10 Conics and Perspectivities

10 Conics and perspectivities

Some people have got a mental horizon of radius zero and call it their point of view. Attributed to David Hilbert

In the last chapter we treated conics more or less as isolated objects. We deﬁned points on them and lines tangent to them. Now we want to investigate various geometric and algebraic properties of conics. In particular, we will see how we can treat conics on the level of bracket algebra.

10.1 Conic through ﬁve points

We start by calculating a conic through a given set of points. For this consider the quadratic equation that deﬁnes a conic.

a x2 + b y2 + c z2 + d xy + e xz + f yz = 0. · · · · · · This equation has six parameters a, . . . , f 1. Multiplying all of them simultaneously by the same non-zero scalar leads to the same conic. Thus the parameter vector (a, . . . , f) behaves like a vector of homogeneous coordinates. Counting degrees of freedom shows that in general it will take five points to uniquely determine a conic. To find the parameters for a conic through five points pi = (xi, yi, zi); i = 1, . . . , 5 we simply have to solve the following linear system of equations:

1 Compared to Section 9.1 we have relabeled the parameters and put the factor of 2 of the mixed terms into the parameters 164 10 Conics and perspectivities

2 2 2 a x1 y1 z1 x1y1 x1z1 y1z1 0 2 2 2 b x2 y2 z2 x2y2 x2z2 y2z2   0  2 2 2  c   x3 y3 z3 x3y3 x3z3 y3z3 = 0 2 2 2 ·  d   x4 y4 z4 x4y4 x4z4 y4z4     0   2 2 2   e     x y z x5y5 x5z5 y5z5     0   5 5 5   f            If this system has a full rank of 5 then there is an (up to scalar multiple) unique solution (a, . . . , f) that deﬁnes the corresponding conic. If more than 3 points are simultaneously collinear of if two points coincide the rank of the system may be lower than 5. This corresponds to the situation that there are more than one conic passing through the given set of points. This method of determining the parameter vector (a, . . . , f) is mathematically elegant however it is computationally expensive. We ﬁrst have to calculate the squared parameters and then have to solve a 5 times 6 system of equalities. There is also another way to calculate such a conic more or less directly. This way will also give us additional structural insight into the geometry and underlying algebra of a conic. In preparation we have to understand how to calculate a degenerate conic that consists of two lines with homogeneous coordinates g and h. A conic must be represented by a quadratic form pT AP p = 0 that vanishes if p is on either of the lines. The (non-symmetrized) matrix of such a quadratic form is simply given by

A = ghT .

This can be easily seen since the quadratic form

pT Ap = pT (ghT )p = (pT g)(hT p) = p, g p, h ! "! " vanish if one of the two scalar products on right side vanish. This in turn corresponds geometrically to the situation in which p is on g or on h. Assume that the line g is spanned by two points labeled 1 and 2 and that line h is spanned by two points labeled 3 and 4. Then we have g = 1 2 and h = 3 4. The quadratic form becomes × × p, 1 2 p, 3 4 = 0. ! × "! × " We may as well express this term as the product of two determinants

[p, 1, 2][p, 3, 4] = 0.

Each factor describes a linear condition on the point p. The product calculates the conjunction between the two expressions. Now, assume that we want to describe the set of conics that passes through for points 1, . . . , 4 in general position. Clearly, there are many conics that satisfy this condition. The corresponding system of linear equations consists of four equations in six variables. Hence the solution space will be two- dimensional. One of these two degrees of freedom goes into the homogeneity of 10.1 Conic through ﬁve points 165

1 4

PSfrag replacements PSfrag replacements

1 2 4 3 2 3

Fig. 10.1. Bundles of conics though four points. Three degenerate special cases.

the conic parameters. Therefore we have a bundle of geometric solutions with one degree of freedom. Figure 10.1 (left) illustrates such a bundle of conics. Among these conics there are three degenerate conics, each of them passing through a pair of lines spanned by the four points. In Figure 10.1 (right) these pairs of lines are marked by identical colors. They correspond to the following four quadratic forms:

[p, 1, 2][p, 3, 4] = 0, [p, 1, 3][p, 2, 4] = 0, [p, 1, 4][p, 2, 3] = 0.

