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16 The complex projective

Now we will to study the simplest case of a complex : the complex . We will see that even this case has already very rich geometric interpretations. The close relation of complex arithmetic operation allows us to express geometric properties by nice algebraic structures. In par- ticular, this case will be the first example of a projective space in which we will be properly able to deal with circles.

1 16.1 CP

Let us recall how we introduced the . We took the one- dimensional space R considered it as a euclidean line and added one point at infinity. Topologically we obtained a circle. The best way to express the elements of the real projective line algebraically was to introduce . For this we associated to each number x R the vector (x, 1)T and identified non-zero scalar multiples. Finally we ident∈ified the vector (1, 0)T (and all its non-zero multiples) with the point at infinity. We ended up with the space 2 T 1 R (0, 0) RP = − { }. R 0 − { } We will do exactly the same for the complex numbers! To obtain the complex projective line we start with all the numbers in C. We associate every number z C with the vector (z, 1)T and identify non-zero scalar multiples. By this we∈ associate all vectors of the form (a, b)T ; b = 0 to a number in # C. What is left is the vector (1, 0)T and all its non-zero multiples. They will 1 represent a unique point at infinity. All in all we obtain the space CP defined by 2 T 1 C (0, 0) CP = − { }. C 0 − { } 248 16 The complex projective line

This space is isomorphic to all all complex numbers together with one point at infinity. What is the dimension of this space? In a sense this depends on the point of few. From the perspective of real numbers already the complex C is a two-dimensional object, since it requires two real parameters to specify the 1 objects of C. Hence one would say that also CP is a real-two-dimensional object since it differs from C just by one point. On the other hand from a complex perspective C is just a one-dimensional object. It contains just one (complex) parameter. As R is the line one could consider C as 1 the complex number line. Thus C is complex-one-dimensional, and so is CP . 1 This is the reason why we call the space CP the complex projective line! There is another issue important to mention in this context. Identifying vectors that only differ by a non-zero multiple this time also includes mul- tiplication by complex numbers. Thus (1, 2)T , (3i, 6i)T , (2 + i, 4 + 2i)T all represent the same point. For every point represented by (a, b)T with b = 0 # we can reconstruct the corresponding number of C by multiplication with 1/b. The dehomogenized number is then a/b C. We will also frequently identify 1 ∈ CP with the space C := C . As in the case of real numbers we use the standard rules for arithmeti∪c {o∞pe}rations with : ∞ ! 1/ = 0; 1/0 = ; 1 + = ; + = . ∞ ∞ ∞ ∞ ∞ ∞ ∞

16.2 Testing geometric properties

We will know consider identify the finite part of the complex projective (i.e. C) with the Euclidean plane R2 and investigate how certain geometric properties can be expressed in terms of algebraic expressions in C. Most of these tests will however not correspond to projectively invariant properties. We will first express the property that two vectors associated to two com- plex numbers z1 and z2 point into the same (or in the opposite) direction. For this we simply have to calculate the quotient z1/z2. Using polar coordinates we get (if z = 0) 2 #

iψ1 iψ2 i(ψ1 iψ2) z1/z2 = (r1e /r2e ) = (r1/r2)e − .

If the two vectors point in to the same direction we have ψ1 = ψ2 and the above expression is a positive real number. If they point in opposite directions iπ we have ψ = π + ψ and (since e− = 1) the above number is real and 1 2 − negative. In other word we could say that in the 0 z1 and z2 are collinear if the quotient z1/z2 is real (provided z2 = 0). Using complex conjugation we can even turn this into an equality, since# z is real if and only if z = z. We get z and z point into the same or opposite direction z /z = z /z . 1 2 ⇐⇒ 1 2 1 2 16.2 Testing geometric properties 249

C

PSfrag replacements B PSfrag replacements z2 A B z1 C z1 z2 z = r eiψ A ·

Fig. 16.1. Testing simple geometric properties.

