10 Conics and perspectivities

10

Conics and perspectivities

Some people have got a mental horizon of radius zero andcall it their point of view.

Attributed to David Hilbert

In the last chapter we treated conics more or less as isolated objects. Wedefined points on them and lines tangent to them. Now we want to investigatevarious geometric and algebraic properties of conics. In particular, we will seehow 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 considerthe 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, . . . , f1. Multiplying all of them simultane-ously by the same non-zero scalar leads to the same conic. Thus the parametervector (a, . . . , f) behaves like a vector of homogeneous coordinates. Countingdegrees of freedom shows that in general it will take five points to uniquelydetermine a conic. To find the parameters for a conic through five pointspi = (xi, yi, zi); i = 1, . . . , 5 we simply have to solve the following linearsystem of equations:

1 Compared to Section 9.1 we have relabeled the parameters and put the factor of2 of the mixed terms into the parameters

164 10 Conics and perspectivities

x21 y2

1 z21 x1y1 x1z1 y1z1

x22 y2

2 z22 x2y2 x2z2 y2z2

x23 y2

3 z23 x3y3 x3z3 y3z3

x24 y2

4 z24 x4y4 x4z4 y4z4

x25 y2

5 z25 x5y5 x5z5 y5z5

·

abcdef

=

00000

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 than3 points are simultaneously collinear of if two points coincide the rank of thesystem may be lower than 5. This corresponds to the situation that there aremore than one conic passing through the given set of points. This method ofdetermining the parameter vector (a, . . . , f) is mathematically elegant how-ever it is computationally expensive. We first have to calculate the squaredparameters 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 andunderlying algebra of a conic. In preparation we have to understand how tocalculate a degenerate conic that consists of two lines with homogeneous coor-dinates g and h. A conic must be represented by a quadratic form pT APp = 0that vanishes if p is on either of the lines. The (non-symmetrized) matrix ofsuch 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 turncorresponds 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 thatline h is spanned by two points labeled 3 and 4. Then we have g = 1 × 2 andh = 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 calculatesthe conjunction between the two expressions.

Now, assume that we want to describe the set of conics that passes throughfor points 1, . . . , 4 in general position. Clearly, there are many conics thatsatisfy this condition. The corresponding system of linear equations con-sists 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

14

23

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

the conic parameters. Therefore we have a bundle of geometric solutions withone degree of freedom. Figure 10.1 (left) illustrates such a bundle of conics.Among these conics there are three degenerate conics, each of them passingthrough a pair of lines spanned by the four points. In Figure 10.1 (right) thesepairs of lines are marked by identical colors. They correspond to the followingfour 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 equationforms again a conic. This conic passes through all four points 1, . . . , 4 sinceboth summands vanish on these points. If λ and µ run trough all possiblevalues we obtain all the conics in the bundle through the four points. Applyingthe technique of Plucker’s µ (compare Section 6.3) we can adjust these values

14

2

3

q

Fig. 10.2. Constructing a conic through five points.


such that the resulting conic passes through another given point q. For thiswe 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 isquadratic in each of the six involved points. Figure 10.2 illustrates the situa-tion. We can also interpret it as a bracket condition encoding the (projectivelyinvariant) 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 nextSection.

Before this we will give the procedure for calculating calculate the sym-metric matrix for the conic through the five points 1, 2, 3, 4, q. We give it as akind of simple computer program:

1: g1 := 1 × 3;

2: g2 := 2 × 4;

3: h1 := 1 × 3;

4: h2 := 2 × 4;

5: G := g1gT2 ;

6: H := h1hT2 ;

7: M := qT HqG − qT GqH ;

8: A := M + MT ;

The matrix A assigned in the last line of the program contains the sym-metrized 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 thisequation is highly symmetric. For each bracket in one term its complement(the bracket consisting of the other three letters) is in the other term. Thesymmetry becomes a bit more transparent if we rewrite the equation withnew points labels:

10.2 Conics and cross ratios 167

1

2 3 4

q

p

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 rewritingequation (∗). We assume that the conic is non-degenerate and that none ofthe determinants vanishes. In this case we can rewrite (∗) to the form

[q, 1, 4][q, 2, 3]

[q, 1, 3][q, 2, 4]=

[p, 1, 4][p, 2, 3]

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

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

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

[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. Thesituation 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 distinctfrom the other four points. Then the cross ratio (1, 2; 3, 4)p is independent ofthe special choice of p.