A linear combination of two of these forms (say the last two)

λ[p, 1, 3][p, 2, 4] + µ[p, 1, 4][p, 2, 3] = 0

generates again a quadratic form. The set of points p satisfying this equation forms again a conic. This conic passes through all four points 1, . . . , 4 since both summands vanish on these points. If λ and µ run trough all possible values we obtain all the conics in the bundle through the four points. Applying the technique of Plu¨cker’s µ (compare Section 6.3) we can adjust these values

1 4

PSfrag replacements q

3 2

Fig. 10.2. Constructing a conic through ﬁve points. 166 10 Conics and perspectivities such that the resulting conic passes through another given point q. For this we have to simply choose

λ = [q, 1, 4][q, 2, 3]; µ = [q, 1, 3][q, 2, 4]. − The resulting conic equation can be written as

[q, 1, 4][q, 2, 3][p, 1, 3][p, 2, 4] [q, 1, 3][q, 2, 4][p, 1, 4][p, 2, 3] = 0. − Observe that this equation is a multi-homogeneous bracket polynomial that is quadratic in each of the six involved points. Figure 10.2 illustrates the situation. We can also interpret it as a bracket condition encoding the (projectively invariant) property that six points 1, . . . , 4, p, q are on a conic (compare Sec- tion 6.4 and Section 7.3). We will come back to this interpretation in the next Section. Before this we will give the procedure for calculating calculate the symmetric matrix for the conic through the ﬁve points 1, 2, 3, 4, q. We give it as a kind of simple computer program: 1: g := 1 3; 1 × 2: g := 2 4; 2 × 3: h := 1 3; 1 × 4: h := 2 4; 2 × T 5: G := g1g2 ; T 6: H := h1h2 ; 7: M := qT HqG qT GqH; − 8: A := M + M T ; The matrix A assigned in the last line of the program contains the symmetrized matrix.

10.2 Conics and cross ratios

Let us come back to the equation

[q, 1, 4][q, 2, 3][p, 1, 3][p, 2, 4] [q, 1, 3][q, 2, 4][p, 1, 4][p, 2, 3] = 0, ( ) − ∗ which characterizes whether six points are on a conic. First observe that this equation is highly symmetric. For each bracket in one term its complement (the bracket consisting of the other three letters) is in the other term. The symmetry becomes a bit more transparent if we rewrite the equation with new points labels: 10.2 Conics and cross ratios 167

2 3 4 1

PSfrag replacements

Fig. 10.3. Four points on a conic seen from other points of a conic.

[A, B, C][A, Y, Z][X, B, Z][X, Y, C] [A, B, Z][A, Y, C][X, B, C][X, Y, Z] = 0. −

There is another important observation that we can make by rewriting equation ( ). We assume that the conic is non-degenerate and that none of the determ∗inants vanishes. In this case we can rewrite ( ) to the form ∗ [q, 1, 4][q, 2, 3] [p, 1, 4][p, 2, 3] = . [q, 1, 3][q, 2, 4] [p, 1, 3][p, 2, 4]

Both sides of the equation represent cross ratios. The left side is a cross ratio of the lines p1, p2, p3, p4 the right side of the equation is a cross ratio of the lines q1, q2, q3, q4. We abbreviate

[q, 1, 4][q, 2, 3] (1, 2; 3, 4) := . q [q, 1, 3][q, 2, 4]

This is the cross ratio of 1, 2, 3, 4 as “seen from” point q. Thus equation ( ) may be restated as ∗ (1, 2; 3, 4)q = (1, 2; 3, 4)p. Point p and point q see the points 1, 2, 3, 4 under the same cross ratio. The situation is shown in Figure 10.3 We summarize this in a theorem:

Theorem 10.1. Let 1, 2, 3, 4, p be ﬁve points on a conic such that p is distinct from the other four points. Then the cross ratio (1, 2; 3, 4)p is independent of the special choice of p.

We will later on see that this theorem is very closely related to the so called exterior angle theorem for circles which states that in a circle a ﬁxed secant is seen from an arbitrary point on the circle under the same angle (modulo π). 168 10 Conics and perspectivities

The last theorem enables us to speak of the cross ratio of four points on a fixed conic as long as no more than two of the points (1, 2, 3, 4) coincide and we can speak of a cross ratio at all. For this we simply chose an arbitrary point p that does not coincide with 1, 2, 3, 4 and take the cross ratio (1, 2; 3, 4)p. The theorem is useful under many aspects. In particular it is useful to parameterize classes of objects. We will investigate two of these applications. First assume that the points 1, 2, 3, 4 are fixed. The last theorem states that for a fixed conic the value of (1, 2; 3, 4)p is invariant of the choice of p. Thus it can be considereCd as a characteristic number that singles out the specific conic from all other conics through the four points. Thus we can take this number Cas a kind of coordinate for the conic within the one-dimensional bundle of conics through 1, 2, 3, 4. In fact if we do so the three special degenerate conics in this bundle (compare Figure 10.1) correspond to the values 0, 1 and . ∞ In the second application we fix the conic itself as well as the the position of the points 1, 2, 3. The point p may be an arbitrary point on the conic whose exact position is not relevant for the calculations as long as it dies not coincide with the other points. If point 4 takes all possible positions on the conic, then the value of (1, 2; 3, 4)p takes all possible values of R , since the line p, 4 takes all possible positions through p. Thus we can u∪s{e∞th}e cross ratio (1, 2; 3, 4)p to characterize the position of 4 with respect to 1, 2, 3 on the conic. The three special values 0, 1 and are assumed when 4 is identical to 1, 3, 2, respectively. In this setup we may∞consider the conic itself as a model of the real projective line. The three points 1, 2, 3 above play the role of a projective basis on this line with respect to which we measure the cross ratio. In this model it is obvious that the topological structure of the real projective line is a circle.