We can use a similar test to decide whether three arbitrary distinct num- bers correspond to collinear points. Let A, B and C be three points in the complex plane. We represent these points by their corresponding complex numbers. Then these three points are collinear if the quotient (B A)/(C A) is real (provided the denominator does not vanish). This can is a−n immed−iate consequence of our previous considerations since this quotient simply com- pares the directions of the vectors B A and C A. Analogously to the last statement we get: − −

A, B, C are collinear (B A)/(C A) = (B A)/(C A). ⇐⇒ − − − − Unfortunately, the last two expressions do not fit into our concepts of projective invariants. The next one, however, will. We will describe whether four points lie commonly on a circle. For this we first need a well known theorem of elementary geometry: the peripheral angle theorem. This theorem states that if we have a circle and a secant from A to B on this circle. Then all points on the circle on one side of the secant “see” the secant under the same angle. The two (invariant) angles on the left and on the right side of the secant sum up to an angle of π. Figure 16.2 illustrates this theorem. We omit a proof here (it can be found in many textbooks of elementary geometry) but we formulate the theorem on the level of complex numbers. For this we denote by ∠A(B, C) the counterclockwise angle between the vectors B A and C A. We can summarize both cases of the peripheral angle theorem−by measuri−ng angle differences modulo multiples of π. In such a version the peripheral angle theorem reads:

Theorem 16.1. Let A and B, C, D be four points on a circle embedded in the complex plane. Then the angle difference ∠C(A, B) ∠D(A, B) is a multiple of π −

We postpone a proof of this version of the peripheral angle theorem until we have the possibility to prove it by a simple projective argument. If C and D 250 16 The complex projective line

α α C

α D α

PSfrag replacements PSfrag replacements

A α B π α − A C π α − π α B D −

Fig. 16.2. The peripheral angle theorem.

are on the same side of the secant through A and B both angles are equal and the difference is 0 π. If they are on opposite sides of the secant the difference is +π or π depen· ding on the orientation. We wi−ll interpret this theorem in the light of complex numbers. The angle ∠C (A, B) can be calculated as the angle ψ1 of the following complex number C A − = r eiψ1 . C B 1 − Similarly the angle ∠D(A, B) can be calculated as angle of D A − = r eiψ2 . D B 2 − We can get the difference of the angles simply by dividing these two numbers. We get C A D A i(ψ ψ ) − − = (r /r )e 1− 2 . C B D B 1 2 − − Since the angle difference is"a multiple of π by the peripheral angle theorem this number must be real. Now, the amazing fact is: the expression on the left is nothing else but a cross-ratio in the complex . Thus we can say that if the four points are on a circle, this cross-ratio (A, B; C, D) is a real number. The converse is also “almost” true we only have to include the special case that the circle may have infinite radius and degenerates to a line. It is easy to check that if A, B, C, D are collinear the cross-ratio is real, as well. In our considerations we have to consider as real number as well. The cross-ratio assumes this value if either C = B o∞r D = A. All in all we obtain the following beautiful theorem that highlights the close relation of complex and the geometry of circles. 1 Theorem 16.2. Four points in CP are cocircular or collinear if and only if the cross-ratio (A, B; C, D) is in R . ∪ {∞} 16.3 Projective transformations 251 16.3 Projective transformations

1 A projective transformation in CP can (as in the real case) be expressed T 2 by a matrix multiplication. If the vector (z1, z2) C represents a point in 1 ∈ CP by homogeneous coordinates. Then a projective transformation can be expressed as 1 1 τ: CP CP → z1 a b z1 z2 )→ c d z2 # $ # $# $ As usual the matrix must be non-degenerate. All entries can be complex. Ma- trices differing only by a non-zero scalar multiple represent the same projective transformation. Since the matrix has for complex parameters and scalar mul- tiples will be identified we have three complex degrees of freedom (or six real degrees of freedom, if one prefers). A complex projective transformation is uniquely determined by determining three pairs of images and pre-images. The same method as introduced in the proof of Theorem 3.4 can be used to obtain the concrete matrix if three pre-images and images are given. 1 As all projective transformations the projective transformations of CP leave cross-ratios invariant. In particular if a cross-ratio of four points A, B, C, D is real then the cross-ratio of the image points τ(A), τ(B), τ(C), τ(D) will be real again. Combing this fact with Theorem 16.2 we obtain the following remarkable fact that by projective transformations circles and lines are trans- formed to circles and lines.

1 Theorem 16.3. Let τ be a projective transformation of CP . Let A, B, C, D be four points on a circle or a line then the images τ(A), τ(B), τ(C), τ(D) will also be on a circle or on a line.

