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. Thus lengths are no intrinsic concepts of similarity geometry, but angles and circles are. Ratios of lengths is also concept of similarity geometry. In fact, in the context of geometric theorems talking about similarity ge- ometry is usually more appropriate then talking about Euclidean geometry. The only difference between the two geometries is that in Euclidean geometry we can actually measure a length whereas in similarity geometry we can only compare lengths. Theorems of Euclidean geometry however are most often not formulated on the level of concrete lengths. Usually they only compare lengths relative to each other (you would not start any reasonable theorem with a sentence like: “Take a segment of length 3cm . . .”). So all the theorems we normally consider as Euclidean theorems are in fact theorems of similarity geometry. We will see that this is exactly the class of theorems governed by I and J. In Section 3.6 when we introduced projective transformations we first started by expressing rotations and translations and scalings (w.r.t. the stan- dard embedding) as multiplication by a 3 3 matrix. There we obtained the following forms for each of the transforma×tions, respectively:

cos(α) sin(α) 0 1 0 tx s 0 0 sin(α) cos(α) 0 ; 0 1 ty ; 0 s 0 . − 0 0 1 0 0 1  0 0 1       17.3 Transformations 265

A matrix that performs an orientation preserving similarity transformation (a combined rotation, translation and scaling) has the general form:

c s a s c b . −0 0 1   Similarly, a general orientation reversing similarity transformation has the form: c s a s c b . 0 −0 1 In both cases we must require c2 + s2 = 0for having a non-zero determinant. & Theorem 17.3. W.r.t the standard embedding orientation preserving similar- ity transformations are exactly those matrices that leave I and J invariant. ori- entation reversing similarity transformations are exactly those matrices that interchange I and J. Proof. Applying an orientation preserving similarity S to I we get

c s a i ic + s i − − − S I = s c b 1 = is + c = (c + is) 1 = (c + is)I. · −0 0 1 ·  0   0   0          The scalar c + is is non-zero since c2 + s2 = 0. Similarly we obtain that S J = (c + is) J. For an orientation reversing& similarity R the calculation is sim· ilar. We get·:

c s a i ic + s i R I = s c b −1 = −is c = ( c is) 1 = ( c is)J. · 0 −0 1 ·  0  − 0−  − − 0 − −         Again similarly we get R J = ( c is)I. Now we conversely assu· me th−at M− is a matrix that leaves I invariant. Since M is assumed to be a matrix that represents a projective transformation of 2 RP we may assume that M has real entries. We then have M I = λI. We now successively determine constraints on the entries of M. We ·first consider the first two entries of the last row of M. i i • • • −1 = λ −1 . x• y• • ·  0   0  •       We get ix + y = 0 Since the entries of the matrix must be real these two entries m−ust be zero. Since the matrix must have a non vanishing determinant the last entry of the last row must be non-zero. We may assume that it is 1 by rescaling the matrix. We now focus on the upper left 2 2 matrix. We have, × 266 17 Euclidean Geometry

u v i i w x • −1 = λ −1 . 0 0 1• ·  0   0        Thus we have ui + v = iλ and wi + x = λ. Subtracting i-times the first equation from−the second−we get: − u iv wi + x = 0. Since the matrix entries are real we must have u = x−an−d v =− w. The remaining two matrix entries can be arbitrary since they are multip−lied by 0. All in all our matrix has the form c s a s c b , −0 0 1 an orientation preserving similarity. A similar calculation shows that if M interchanges I and J we get an orientation reversing similarity. (' The last theorem states that we can characterize similarities by the prop- erty that the pair of points I, J is left invariant. Thus we can in a sense reverse our point of view and{mak}e I and J to the first class citizens and use them to define what “similarity geometry” means. We will now provide a de- tailed example of how one can introduce the concepts of similarity geometry entirely based an two special points I and J. If I and J are fixed we can define similarity transformations to be those projective transformations that leave the pair I, J invariant. A property that is invariant under similarity transformations i{s cal}led a similarity property. It will be our aim to base invariant properties and constructions of similarity ge- ometry directly on I and J without the detour via similarity transformations. In the previous section we already achieved this goal for one specific similarity property: cocircularity. We expressed cocircularity of four points A, B, C, D as a purely projective condition involving these points and the pair I, J . Now we again turn the point of view, make I and J the primary objec{ts an}d define cocircularity by the property of Theorem 17.2. Thus we have: Similarity transformations are those that fix the pair I, J . • A, B, C, D are cocircular if A, B, C, D, I, J lie on a con{ic. } • Based on these definitions we can derive the statement Cocircularity is invariant under similarity transformations. • Proof. A proof of this fact would look as follows: Assume that A, B, C, D are 2 2 cocircular (thus A, B, C, D, I, J lie on a conic ) and assume that τ: RP RP is a similarity transformation. We will prove that τ(A), τ(B), τ(C), τ(D→), are cocircular as well. Since τ is a projective transformation and being on a conic is a projectively invariant property the six points. τ(A), τ(B), τ(C), τ(D), τ(I) and τ(J) are on a conic as well. 17.4 Translating theorems 267

PSfrag replacements PSfrag replacements

A B E J D

E H H C

F G F A D I

G B C I J

Fig. 17.2. Miguel’s theorem and its projective interpretation.