We will later on see that this theorem is very closely related to the so calledexterior angle theorem for circles which states that in a circle a fixed secant isseen from an arbitrary point on the circle under the same angle (modulo π).


The last theorem enables us to speak of the cross ratio of four points on afixed conic as long as no more than two of the points (1, 2, 3, 4) coincide andwe can speak of a cross ratio at all. For this we simply chose an arbitrary pointp 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 toparameterize 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 thatfor a fixed conic C the value of (1, 2; 3, 4)p is invariant of the choice of p. Thus itcan be considered as a characteristic number that singles out the specific conicC from all other conics through the four points. Thus we can take this numberas a kind of coordinate for the conic within the one-dimensional bundle ofconics through 1, 2, 3, 4. In fact if we do so the three special degenerate conicsin 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 positionof the points 1, 2, 3. The point p may be an arbitrary point on the conicwhose exact position is not relevant for the calculations as long as it dies notcoincide with the other points. If point 4 takes all possible positions on theconic, then the value of (1, 2; 3, 4)p takes all possible values of R∪ {∞}, sincethe line p, 4 takes all possible positions through p. Thus we can use the crossratio (1, 2; 3, 4)p to characterize the position of 4 with respect to 1, 2, 3 on theconic. The three special values 0, 1 and ∞ are assumed when 4 is identical to1, 3, 2, respectively. In this setup we may consider the conic itself as a modelof the real projective line. The three points 1, 2, 3 above play the role of aprojective 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 projectiveline is a circle.

10.3 Perspective generation of conics

The considerations of the last section can be reversed in order to create conicsby perspective bundles of lines. For this we consider the points p and q ascenters of two bundles of lines that are projectively related to each other.Forming the intersections of corresponding lines from each bundle creates alocus of points that all have to lie on a single conic.

To formalize this fact (in particular to deal with the special cases) we haveto sharpen our notions on projective transformations slightly.

Definition 10.1. Let l1 and l2 be two distinct lines in LR and o ∈ PR

not incident to l1 or l2. Furthermore let Pl1 and Pl2 be the sets of pointson the two lines, respectively. The map τ :Pl1 → Pl2 defined by τ(p) =meet(l2, join(o, p)) is called a (point-)perspectivity.

We furthermore use the term projective transformation from Pl1 to Pl2

in the following sense. We represent the points on Pli ; i = 1, 2 by a suitablelinear combinations αiai + βibi. If τ :Pl1 → Pl2 can be expressed as

10.3 Perspective generation of conics 169

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

τ(

(α1

β1

)) =

(a bc d

)(α1

β1

)=

(α2

β2

),

then we call τ a projective transformation. Theorem 5.1 established that har-monic maps are projective transformation. In Lemma 4.3 we proved thatperspectivities are particular projective transformations. Dually we can alsospeak about perspectivities of bundles of lines.

Definition 10.2. Let p1 and p2 be two distinct points in PR and o ∈ LR

not incident to p1 or p2. Furthermore let Lp1and Lp2

be the sets of linesthrough the two points, respectively. The map τ :Lp1

→ Lp2defined by τ(l) =

join(p2,meet(o, l)) is called a (line-)perspectivity.

Figure 10.4 shows images for both types of perspectivities. We will alsoconsider projective transformations τ : Lp1

→ Lp2in the corresponding dual

sense to point transformations. Again line-perspectivities are special projec-tive transformations. Now, we will use Theorem 10.1 to prove the followingfact.