Proof. The proof is immediate since the fact that A, B, C, D is on a circle or on a line is characterized by (A, B; C, D) R . After applying the ∈ ∪ {∞} projective transformation the cross ratio is again in R . ∪ {∞} +* For reasons of convenience we will consider lines as vary large circles. One may think of lines as circles with infinite radius or alternatively of circles that pass through the point at infinity . This we may summarize the last theorem in the simple statement that a circ∞le is mapped by a projective transformation 1 1 τ: CP CP again into a circle. → 1 If we identify CP with C we may also express a projective transfor- ∪{∞} mation by simple rational expression in z C . A projective transfor- a b ∈ ∪ {∞} mation represented by the matrix leads via a sequence of homogeniza- c d ! " tion/transformation/dehomogenization to the following :

z a b z az + b az + b z = . )→ 1 )→ c d 1 cz + d )→ cz + d # $ # $# $ # $ 252 16 The complex projective line

Fig. 16.3. Iterated application of a Mo¨bius transformation.

Special care has to be taken if is involved. is mapped to a/c and if the denominator of the ratio is∞zero then the ∞result should be considered as zero. Such rational mapping C C is called a Mo¨bius transformation. Mo¨bius transformations are extensiv→ely studied on complex function theory. Our consideration show that the!y are !nothing else but complex projective 1 transformations in CP . We will briefly study different types of Mo¨bius transformations. We will 1 illustrate them by pictures in the finite part of C of CP . We will go from the most general to more special transformations. Fixpoints of projective trans- formations correspond to the Eigenvectors of the transformation matrices. Unlike the real case in the complex case the fixpoints will always be elements 1 of CP (In the real case it might have happened that a fixpoint is complex). If the two fixpoints are distinct we may specify a Mo¨bius transformation by the position of the two fixpoints and another point and its image. Figure 16.3 illustrates a most general Mo¨bius transformation. The trans- formation τ is defined by the two fixpoints (the green points) and the two red points one red point is mapped by τ to the other. The picture shows the iterated application of τ to one of the red points and the iterated application 1 of τ − to one of the red points. The yellow points represent these images. 1 The iterated application of τ and τ − converges to the two green fixpoints, respectively. Also the iterated images of a circle are shown. Observe that the images are again circles. Iterated application of Mo¨bius transformation may generate pictures of mind-twisting beauty. Figure 16.4 shows an example of another Mo¨bius trans- 16.3 Projective transformations 253

Fig. 16.4. Iterated application of a Mo¨bius transformation. formation (and its inverse) applied to a circle. As before the transformation is defined by two fixed points and another pair of points. The circles pro- duced by the iterated application of the transformation are colored blue and yellow alternatingly. A careful choice of the parameters produces interesting circle-packing patterns with several spiraling structures. Let us have a closer look at different kinds of Mo¨bius transformations that may occur. We will express the transformations in the form z az+b . )→ cz+d a b With non vanishing determinant . In particular if c = 0 and d = 1 the c d % % transformation assumes the simpl%e form% az + b. Thus in particular the linear % % transformations are Mo¨bius trans%form%ations. If furthermore a = 1 we have a simple shift along a translation vector that corresponds to b considered as vector. If b = 0 and a is a real number then z a z is a simple scaling around the origin. If on the other hand b = 0 and a)→is a· number on the unit circle a = eiψ then the transformation represents a rotation around the origin by an angle ψ. If a is neither real nor on the unit circle the transformation results in scaling around the origin combined with a rotation. If we iterate this process we produce spiral traces. The first row of Figure 16.5 represents the cases “a is real”, “a is neither real nor on the unit circle”, “a is on the unit circle”. Each arrow indicates the relation of a particular point and its image under the transformation. It is interesting to study the fixpoints for the simple mapping 254 16 The complex projective line

i 2 π i 2 π z 1.2 z z (1.2 e · · 36 ) z z e · · 36 z "→ · "→ · · "→ ·

(t 1)+(1+t)z (t 1)+(1+t)z (t 1)+(1+t)z − − − z (1+t)+(t 1)z z (1+t)+(t 1)z z (1+t)+(t 1)z "→ − "→ − "→ − i 2 π i 2 π t = 1.2 t = 1.2 e · · 36 t = e · · 36 ·

Fig. 16.5. Prototypes of Mo¨bius transformations.

a 0 z a z. The corresponding matrix is . The Eigenvectors are (0, 1)T )→ · 0 1 # $ and (1, 0)T . Thus these operation leave the point 0 and the point of C fixed. ∞ The second row of Figure 16.5 shows essentially the same transformations!. However, the fixpoints have been moved to other positions (namely 1 and 1). A projective transformation that maps 0 to 1 and maps to +1 −is given for instance by − ∞ 1 + z g(z) = − . 1 + z The inverse of this map is given by