10.3 Perspective generation of conics

The considerations of the last section can be reversed in order to create conics by perspective bundles of lines. For this we consider the points p and q as centers of two bundles of lines that are projectively related to each other. Forming the intersections of corresponding lines from each bundle creates a locus of points that all have to lie on a single conic. To formalize this fact (in particular to deal with the special cases) we have to sharpen our notions on projective transformations slightly. Deﬁnition 10.1. Let l and l be two distinct lines in and o 1 2 LR ∈ PR not incident to l1 or l2. Furthermore let l1 and l2 be the sets of points on the two lines, respectively. The map τP: P deﬁned by τ(p) = Pl1 → Pl2 meet(l2, join(o, p)) is called a (point-)perspectivity. We furthermore use the term projective transformation from to Pl1 Pl2 in the following sense. We represent the points on li ; i = 1, 2 by a suitable linear combinations α a + β b . If τ: canPbe expressed as i i i i Pl1 → Pl2 10.3 Perspective generation of conics 169

PSfrag replacements PSfrag replacements 1 1 2 2 3 3 4 4 q q p p

Fig. 10.4. A point perspectivity and a line perspectivity.

α a b α α τ( 1 ) = 1 = 2 , β c d β β ' 1( ' (' 1 ( ' 2( then we call τ a projective transformation. Theorem 5.1 established that harmonic maps are projective transformation. In Lemma 4.3 we proved that perspectivities are particular projective transformations. Dually we can also speak about perspectivities of bundles of lines. Deﬁnition 10.2. Let p and p be two distinct points in and o 1 2 PR ∈ LR not incident to p1 or p2. Furthermore let p1 and p2 be the sets of lines through the two points, respectively. The maLp τ: L deﬁned by τ(l) = Lp1 → Lp2 join(p2, meet(o, l)) is called a (line-)perspectivity. Figure 10.4 shows images for both types of perspectivities. We will also

consider projective transformations τ : p1 p2 in the corresponding dual sense to point transformations. Again lLine-p→ersLpectivities are special projective transformations. Now, we will use Theorem 10.1 to prove the following fact.

2 Theorem 10.2. Let p and q be two distinct points in RP . Let p and q be L L the sets of all lines that pass through p and q, respectively . Let τ: p q be a projective transformation which is not a perspectivity. Then tLhe p→oinLts meet(l, τ(l)) are all points of a certain conic . C Proof. Let l1, l2, l3, l4 be four arbitrary lines from Lp not through q. Con- sider the points ai = meet(li, τ(li)); i = 1, . . . , 4. Since the two bundles of lines were related by a projective transformation the cross ratio (l1, l2; l3, l4) equals the cross ratio (τ(l1), τ(l2); τ(l3), τ(l4)). This relation can be written as (a1, a2; a3, a4)p = (a1, a2; a3, a4)q. Hence the six points a1, a2, a3, a4, p, q lie on a conic. Since τ is not a perspectivity the points a1, a2, a3 cannot be collinear. (Assume on the contrary that they lie on a line &. Then the image of an arbitrary fourth line l4 must satisfy the relation (l1, l2; l3, l4) = 170 10 Conics and perspectivities

PSfrag replacements PSfrag replacements 1 1 2 2 3 3 4 4 q q p p

Fig. 10.5. Generation of a conic by projective bundles.

(τ(l1), τ(l2); τ(l3), τ(l4)). Hence the intersections of l4 with & and τ(l4) with & must coincide. This means that τ is a perspectivity.) Thus the conic uniquely C defined by p, q, a1, a2, a3 is non-degenerate. Since l4 was chosen to be arbitrary all other intersections a = meet(l, τ(l)) must lie on as well. Conversely, for any point a on p, q there is a line l that joins p aCnd a. The intersection of l and τ(l) mustCb{e on}the conic. Thus this intersection must be point a. +* The last theorem gives us a nice procedure to explicitly generate a conic as a locus of points. The conic is determined by two points p and q and a projective transformation between the line bundles through these two points. For generation of the conic we take a free line from the bundle p and let it sweep through the bundle. All intersections of l and τ(l) form tLhe points of the conic. Dually if we have two lines l1 and l2 whose point sets are connected by a projective transformation τ we can consider a point p freely movable on l1. The lines join(p, τ(p)) forms the set of tangents to a particular conic. Figure 10.5 shows two particularly simple (but still interesting) examples of this generation principle. On the right two bundles of lines are shown where the second one simply arises from shifting and rotating the first one (this a particularly simple projective transformation). The resulting generated conic, that comes from intersecting corresponding lines is a circle. This result could also be derived elementary by using the exterior angle theorem for circles. We will later on see that this theorem is highly related to our conic constructions. The second example shows two sets of equidistant points on two different lines (they are again related by a projective transformation). Joining corresponding lines yields the envelopes of a circle. One should compare these two pictures with Figure 10.4 in which pairs of objects were shown that were related by a perspectivity. This case is the degenerate limit case of the above construction.