Since τ is a similarity transformation we have either τ(I) = I and τ(J) = J or we have τ(I) = J and τ(J) = I. In either case this implies that τ(A), τ(B), τ(C), τ(D), I and J are on a conic which means that τ(A), τ(B), τ(C), τ(D), are cocircular. (' At first sight such a kind of reasoning may seem to be unnaturally compli- cated. However, it is conceptually very nice since we reduced the concept of cocircularity entirely to the introduction of two special points I and J and to projective invariants, without even referring to the particular coordinates of I and J. If we chose special coordinates I = ( i, 1, 0)T and J = (i, 1, 0)T then (w.r.t. the standard embedding) this setup sp−ecializes to our usual picture of similarity transformations and cocircularity.

17.4 Translating theorems

In particular the considerations of the last section show that behind every theorem of similarity geometry (or Euclidean geometry) there lies a projective truth. We will exemplify this by a nice theorem that needs cocircularity as only predicate. Theorem 17.4. Miquel’s Theorem: Let A, B, C, D, E, F, G, H be eight distinct points in the euclidean plane such that the following quadruples are co- circular: (A, B, C, D), (A, B, E, F ), (B, C, F, G), (C, D, G, H), (D, A, H, E). Then (E, F, G, H) will be cocircular as well. Proof. We will present a proof of Miquel’s Theorem based on projective in- variants. The hypotheses of the theorem imply that the following sixtuples lie on common conics: (A, B, C, D, I, J), (A, B, E, F, I, J), (B, C, F, G, I, J), (C, D, G, H, I, J), (D, A, H, E, I, J). This produces the following five bracket equations. 268 17 Euclidean Geometry

[CDJ][ABJ][BCI][ADI] = [ABI][CDI][ADJ][BCJ] [ABI][AEJ][BF J][EF I] = [ABJ][BF I][AEI][EF J] [BCJ][BF I][CGI][F GJ] = [BCI][BF J][CGJ][F GI] [CDI][CGJ][GHI][DHJ] = [CDJ][CGI][GHJ][DHJ] [ADJ][AEI][EHJ][DHI] = [ADI][AEJ][EHI][DHJ]

All brackets in these expressions will be non-zero, since they always involve two distinct finite points and either I or J. Multiplying all left sides and all rights sides and canceling brackets that appear on both sides we derive the equation.

[EF I][F GJ][EHJ][GHI] = [EF J][F GI]EHI][GHJ].

This expression is exactly the condition that also (E, F, G, H, I, J) are on a conic. And this implies the cocircularity of (E, F, G, H). (' Our proof did (except for non-degeneracy assumptions) not refer to the coordinates of I and J. The bracket argument also proves a corresponding purely projective theorem about eight points and six conics, where I and J are located at real and finite positions. The corresponding theorem is shown in Figure 17.2 on the right. The labeling is consistent with the labeling of Miquel’s Theorem on the left.

17.5 More geometric properties

Our next aim is to derive projective conditions for other concepts of simi- larity geometry. We start with perpendicularity. We want to characterize the property that for three points A, B, C the lines AB and AC are perpendicular to each other. If we associate the three points with the corresponding points of the complex number plane A, B and C then we can represent the above relation as an algebraic condition. For this we have to certify that the angle between the complex numbers A( (B and (A C is 90◦. This is the case if and only if − − (A ( B ( ( − iR. A C ∈ ( − ( This in turn translates to the equation: ( ( A B A B − = − . A C −)A C * ( − ( ( − ( 2 As before we reinterpret thi(s equ(ation in(RP (with the help of I and J. We get: [ABI] [ABJ] = . [ACI] − [ACJ] 17.5 More geometric properties 269 PSfrag replacements

I B

L J

l A M

l C m ∞ A

Fig. 17.3. Projective interpretation of orthogonality.