1 1 z g− (z) = − − . 1 + z − 1 Calculating z g(t g− (z)) we obtain a transformation of the form )→ · (t 1) + (1 + t)z z − . )→ (1 + t) + (t 1)z − Such a transformation leaves 1 and +1 invariant and except of this trans- forms the multiplication by th−e factor t to the situation with these new fix- points. It is not an accident that first and last pictures of the second row 16.4 Inversions and Mo¨bius Reflections 255 somehow look like flux pictures from electrodynamics. Mo¨bius transforma- tions play an essential role in the field theory of electrostatic or magnetic charges.

16.4 Inversions and Mo¨bius Reflections

There is one subtle but important point in the theory of transformations in 1 CP which we have neglected so far. Our considerations of Chapter 5 made a 1 difference between harmonic maps and projective transformations. Over RP these two concepts coincide. The fundamental theorem of projective geometry, however, states that harmonic maps are the more general concept. Every har- monic map can be expressed by a field followed by a projective transformation. In fact, unlike R the field C possesses a non-trivial field auto- morphism: complex conjugation – and this is the only one. Thus in addition to projective transformations also complex conjugation z z leave the set of circles invariant. Geometrically this is not very surprising)→since complex con- jugation just resembles a mirror reflection in the real axis. The surprising fact is that such simple operations like reflection not at all covered by projective 1 transformations in CP . In a certain sense all Mo¨bius transformations will be orientation preserving. The characteristic magnitude preserved by Mo¨bius transformations can be considered as a kind of “circle sidedness predicate”. To make this precise we 1 define what we mean by the positive side of a circle in CP . To be precise here we must consider oriented circles defined by three points A, B, C. We denote such an oriented circle by (A, B, C). Such an oriented circle consists of the circle through A, B, C toge,ther with some orientation information. One may think of this orientation as a rotation sense on the boundary such that we tra- verse the three points in the order A, B, C (if the circle degenerates into a line it may happen that one has to path through infinity if one traverses A, B, C in this order). At every point of the circles boundary one may think of an arrow indicating the rotation sense. Now we rotate this arrow counterclockwise by 90◦. The rotated arrows point to side of the that we call its “positive side”. It is important to notice that by this definition the positive side may be either the interior or the exterior of the circle, depending on the order of A, B, C. If the circle degenerates to a line then one of the two halfspaces defined by the line becomes positive, the other one becomes negative, again depending on the order of A, B, C. There is also an easy algebraic characterization of the positive side of the circle We will take this as the formal definition. 1 Definition 16.1. Let A, B, C be three points in CP . The positive side of the circle (A, B, C) defined by A, B, C is the set , 1 p CP the imaginary part of (A, B; C, p) is positive . { ∈ | } 256 16 The complex projective line

Thus the imaginary part of the cross-ratio specifies the sides of the circle. It is positive on the positive side, zero on the boundary and negative on the negative side. The reader may convince himself that this definition agrees with our geometric definition. Now it is immediate to see that we have: 1 Theorem 16.4. Let A, B, C be three points of CP and let τ be a projective 1 transformation in CP . Then τ maps the interior of (A, B, C) to the interior of (τ(A), τ(B), τ(C)). , , Proof. The proof is immediate. Since τ is a projective transformation it does in particular preserve the sign of the imaginary part of cross ratios. +* 1 Thus we may say that projective geometry in CP does not only deal with circles it deals with oriented circles. Projective transformations map the pos- itive sides of oriented circles to positive sides of other oriented circles. Hence there cannot be a projective transformation that leaves the real axis point- wise invariant and interchanges its upper and its lower halfspace. Likewise there cannot be a projective transformation that leaves a circle invariant and interchanges its interior and its exterior. Complex conjugation z z helps to add these kind of transformations to our geometric system. It)→leaves real axis (which is a special “circle” with infinite radius) invariant. But it interchanges its positive and its negative side. A general harmonic map is a field automorphism followed by a projective transformation. Hence we may express those harmonic maps that are not projective transformation in the form

a z + b a b z · ; = 0. )→ c z + d c d # % % · % % % % We will call a map of this kind an anti-M% o¨biu%s transformation. Anti-Mo¨bius transformations preserve cocircularity but they interchange sidedness. In fact, every anti-Mo¨bius transformation has a circle that as a whole stays invariant under the transformation. Its interior and its exterior will be interchanged. We will not prove this here however we will restrict to a few important examples of such maps with important geometric significance. One of those maps we already encountered z z is the reflection in the real axis. Another important map is of this kind i)→s given by