Remark 10.1. The construction underlying Theorem 10.2 also demonstrates that the sets of points on a non-degenerate conic can be polynomially parameterized (in homogeneous coordinates). For this consider two points p, q 10.4 Transformations and conics 171 on the conic. And the two corresponding bundles of lines together with the corresponding projective transformation τ. We introduce a projective basis on each of the two bundles together with a suitable homogeneous coordinati- sation (say we represent lines from the ﬁrst bundle by λpl1 + µpl2 and points from the second bundle by λqm1 + µqm2.) The projective transformation τ λ a b λ a b can be written as q = p for a suitable matrix . Thus the µq c d µp c d points on the coni'c ha(ve ho'mog(en'eou(s coordinates ' (

(tl + (1 t)l ) ((at + b(t 1))m + (ct + d(t 1))m ). 1 − 2 × − 1 − 2 Here t is a parameter that runs through all elements of R from to + . By this we get all points of the conic except for the one corre−sp∞onding∞to t = . The above formula is simply a polynomial function. A∞similar statement is no longer true for curves of higher degree. In general they cannot be parameterized by rational or even polynomial functions.

10.4 Transformations and conics

In this section we will deal with two types of transformations. Those who change the shape of a conic (there we will study how we can derive the equation of the transformed conic) and those that leave the conic invariant. For them we will have a look at the transformation group generated by these transformations. The ﬁrst task is simple. What happens to a conic under a projective trans- 2 2 formation τ: RP RP . The transformation is best understood if we write the conic equation→in matrix form

pT Ap = 0.

Now assume that we apply a projective transformation τ that is expressible as multiplication by an invertible 3 3 matrix T . Thus a point p on the conic becomes transformed to a point T p.×Such a point should be in the transferred conic. This implies that the equation of the transformed conic is

1 T 1 (T − p) A(T − p).

Thus we obtain the matrix of the transferred conic as

1T 1 T − AT − .

Analogously the equation of the dual conic lT Bl = 0 transfers to

T (T T l) B(T T l) and the matrix of the transformed dual conic becomes T BT T . 172 10 Conics and perspectivities

Let us turn to the more interesting task of studying all those projective transformations that leave a given fixed conic invariant. Such a transformation must map points on to points on . We hCere discuss the non-degenerate case only and postpone tChe degenerate Ccase to later chapters. The key to the classification of such transformations is Theorem 10.1 which allows us to identify the points on a conic with the points on a projective line and to associate a cross ratio to quadruples of such points. Our aim is to prove that a projec- 2 2 tive transformation τ: RP RP that leaves invariant induces a projective transformations on (cons→idered as a projectiCve line). For the following considerations we fix a nCon-degenearate conic and identify it with the projective line. As indicated in Section 10.2 we will Cspeak of the cross ratio (1, 2; 3, 4) C of four points on which is (1, 2; 3, 4)p. for a suitably non-degenerate choice of p. C

2 2 Theorem 10.3. Let τ: RP RP be a projective transformation that leaves invariant. Then the restric→tion of τ to is a projective transformation on C C C Proof. Let 0, 1 and three distinct points on . The position of an arbitrary point x on i∞s uniquely determined by tChe value of the cross ratio C 2 2 (0, ; 1, x) . Let τ: RP RP be a projective transformation that leaves inva∞riant. WC e will prove →that the position of τ(x) is already deﬁned by thCe positions of τ(0), τ(1) and τ( ) and that we have in particular ∞ (0, ; 1, x) = (τ(0), τ( ); τ(1), τ(x)) . ∞ C ∞ C For this let p on be chosen such that p does not coincide with the points 0, 1, , or x. ThenCτ(p) will automatically not coincide with τ(0), τ(1), τ( ), or ∞τ(x). Since τ is a projective transformation we have ∞