Theorem 10.2. Let p and q be two distinct points in RP2. Let Lp and Lq be

the sets of all lines that pass through p and q, respectively . Let τ :Lp → Lq

be a projective transformation which is not a perspectivity. Then the pointsmeet(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 oflines 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 writtenas (a1, a2; a3, a4)p = (a1, a2; a3, a4)q. Hence the six points a1, a2, a3, a4, p, qlie on a conic. Since τ is not a perspectivity the points a1, a2, a3 cannot becollinear. (Assume on the contrary that they lie on a line &. Then the im-age of an arbitrary fourth line l4 must satisfy the relation (l1, l2; l3, l4) =


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 C uniquelydefined by p, q, a1, a2, a3 is non-degenerate. Since l4 was chosen to be arbitraryall other intersections a = meet(l, τ(l)) must lie on C as well. Conversely, forany point a on C{p, q} there is a line l that joins p and a. The intersection ofl and τ(l) must be on the conic. Thus this intersection must be point a. *+

The last theorem gives us a nice procedure to explicitly generate a conicas a locus of points. The conic is determined by two points p and q and aprojective transformation between the line bundles through these two points.For generation of the conic we take a free line from the bundle Lp and let itsweep through the bundle. All intersections of l and τ(l) form the points ofthe conic. Dually if we have two lines l1 and l2 whose point sets are connectedby a projective transformation τ we can consider a point p freely movable onl1. 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) examplesof this generation principle. On the right two bundles of lines are shown wherethe second one simply arises from shifting and rotating the first one (this aparticularly simple projective transformation). The resulting generated conic,that comes from intersecting corresponding lines is a circle. This result couldalso be derived elementary by using the exterior angle theorem for circles. Wewill 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 correspondinglines yields the envelopes of a circle. One should compare these two pictureswith Figure 10.4 in which pairs of objects were shown that were related by aperspectivity. This case is the degenerate limit case of the above construction.

Remark 10.1. The construction underlying Theorem 10.2 also demonstratesthat the sets of points on a non-degenerate conic can be polynomially pa-rameterized (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 thecorresponding projective transformation τ . We introduce a projective basison each of the two bundles together with a suitable homogeneous coordinati-sation (say we represent lines from the first bundle by λpl1 + µpl2 and pointsfrom the second bundle by λqm1 + µqm2.) The projective transformation τ

can be written as

(λq

µq

)=

(a bc d

)(λp

µp

)for a suitable matrix

(a bc d

). Thus the

points on the conic have homogeneous coordinates

(tl1 + (1 − t)l2) × ((at + b(t − 1))m1 + (ct + d(t − 1))m2).

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 corresponding tot = ∞. The above formula is simply a polynomial function.

A similar statement is no longer true for curves of higher degree. In generalthey 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 whochange the shape of a conic (there we will study how we can derive the equa-tion of the transformed conic) and those that leave the conic invariant. Forthem we will have a look at the transformation group generated by thesetransformations.

The first task is simple. What happens to a conic under a projective trans-formation τ : RP

2 → RP2. 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 expressibleas multiplication by an invertible 3× 3 matrix T . Thus a point p on the conicbecomes transformed to a point Tp. Such a point should be in the transferredconic. This implies that the equation of the transformed conic is

(T−1p)TA(T−1p).

Thus we obtain the matrix of the transferred conic as

T−1TAT−1.

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

(T T l)TB(T T l)

and the matrix of the transformed dual conic becomes TBT T .


Let us turn to the more interesting task of studying all those projectivetransformations that leave a given fixed conic C invariant. Such a transforma-tion must map points on C to points on C. We here discuss the non-degeneratecase only and postpone the degenerate case to later chapters. The key to theclassification of such transformations is Theorem 10.1 which allows us to iden-tify the points on a conic with the points on a projective line and to associatea cross ratio to quadruples of such points. Our aim is to prove that a projec-tive transformation τ : RP

2 → RP2 that leaves C invariant induces a projective

transformations on C (considered as a projective line). For the following con-siderations we fix a non-degenearate conic C and identify it with the projectiveline. As indicated in Section 10.2 we will speak of the cross ratio (1, 2; 3, 4)Cof four points on C which is (1, 2; 3, 4)p. for a suitably non-degenerate choiceof p.