Slightly reordering the terms we get: [ABI][ACJ] 1 = = (B, C; I, J) . − [ACI][ABJ] A We can formulate this characterization in the following result. Theorem 17.5. The lines AB and AC are orthogonal if and only if the pairs of lines (AB, AC) and (AI, AJ) are in harmonic position. Alternatively we can directly speak of orthogonal lines by considering their intersections with the line at infinity in relation to I and J

Theorem 17.6. Two lines l and m are orthogonal if and only if their inter- sections with the line at infinity L = l l and M = m l are in harmonic position with I and J. × ∞ × ∞

Proof. Assume that l and m are orthogonal and that A is their intersection. Let B be a point on l and let C be a point on m. The cross-ratio (B, C; I, J)A is the same as the cross-ratio (L, M; I, J). Hence by Theorem 17.5 the point pairs L, M and I, J are in harmonic position. Co{nverse}ly, ass{ume }that L, M and I, J are in harmonic position. Then l and m cannot be parallel s{ince o}therwi{se th}e cross-ratio of the four infinite points would be 1. Thus we may assume that l and m have a finite intersection A. After again introducing two auxiliary points B and C on the two lines Theorem 17.5 proves the claim. (' The next geometric property we want to translate is closely related to characterizing circles by their midpoint and a point on the boundary.

Theorem 17.7. If the distance from A to B equals the distance from A to C, then [ABI][ACI][CBJ]2 = [ABJ][ACJ][CBI]2. 270 17 Euclidean Geometry

B

A

PSfrag replacements C

Fig. 17.4. Projective interpretation of orthogonality.

Proof. Again we first consider the situation realized in the complex number plane C. If AB = AC then the three points A, B, C form an isosceles tri- | | | | angle. In this this case the angle ∠B (A, C) is the same as the angle ∠C (B, A). This means that we have

A B B C − − R. C B A C ∈ ( − ( + ( − ( This is equivalent to ( ( ( ( (A B)(A C) (A B)(A C) − − = − − (C B)(B C) )(C B)(B C)* ( − ( ( − ( ( − ( ( − ( 2 Translated to RP t(his r(eads(as: (This is eq(uiva(lent(to (

[ABI][ACI][CBJ]2 = [ABJ][ACJ][CBI]2

Which is the desired equation. (' Observe that also the characterization in the last theorem turns out to be a projectively invariant expression. However, it is only a necessary condition for AB = AC . The point A occurs quadratically in this expression. Thus we m| ay|exp|ect |a conic section as locus for A if B and C are fixed. In fact this conic consists of two lines. One is the median of B and C the other this the join of B and C. The formula becomes zero if A is on one of this lines. Thus the case of A being the midpoint of a circle through B and C is only one situation when the above expression vanishes. Being on the join of B and C is the other. Later in this section we will also learn about a necessary and sufficient characterization of AB = AC . | | | | 17.6 Laguerre’s Formula 271 17.6 Laguerre’s Formula

So far we only used I and J to express properties that stay invariant un- der similarity transformations. We can even go one step further and perform measurements using I and J. In Section 4 we learned that the simplest way to extract projectively in- variant data from point configurations was to calculate cross-ratios. We will now use cross ratios in combination with I and J to mimic measurements in a projective setup. The key result (which has also many beautiful general- izations, as we will see later) is Laguerre’s formula. It was found in 1851 by Edmond Laguerre when he was just 19 years old. It allows to measure the angle between two lines. Before we state Laguerre’s formula we will clarify what exactly we mean by the angle between two lines l and m. Let us assume first that these two lines intersect in a single finite point O of the Euclidean plane. By the angle ∠(l, m) from l to m we mean the angle by which l has to be rotated counterclockwise around O until it coincides with m. By this angles between two lines will always lie in the open interval between 0 and π. If lines coincide or are parallel then the angle between them is 0. Alternatively one could say that the angle between l and m corresponds to the angle about which l has to be rotated that its direction coincides with m. One might ex- pect that angles should be measured between 0 and 2π. However this makes no sense for unoriented lines. Now we are ready to state Laguerres formula

2 Theorem 17.8. Let l and m be two finite lines of RP and let L = l l and M = m l the corresponding intersections with the line at infini×ty.∞Then the angle×be∞tween l and m is (modulo π) 1 ln (M, L; I, J) . 2i · Proof. The proof of this surprising ,result is stra-ight forward using the method 2 T of transferring geometric properties from RP to C. Let l = (l1, l2, l3) and T m = (m1, m2, m3) be the homogeneous coordinates of the lines. Then L and T M have the coordinates L = (l2, l1, 0) and M = (m2, m1, 0). The first two entries of these vectors are no−rmal vectors to the two l−ines. The angle of these normal vectors (modulo π) is exactly the angle between the two lines. We translate the normal vectors to two corresponding complex numbers

z = l i l = r eiψl and z = m i m = r eiψm . l 2 − · 1 l· m 2 − · 1 m· The angle ψm ψl modulo π is exactly the angle we are looking for. We can extract this an−gle by forming the following ratio:

iψm iψl iψm iψl zm zl rm e rl e− e e− 2i(ψ ψ ) · = · · · = · = e m− l z z r e iψm r eiψl e iψm eiψl m · l m· − · l· − · In the last expression the absolute values of zl and zm cancel since each number (or its conjugate) is used in the numerator and in the denominator. 272 17 Euclidean Geometry