(0, ; 1, x) = (τ(0), τ( ); τ(1), τ(x)) . ∞ p ∞ τ(p) The special choice of the position of p guarantees that the cross ratios are well defined. Now p as well as τ(p) are on . The other four image points are also on . Thus the above two cross ratiosC are the cross ratios are the cross ratios (0C, ; 1, x) on and (τ(0), τ( ); τ(1), τ(x)) on . Thus these two cross ratio∞s must Cbe eqCual as claimed.∞This implies tChat tChe restriction of τ to must be a projective transformation. C +* The proof of the last theorem was algebraically simple but conceptually in- 2 teresting. It relates a projective transformation on RP that leaves invariant to its action on itself. With our concept of representing the proCjective line we see that in thCis world τ induces nothing elCse but a 1-dimensional projective transformation. In the theorem it was crucial that the value of the cross ratio of four points seen from a fifth point p is independent of the choice of p. This allowed us to relate the image seen from p to the image seen form τ(p). We can also take the opposite define a projective transformation that leave invariant by explicitly giving the images of four suitably chosen points on . C C 10.4 Transformations and conics 173

d τ(d) τ(c)

PSfrag replacements

τ a a τ(b) c → b c PSfrag replacements d τ(a)

Fig. 10.6. A transformation that leaves a conic invariant.

Theorem 10.4. Let a, b, c, d, and a", b", c", d" be two quadruples of distinct points on a non-degenerate conic such that (a, b; c, d) = (a", b"; c", d") . Then C 2C 2 C there exists a unique projective transformation τ: RP RP with τ(a) = a", → τ(b) = b", τ(c) = c", τ(d) = d" which furthermore leaves invariant. C Proof. The transformation τ is uniquely determined by the pre-image points a, b, c, d and the image points a", b", c", d". Thus we only have to show that τ indeed leaves invariant. Since a non-degenerate conic is uniquely determined by ﬁve pointsCon it it suﬃces to prove that there exists one more point p on whose image τ(p) is also on . For this let p be an arbitrary point distinct fCrom the points a, b, c, d. Thus Cwe have

(a", b"; c", d") = (a, b; c, d) = (a, b; c, d)p = C C (τ(a), τ(b); τ(c), τ(d))τ(p) = (a", b"; c", d")τ(p). The third equation holds since τ is a projective transformation. The fact that (a", b"; c", d") = (a", b"; c", d")τ(p) shows that τ(p) also must lie on the conic . C C +* Figure 10.6 shows a circle before and after a projective transformation that 2 2 leaves the circle invariant. The transformation τ: RP RP is determined by the image of the four red points. The position of the fo→ur image points cannot be chosen arbitrarily. They must have the same cross ratio with respect to the circle as the four pre-image points. In the situation shown in the picture the four points are in harmonic position with respect to the circle. The white points in the pre-image circle (left) map to the white points in the image circle (right). The lines and the central point indicate how the interior of the circle is distorted by τ. 1 We can also make the relation of to the projective line RP more explicit and relate the points of to the line Cbundle of lines through a point p . C Lp ∈ C 174 10 Conics and perspectivities

Such a line bundle considered as a set of lines is by duality a representation of the projective line. We can explicitly relate every point on to a line in C p: Each line is associated to its intersection with different from p. There is Lone line in the bundle that has to be treated separCately. The tangent through p is associated to p itself. (This reflects the limit situation when the point on approaches p). We can express this relation by a bijective map φp: p fCrom to the bundle of lines through p. Now the last theorem states tCha→t tLhe C projective transformation τ induces a projective transformation τp : p p in this line bundle via: L → L 1 τp(l) := φp(τ(φp− (l))). The reader is invited to convince himself that the limit case of the tangent through p fits seamlessly into this picture. Figure 10.7 illustrates the relation of the points on the conic to the line bundle. In addition to the line bundle the picture also shows an additional line & that is intersected with every line of the bundle. So the points on the conic are also in one-to-one correspondence to the points on &. The picture exemplifies also how this relation of points on the conic to points on the line is closely related to the classical stereographic projection, a relation that will become much more important later. It is kind of remarkable how important it is that the point p is really placed on the conic. If it were inside the conic we would get a two-to-one relation between points on the conics and lines in the bundle p. It point p were outside the conic not all lines of the bundle would intersecLt the conic at all. An intersection of a line and a conic corresponds to solving a quadratic equation. The fact that we consider a bundle at a point on the conic implies that we already know one of the two solutions of this quadratic equation. Thus solving the quadratic equation in principle can be reduced to a linear problem by factoring out the already known solution. The linearity is the deeper reason why there is a one-to-one correspondence of and the lines in the bundle. C Let us close this section with a remark on the group structure of the set of those transformations that leave invariant. Theorem 10.3 can be interpreted in the following way: The group oCf all projective transformations that leaves a non-degenerate conic invariant is isomorphic to the group of transformations 1 of RP .