Theorem 10.3. Let τ : RP2 → RP

2 be a projective transformation that leavesC invariant. Then the restriction of τ to C is a projective transformation on C

Proof. Let 0, 1 and ∞ three distinct points on C. The position of an arbi-trary point x on C is uniquely determined by the value of the cross ratio(0,∞;1,x)C . Let τ : RP

2 → RP2 be a projective transformation that leaves C

invariant. We will prove that the position of τ(x) is already defined by thepositions of τ(0), τ(1) and τ(∞) and that we have in particular

(0,∞;1,x)C = (τ(0), τ(∞); τ(1), τ(x))C .

For this let p on C be chosen such that p does not coincide with the points 0,1, ∞, or x. Then τ(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)p = (τ(0), τ(∞); τ(1), τ(x))τ(p) .

The special choice of the position of p guarantees that the cross ratios arewell defined. Now p as well as τ(p) are on C. The other four image points arealso on C. Thus the above two cross ratios are the cross ratios are the crossratios (0,∞;1,x)C on C and (τ(0), τ(∞); τ(1), τ(x))C on C. Thus these twocross ratios must be equal as claimed. This implies that the restriction of τto C must be a projective transformation. *+

The proof of the last theorem was algebraically simple but conceptually in-teresting. It relates a projective transformation on RP

2 that leaves C invariantto its action on C itself. With our concept of C representing the projective linewe see that in this world τ induces nothing else but a 1-dimensional projectivetransformation. In the theorem it was crucial that the value of the cross ratioof four points seen from a fifth point p is independent of the choice of p. Thisallowed 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 leaveC invariant by explicitly giving the images of four suitably chosen points on C.

10.4 Transformations and conics 173

a

b

c

d

τ→

τ (a)

τ (b)

τ (c)

τ (d)

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 distinctpoints on a non-degenerate conic C such that (a, b; c, d)C = (a′, b′; c′, d′)C . Thenthere exists a unique projective transformation τ : RP

2 → RP2 with τ(a) = a′,

τ(b) = b′, τ(c) = c′, τ(d) = d′ which furthermore leaves C invariant.

Proof. The transformation τ is uniquely determined by the pre-image pointsa, b, c, d and the image points a′, b′, c′, d′. Thus we only have to show that τindeed leaves C invariant. Since a non-degenerate conic is uniquely determinedby five points on it it suffices to prove that there exists one more point p onC whose image τ(p) is also on C. For this let p be an arbitrary point distinctfrom the points a, b, c, d. Thus we have

(a′, b′; c′, d′)C = (a, b; c, d)C = (a, b; c, d)p =(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 factthat (a′, b′; c′, d′)C = (a′, b′; c′, d′)τ(p) shows that τ(p) also must lie on theconic C. *+

Figure 10.6 shows a circle before and after a projective transformation thatleaves the circle invariant. The transformation τ : RP

2 → RP2 is determined by

the image of the four red points. The position of the four image points cannotbe chosen arbitrarily. They must have the same cross ratio with respect tothe circle as the four pre-image points. In the situation shown in the picturethe four points are in harmonic position with respect to the circle. The whitepoints 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 circleis distorted by τ .

We can also make the relation of C to the projective line RP1 more explicit

and relate the points of C to the line bundle Lp of lines through a point p ∈ C.


Such a line bundle considered as a set of lines is by duality a representationof the projective line. We can explicitly relate every point on C to a line inLp: Each line is associated to its intersection with C different from p. There isone line in the bundle that has to be treated separately. The tangent throughp is associated to p itself. (This reflects the limit situation when the point onC approaches p). We can express this relation by a bijective map φp: C → Lp

from C to the bundle of lines through p. Now the last theorem states that theprojective transformation τ induces a projective transformation τp : Lp → Lp

in this line bundle via:τp(l) := φp(τ(φ−1

p (l))).

The reader is invited to convince himself that the limit case of the tangentthrough p fits seamlessly into this picture.