Our considerations of Section 17.1 show that we have

zl = [L, I, l ]; zl = [L, J, l ]; zm = [M, I, l ]; zm = [M, J, l ]. ∞ ∞ ∞ ∞ Using these determinants to express the above expression we get

[M, I, l ][L, J, l ] zm zl 2i(ψ ψ ) (M, L; I, J) = ∞ ∞ = · = e m− l . [M, J, l ][L, I, l ] zm zl ∞ ∞ · Resolving for the desired angle we get: 1 ψ ψ = ln (M, L; I, J) m − l 2i · , - which is exactly Laguerre’s formula. (' Laguerre’s formula relates the properties of angles to the properties of cross ratios in a surprising way. We will collect a few of these properties: Measurement modulo π: The fact that angles between lines are measured modulo π is reflected by the fact that the natural logarithm function is only unique modulo a factor of 2πi (we have ea = e(a + 2πi)) together with the 1 factor 2i in Laguerre’s formula. Real lines generate real angles: At first it is surprising that Laguerre’s formula indeed produces only real values. If fact, if the lines have real coordi- [M,I,l ][L,J,l ] nates then the numerator and the denominator of (M, L; I, J) = [M,J,l∞][L,I,l∞] are complex conjugates. Dividing two conjugate numbers produces a∞compl∞ex number on the unit circle. Its logarithm is purely imaginary. This is compen- 1 sated by the factor 2i . Interchange of lines reverses angle: We must have ∠(l, m) = ∠(m, l) modulo π. Interchanging L and M transforms the cross-ratio to its−inverse: (L, M; I, J) = 1/(M, L; I, J). Since ln(a) = ln(1/a) modulo 2πi this pro- duces the desired sign change. − Angles are additive: If we have three lines h, l, m we must have ∠(l, m)+ ∠(m, h) = ∠(l, h) modulo π. This formula expresses the multiplicativity of the cross ratio. We have [M, I, l ][L, J, l ] [H, I, l ][M, J, l ] (M, L; I, J)(H, M; I, J) = ∞ ∞ ∞ ∞ [M, J, l ][L, I, l ] · [H, J, l ][M, I, l ] ∞ ∞ ∞ ∞ [H, I, l ][L, J, l ] = ∞ ∞ [H, J, l ][L, I, l ] ∞ ∞ = (H, L; I, J)

By ln(a b) = ln(a) + ln(b) modulo 2πi this translates to additivity of angles. · 17.7 Distances 273

Orientation preserving similarities leave angles invariant: If τ is an orientation preserving similarity we must have ∠(l, m) = ∠(τ(l), τ(m)) (here we interpret τ(l) as the corresponding action of τ on a line by multiplication of the inverse transformation matrix). To see this identity we calculate

(M, K, I, J) = (τ(M), τ(K); τ(I), τ(J)) = (τ(M), τ(K); I, J).

Hence the angle is the same after the transformation. Orientation reversing similarities reverse angles: If τ is an orientation reversing similarity we must have (l, m) = (τ(l), τ(m)). We get: ∠ −∠

(M, K, I, J) = (τ(M), τ(K); τ(I), τ(J)) = (τ(M), τ(K); J, I) = 1/(τ(M), τ(K); I, J).

Which produces (via the logarithm) the reversed angle. One should also observe that Laguerre’s formula contains as a special case the characterization of right angles. If l and m are orthogonal we have π iπ ∠(l, m) = 2 . Since e = 1 this translates via Laguerre’s formula to the condition (M, L; I, J) = 1−. This is exactly our characterization of Theorem 17.6. −

17.7 Distances

Using I and J we can also express distances between two points P and Q in Euclidean geometry. This formula is a little bit “tricky” and we will present it her without a strictly formal proof. First of all we cannot expect to express the distance purely as a projectively invariant formula only involving P , Q, I and J, since distance is not an invariant of similarity geometry. The only thing we can hope for is that we obtain a formula that compares the distance of P and Q to the distance of two reference points A and B. Thus we will compute P,Q a formula for | | . If the distance A, B is normalized to be the unit length A,B | | we will have a |form| ula for the distance of two arbitrary points. We will finally have an invariant expression in the six points A, B, P, Q, I, J. 2 For two points P = (p1, p2) and Q = (q1, q2) in the Euclidean plane R we usually calculate the distance via Pythagoras’s Theorem.