10.5 Hesses “U¨ bertragungsprinzip”

The last sections made it clear that we can identify a non-degenerate conic with a projective line. In this section we will go even one step further. We will demonstrate a way how one can interpret arbitrary lines and points of 2 RP by suitable objects of the projective line. This allows us to represent 2 statements in the two-dimensional world of RP by corresponding statements PSfrag replacements

10.5 Hesses “U¨bertragungsprinzip” 175

" φp(d) φp(h) φp(e) φp(f) φp(g) φp(h) φp(i) φp(j) φp(c)

f # e g h d i c

b j

a k φp(p) p

Fig. 10.7. Generation of a conic by projective bundles.

of certain objects on the projective line. The idea of this translation goes back to an article of Otto Hesse from 1866. Hesse was mainly interested in questions of invariant theory and studied several ways to linearize objects if higher degree. In his works around 1866 he was interested in generalizing the concept of duality. Duality allows us to derive for every theorem of projective geometry a corresponding dual theorem just by applying a dictionary that translates “point” by “line”, “line” by “point”, “intersection” by “meet”, and so forth. In the same spirit Hesse formulated a principle that allowed it to derive a 1-dimensional theorem from any two-dimensional theorem of projective geometry. He coined his principle by the term “U¨bertragungsprinzip”. A reasonable translation of this term could be “principle of transfer”. His work had far reaching consequences. It was used by Klein in his famous Erlanger Programm to demonstrate the concept of equivalent geometries. It inspired further work and many interesting generalizations. Some of these generalizations had important impact on the classification of Lie algebras or even on quantum theory. Within the present book we will use the transfer principle for deriving elegant bracket expressions for geometric configurations involving conics and lines. 2 In his original work Hesse related Points in RP to solutions of one- dimensional quadratic forms. We will take a slightly more visual approach that allows us to represent the solutions of the quadratic forms directly as intersections of a conic with a line. As before we consider a non-generate conic 2 as an image of a projective line. Now to a line l in RP we associate its Ctwo points of intersection with . A word of caution is necessary. First of all not all lines will have two inCtersections with . This corresponds to the situation that Hesse studied solutions of arbitraryC quadratic forms with real coefficients. There may be two real solutions, two complex solutions (which are conjugates) or one (double) real solution. The three cases correspond to the situations where the line intersects in two, in no or in one point, respectively. To state Hesse’s ideas in full generality we have to also deal with the complex 176 10 Conics and perspectivities solutions. This will be our first careful investigation of complex situations in projective geometry. Thus to treat Hesses transfer principle properly we must 1 1 talk about CP instead of RP . However, the only objects we have to consider are pairs of points (p, q) which are either both real or complex conjugates (p = q) or coincide (p = q). For the following considerations one may either consider these complex elements (all algebraic considerations work straight forward) or one may assume (for convenience) that the conic is large enough such that all lines under consideration intersect it in at least one point. A line l that intersects the conic in two (real or complex) points p , and C 1 p2 is represented by the pair (l) := (p1, p2). If l is tangent to the conic at point p we represent it by theHpCair (l) := (p, p). In all our considerations related to Hesse’s transfer principle tHheC order of the points within such a pair will be irrelevant. Nevertheless it is important to speak of pairs rather than sets to cover also the situation of a double point (p, p). If lines are represented by pairs of points what is the corresponding rep- 2 resentation of a point of RP ? In Hesse’s transfer principle points would be represented by projective transformations on the projective line that are furthermore involutions (i.e. τ 2 = id). Such a transformation is derived in the following way. For a point p not on we take two arbitrary distinct lines l and m C through p that intersect and consider the pairs of points (l) = (a1, a2) and C HC (m) = (b1, b2). These four points are distinct and since they lie on a non- HdeCgenerate conic no three of them are collinear. Thus there is a unique projec- 2 2 tive transformation τ: RP RP that simultaneously interchanges a1 with → a2 and b1 with b2. In particular this transformation leaves l, m and p invariant. Furthermore we have (by Theorem 4.2) (a1, a2; b1, b2) = (a2, a1; b2, b1) . This in turn implies by Theorem 10.3 that τ leaves theC conic invarianCt. Such a projective transformation induces by Theorem 10.2 a coCrresponding 1 transformation τp on considered as RP . This is the object to which p is C translated. The crucial fact on the definition of τp is that it only depends on the choice of p but it is independent of the particular choice of l and m. We will not do this here. The reason for this is that we want to bypass a certain technical problem related to expressing a point p by a projective transformation τp. If the point p is on the conic then the above construction does not lead to a proper C projective transformation, since a1 and b1 (or b2) become identical. Instead of introducing a concrete object that represents a point we will characterize concurrence of lines k, l, m directly by a relation of the corresponding point pairs (k), (l)and (m). This characterization also covers the degenerate caHseCs in whHCich the cHoiCncident point lies on . C 2 Theorem 10.5. Let be a conic and let k, l, m be lines of RP . To exclude the complex case we aCssume that they intersect or touch the conic. If k, l, m are concurrent then ( (k); (l); (m)) form a quadrilateral set. HC HC HC We will prove this theorem by restriction to a remarkable special case by a suitable projective transformation. This special case was communicated by 10.5 Hesses “U¨bertragungsprinzip” 177