1 1 ι: CP CP z → 1 )→ z This map is known as inversion in the unit circle. It leaves points on the complex unit pointwise circle invariant and interchanges the interior and the exterior of this circle. The map ι is an involution since ι(ι(z)) = z as an easy computation shows. The origin is mapped to the point . This operation is sometimes also called circle-reflection. However, one shou∞ld be aware that this operation does not represent a map that mimics optical reflection in a circle. 16.5 Grassmann Plu¨cker Relations 257

Reflections in other circles can also be expressed easily by using conjugation with a Mo¨bius transformation τ. If τ maps the unit circle to another circle C 1 then z τ(ι(τ − (z))) is the inversion in the circle C. The)→area of circle inversion is a very rich geometric field and by our con- siderations can be considered as very closely related to projective geometry. However, we will not go into details here. The interested reader is referred to the vast literature on inversive and circle geometry.

16.5 Grassmann Plu¨cker Relations

We now come back to our main track and investigate what conclusions 1 we could draw from bracket expressions in CP . We will first investigate 1 Grassmann-Plu¨cker Relations. As over the reals in CP the rank 2 three term Grassmann-Plu¨cker Relation holds. For any points quadruple of points 1 A, B, C, D CP we have ∈ [AB][CD] [AC][BD] + [AD][BC] = 0.. − There is an amazing consequence that we can draw from this formula that has a completely Euclidean interpretation. For this by AB we denote the distance prom a point A to a point B. | | Theorem 16.5. (Ptolemy’s Theorem) Let A, B, C, D be four points in the Euclidean plane. Then we have

AB CD + AD BC AC BD . | || | | || | ≥ | || | Equality holds if and only if the four points are on a common circle or line in the A, B, C, D.

Proof. First note the striking resemblance of Ptolemy’s formula and the Grass- mann Plu¨cker Relations. The brackets simply seem to be replaced by dis- tances. In fact we will see that Ptolemy’s Theorem may be considered as a kind of “shadow” of the Grassmann Plu¨cker Relation. For this we start with the relation [AB][CD] [AC][BD] + [AD][BC] = 0. − α γ β We represent each &of th'e( thr)ee s&um'm(and) by&a si'n(gle)letter. Thus the relation simply reads α γ + β = 0. Rewriting this we get α + β = γ. Now these three variables are co−mplex numbers. In polar coordinates they can be written as iψα iψ iψγ rαe , rβ e β , rγ e . The length rα is just the product of the lengths of [AB] and [CD]. If we assume that the points are embedded in the standard homogenization with [XY ] = X Y we obtain that r = AB CD . Similarly − α | || | we have rβ = AD BC and rγ = AC BD . By the triangle inequality we we get from α +| β =|| γ. |The inequal|ity || | 258 16 The complex projective line

A B B PSfrag replacements a A D b C c PSfrag replacements D C

D C C

Fig. 16.6. Ptolemy’s and Pythagoras’s Theorem.

α + β γ . | | | | ≥ | | This is just Ptolemy’s expression. Equality is obtained in this expression if all three vectors point into the same direction. In this case we have

[AB][CD] α + [AD][BC] β + = R and = R [AC][BD] γ ∈ [AC][BD] γ ∈

Thus in the case of equality these two cross ratios are real and positive. This is the case if and only if the four points are cocircular in this order. +* We will briefly have a look at a very special case of this theorem. Assume that A, B, C, D are the points of an rectangle in this order. The four points of a rectangle are cocircular. Furthermore opposite sides have same lengths and the diagonals have the same length. Thus the expression

AB CD + AD BC = AC BD . | || | | || | | || | may be written as AB 2 + BC 2 = AC 2. | | | | | | If we denote the sides of the rectangle by a and b and its diagonal by c then we get a2 + b2 = c2. This is just Pythagoras’s Theorem! In other words Pythagoras’s Theorem may be considered as a shadow of a Grassmann-Plu¨cker Relation. Algebraically it has exactly the same shape.