P, Q = (p p )2 + (q q )2. | | 1 − 2 1 − 2 We will first express this form.ula in terms of determinants and than enlarge this formula to get a projectively invariant expression. Again we assume that 274 17 Euclidean Geometry

P and Q have homogeneous coordinates w.r.t. the standard embedding. Thus T T we have P = (p1, p2, 1) and Q = (q1, q2, 1) . We now consider the expression [P, Q, I][P, Q, J] expanding this term, we get . p q i p q i 1 1 − 1 1 p2 q2 1 p2 q2 1 = ((q p ) + i(q p )) ((q p ) i(q p )) /' '' ' 1 − 1 2 − 2 · 1 − 1 − 2 − 2 0' 1 1 0 '' 1 1 0 ' 0' '' ' . 0' '' ' 1' '' ' = (p p )2 + (q q )2 ' '' ' 1 − 2 1 − 2 = P, Q . |. | This is exactly the desired distance. Unfortunately the expression

[P, Q, I][P, Q, J] is not at all a projective invar.iant. The following expression however is:

[P, Q, I][P, Q, J][A, I, J][B, I, J] = P Q .[A, B, I][A, B, J][P, I, J][Q, I, J] | | This expression in.deed is a projective invariant (each letter occurs with the same power in numerator and denominator) Furthermore for the case of the standard embedding and A, B = 1 we get exactly the right distance function (as the following compari|son o|f terms shows):

P,Q 2i 2i | | − − [P, Q, I][P, Q, J] [A, I, J] [B, I, J] = P, Q . .2 [A, B,3I4][A, B, J5]2[P,3I4, J5]2[Q,3I4, J5] | | 1 2i 2i . − − All in all we get:4 52 3 4 52 3 4 52 3 Theorem 17.9. The Euclidean distance of two points P and Q can be calcu- lated by [P, Q, I][P, Q, J][A, I, J][B, I, J] , .[A, B, I][A, B, J][P, I, J][Q, I, J] if A, B = 1 is a refer.ence length. | | It is also instructive to inspect this formula more closely. The occurrence of the square-root function expresses that it only makes sense to speak of distances up to a global sign. We can also compare the last result to the formula derived in Theorem 17.7, where we expressed a necessary condition for the property AB = AC . Theorem 17.9 contains this situation as special case. If we set P |= A| an|d Q|= C and square the expression of Theorem 17.9 we get 17.7 Distances 275

2 [A, C, I][A, C, J][A, I, J][B, I, J] 1 = ) .[A, B, I][A, B, J][A, I, J][C, I, J]* [A., C, I][A, C, J][A, I, J]2[B, I, J]2 = [A, B, I][A, B, J][A, I, J]2[C, I, J]2

[A, C, I][A, C, J][B, I, J]2 = . [A, B, I][A, B, J][C, I, J]2

This is also a necessary condition of AB = AC . In the formula | | | | [A, C, I][A, C, J][B, I, J]2 1 = [A, B, I][A, B, J][C, I, J]2 the points A, B and C occur quadratically. If we fix two of the points then the locus for the remaining one must be a conic. The conic for B is the circle with center A and perimeter point C. Similarly the conic for B is the circle with center A and perimeter point B. The conic for A turns out to consist of two lines. One of the lines is the median of B and C the other one is the line at infinity (as a simple calculation shows). If we combine the equation

[A, C, I][A, C, J][B, I, J]2 = [A, B, I][A, B, J][C, I, J]2

With the equation of Theorem 17.7

[A, B, I][A, C, I][C, B, J]2 = [A, B, J][A, C, J][C, B, I]2 by multiplying left an right sides we get

[A, C, I]2[C, B, J]2[B, I, J]2 = [A, B, J]2[C, B, I]2[C, I, J]2.

Taking the square-root on both sides We arrive at

[A, C, I][C, B, J][B, I, J] = [A, B, J][C, B, I][C, I, J]. ± Choosing the right sign in this expression turns out to be a necessary and sufficient condition for AB = AC . The formula that characterizes this case is: | | | | [A, C, I][C, B, J][B, I, J] = [A, B, J][C, B, I][C, I, J]. − Observe that A occurs linearly in this formula and both B and C occur quadratically.