l1 (l1) = (q, q) H (l1) = (p1, p2) H PSfrag replacements p2

p1 (l1) = (p, p) H

Fig. 10.8. Hesse’s transfer principle for lines. Each line is associated to a pair of points. In case the line does not intersect the conic the points are complex and conjugates.

Yuri Matiyasevich (private communication) who discovered this remarkable conﬁguration as a high-school student. Matiyasevich’s conﬁguration is a kind of geometric gadget for performing multiplications. He used this gadget to give a geometric construction for the prime numbers. We formulate it in the real euclidean plane:

Lemma 10.1. Let x and y be two real numbers. The join of the points ( x, x2) and (y, y2) crosses the y-axis at the point (0, x y). − · Proof. We can proof this by direct calculation when we show that the three points are collinear.

x y 0 − det x2 y2 xy = x y2 + y xy ( x) xy y x2 = 0.  1 1 1  − · · − − · − ·   +* Figure 10.9 gives an impression of how the parabola-multiplication-device works. For our purposes we must also cover the degenerate case y = x. Then the join becomes a tangent and we obtain: −

Lemma 10.2. Let x be a real number. Then the tangent at ( x, x2) to the parabola y = x2 crosses the y-axis at the point (0, x2). − − Proof. Also this can easily checked by direct calculation. The tangent has slope 2x Hence the tangent has the equation f(t) = a 2x.t resolving for a gives −x2 = a 2x( x). Thus a must be x2. − − − − − +* 178 10 Conics and perspectivities

2 PSfrag replacements 1 a 5 4 3 2 1 0 1 2 3 4 5 6

Fig. 10.9. Multiplying by a parabola.

Now, what has Matiyasevich’s gadget to do with Hesse’s transfer principle. The parabola plays the role of the conic. The points on the conic are vertically 1 projected onto the x-axis. Thus the x-axis is the representation of RP that is isomorphic to the points on the conic (the unique infinite point of the parabola corresponds to the point at infinity of the x-axis). The line shown in Figure 10.9 intersects the conic in two points (the green and the blue one). They are associated to their x-value by the projection. Thus the green and blue point on the x axis corresponds to the Hesse-pair that represents the line. Now we are ready to prove Theorem 10.5 (which is essentially Hesse’s transfer principle) as a simple Application of Matiyasevich’s construction. Proof of Theorem 10.5: Since three tangents of a conic never intersect in one point at least one of the lines must meet the conic iCn two points. After a suitable projective transformation we may assume that the conic p is the parabola y = x2 (in Euclidean coordinates) and that one of the lines (say k) is the y-axis. We identify the x-axis together with its point at infinity with the RP 1 associated to the conic. The corresponding mapping goes via ∞vertical projection. Thus k is mapped to (k) = (0, ). Now assume that the other two lines l and m intersect the yH-aCxis at the∞same point as required by the theorem. Let the corresponding point pairs on the x-axis be (l) = (lx, ly) HC and (m) = (mx, my). Since the two lines in the theorem intersect the y-axis in thHe Csame point we can consider them as an two instances of Matiyasevich’s construction and we get: ( l ) (l ) = ( m ) (m ). − x · y − x · y This expression can be used to prove the corresponding quadset relation. For this we introduce homogeneous coordinates l l m m 0 1 x , y , x , y , , 1 1 1 1 1 0 ' ( ' ( ' ( ' ( ' ( ' ( 10.5 Hesses “U¨bertragungsprinzip” 179

Fig. 10.10. Hesse’s transfer principle as incidence theorem. and calculate the characteristic quadset equation of Section 8.2. For the six points (l , l ; m , m ; 0, ) being a quadset we must show x y x y ∞ [l , ][m , l ][0, m ] = [l , m ][m , ][0, l ]. x ∞ x y y x y x ∞ y This expands to l 1 m l 0 m l m m 1 0 l x x y y = x y x y . 1 0 1 1 1 1 1 1 1 0 1 1 ) )) )) ) ) )) )) ) ) )) )) ) ) )) )) ) ) )) )) ) ) )) )) ) Expanding the) deter))minants))yields:) ) )) )) ) ( 1)(m l )( m ) = (l m )( 1)( ly) − x − y − y x − y − − which reduces to m m + l m = l l + m ly. − x y y y − x y y Subtracting lymy on both sides leaves us exactly with the identity proved by Matiyasevich’s equation. +* With Theorem 10.5 we reduced the essence of Hesse’s transfer principle to an incidence theorem in the projective plane. Lines are represented by pairs of points. Three lines intersect if the corresponding three pairs of points form a quadrilateral set. Figure 10.10 summarizes the essence of Hesse’s transfer principle as an incidence theorem. The green lines are the three lines that intersect. The six points of intersection seen from one point on the boundary of the conic generate a line bundle that must form a quadrilateral set. The red part of the ﬁgure certiﬁes the quadset relation by the construction given in Figure 8.2. 180 10 Conics and perspectivities

