these four points with I and J. Usually one does not see I and J since they are complex (and at infinity). 17.3 Transformations Before we discuss further examples of expressing Euclidean properties in pro- jective terms we will have a brief look at the “philosophy” behind the approach of the last section. Our way of expressing cocircularity by a projectively in- variant expression did heavily rely on the standard embedding of Euclidean 2 1 geometry into projective geometry (as well in RP as in CP ). This standard embedding caused the particular choice of the coordinates for I and J. The crucial property of I and J is that they, considered as a pair, remain invari- ant under certain transformations (in particular under Euclidean transforma- tions). At this point we have to be a little careful to exactly specify which groups of transformations we consider. Roughly speaking, the geometrically 2 relevant groups of transformations in RP form a hierarchical system. In order of generality the groups relevant for us are projective transformations, affine transformations, similarity transformations, Euclidean transformations. The following table lists transformations and invariant properties that belong to these different subgroups the projective transformations. 264 17 Euclidean Geometry projective affine similarity Euclidean Transformations : general projective shear • scaling • • rotation • • • reflection • • • • translation • • • • • • • • Invariants : cross-ratio ratios of length • • • • angles • • • distances • • • We speak of projective geometry if we only consider properties that remain invariant under projective transformations. We speak of affine geometry if we only consider properties that remain invariant under affine transformations, and so on. Thus Euclidean geometry deals with properties that remain invari- ant under rotation, translation and reflection. Our considerations will now deal with similarity geometry. The difference to Euclidean transformations is that in addition to rotation, translation and reflection also scaling is allowed.