5 3 PSfrag replacements 1

4 6 2

Fig. 10.11. Pascal’s Theorem

10.6 Pascal’s and Brianchon’s Theorem

No exposition on conics would be complete without a treatment of Pascal’s Theorem. This theorem was discovered already in 1640 by the famous Blaise Pascal and can be considered as a generalization of Pappus’s Theorem. Fig- ure 10.11 shows an instances of this theorem. Theorem 10.6. If 1, . . . , 6 are six points on a conic then the three intersections of opposite sides the hexagon (1, 2, 3, 4, 5, 6) are collinear. Proof. We already presented proofs of this theorem in Chapter 1. However, this time we want to add another prove which is a simple application of Hesse’s transfer principle. We may assume that the three intersection points are distinct, since otherwise they are trivially collinear. For the labeling refer to Figure 10.11. In order to apply the transfer principle we will simply express the three inner intersections of Pascal’s Theorem as quadrilateral set conditions. Since the six points 1, . . . , 6 all lie on the conic we can identify them (applying 1 the transfer principle) with points in RP . We will also need two more points on the conic, namely the intersections X and Y with the central conclusion line. Also they are considered as points in RP 1. Now the fact that 12, 45, XY , meet in a point is equivalent to the condition that (1, 2; 4, 5; X, Y ) forms a quadrilateral set. This corresponds to the algebraic condition [1Y ][52][X4] = [14][5Y ][X2]. Similarly the fact that 34, 16, XY , meet in a point can be encoded by the equation: [3Y ][14][X6] = [36][1Y ][X4]. Multiplying both left and right sides and canceling brackets that appear on both (the distinctness of the intersection points implies that they are non-zero) sides leaves us with: 10.7 Harmonic points on a conic 181

Fig. 10.12. Brianchon’s Theorem

[52][3Y ][X6] = [5Y ][36][X2] which implies that 32, 56 and XY meet in a point and thus proves the theorem. +* For reasons of completeness we also mention the dual of Pascal’s theorem. It is named after Charles Julien Brianchon and was discovered in 1804 (more than 150 years after Pascal’s Theorem!).

Theorem 10.7 (Brianchons Theorem). Let 1, . . . , 6 be six tangents to a conic (considered as the sides of a hexagon). Then the joins of opposite hexagon vertices meet in a point (see Figure 10.12).

Pascal’s Theorem also holds in limit cases in which one upto three consecutive points of the hexagon (1, . . . , 6) coincide. The join of two such consecutive points then becomes a tangent to the conic. We refer the reader to Section 1.4 for examples of such limit situations.

10.7 Harmonic points on a conic

As a (for now) ﬁnal application if Hesse’s transfer principle we wan to show that it is extremely simple to construct a harmonic point on a non-degenerate conic. For this again we identify the conic with the projective line. If three points a, b, c on are given we want to conCstruct a fourth point d such that (a, b; c, d) = 1Cholds. The construction us shown in Figure 10.13 and just consists oCf two−tangents at a and b and a join of their intersection to c. By Hesses transfer principle applied to this situation we get that (a, a; b, b; c, d) is a quadrilateral set (the tangents correspond to the double points(a, a) and (b, b)). This means that 182 10 Conics and perspectivities

PSfrag replacements

b d

Fig. 10.13. Construction of a harmonic quadruple (a, b; c, d) = 1 −

[ab][bd][ca] = [ad][ba][cb].

Dividing one term by the other and canceling the bracket [a, b] gives:

[bd][ca] = 1. [ad][cb] −

Which is after a slight reordering of the letters easily recognized as the condition for (a, b; c, d) being harmonic. It is an amazing fact that the construction of a harmonic point on a conic turns out to be even simpler than the corresponding task on a line. This reflects on the one hand the fundamental importance of conics and on the other hand the fact that conics are closely related to involutions and involutions are closely related to harmonic sets. In particular if we fix a and b and consider 1 1 the construction of point d as a function τ: RP RP with τ(c) = d, then this map τ turns out to be a projective involution→with fixpoints a and b.