Figure 10.7 illustrates the relation of the points on the conic to the linebundle. In addition to the line bundle the picture also shows an additionalline & that is intersected with every line of the bundle. So the points on theconic are also in one-to-one correspondence to the points on &. The pictureexemplifies also how this relation of points on the conic to points on the lineis closely related to the classical stereographic projection, a relation that willbecome much more important later. It is kind of remarkable how important itis that the point p is really placed on the conic. If it were inside the conic wewould get a two-to-one relation between points on the conics and lines in thebundle Lp. It point p were outside the conic not all lines of the bundle wouldintersect the conic at all. An intersection of a line and a conic corresponds tosolving a quadratic equation. The fact that we consider a bundle at a pointon the conic implies that we already know one of the two solutions of thisquadratic equation. Thus solving the quadratic equation in principle can bereduced to a linear problem by factoring out the already known solution. Thelinearity is the deeper reason why there is a one-to-one correspondence of Cand the lines in the bundle.

Let us close this section with a remark on the group structure of the set ofthose transformations that leave C invariant. Theorem 10.3 can be interpretedin the following way: The group of all projective transformations that leaves anon-degenerate conic invariant is isomorphic to the group of transformationsof RP

1.

10.5 Hesses “Ubertragungsprinzip”

The last sections made it clear that we can identify a non-degenerate conicwith a projective line. In this section we will go even one step further. Wewill demonstrate a way how one can interpret arbitrary lines and points ofRP

2 by suitable objects of the projective line. This allows us to representstatements in the two-dimensional world of RP

2 by corresponding statements

10.5 Hesses “Ubertragungsprinzip” 175

a

b

c

d

ef g

h

i

j

k

p

φ"p(c)

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

φp(p)

#

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

of certain objects on the projective line. The idea of this translation goesback to an article of Otto Hesse from 1866. Hesse was mainly interested inquestions of invariant theory and studied several ways to linearize objects ifhigher degree. In his works around 1866 he was interested in generalizing theconcept of duality. Duality allows us to derive for every theorem of projectivegeometry a corresponding dual theorem just by applying a dictionary thattranslates “point” by “line”, “line” by “point”, “intersection” by “meet”, andso forth. In the same spirit Hesse formulated a principle that allowed it toderive a 1-dimensional theorem from any two-dimensional theorem of projec-tive geometry. He coined his principle by the term “Ubertragungsprinzip”. Areasonable translation of this term could be “principle of transfer”.

His work had far reaching consequences. It was used by Klein in his famousErlanger Programm to demonstrate the concept of equivalent geometries. Itinspired further work and many interesting generalizations. Some of thesegeneralizations had important impact on the classification of Lie algebras oreven on quantum theory. Within the present book we will use the transferprinciple for deriving elegant bracket expressions for geometric configurationsinvolving conics and lines.

In his original work Hesse related Points in RP2 to solutions of one-

dimensional quadratic forms. We will take a slightly more visual approachthat allows us to represent the solutions of the quadratic forms directly as in-tersections of a conic with a line. As before we consider a non-generate conicC as an image of a projective line. Now to a line l in RP

2 we associate itstwo points of intersection with C. A word of caution is necessary. First ofall not all lines will have two intersections with C. This corresponds to thesituation that Hesse studied solutions of arbitrary quadratic forms with realcoefficients. There may be two real solutions, two complex solutions (which areconjugates) or one (double) real solution. The three cases correspond to thesituations 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


solutions. This will be our first careful investigation of complex situations inprojective geometry. Thus to treat Hesses transfer principle properly we musttalk about CP

1 instead of RP1. 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 eitherconsider these complex elements (all algebraic considerations work straightforward) or one may assume (for convenience) that the conic is large enoughsuch that all lines under consideration intersect it in at least one point.

A line l that intersects the conic C in two (real or complex) points p1, andp2 is represented by the pair HC(l) := (p1, p2). If l is tangent to the conic atpoint p we represent it by the pair HC(l) := (p, p). In all our considerationsrelated to Hesse’s transfer principle the order of the points within such a pairwill be irrelevant. Nevertheless it is important to speak of pairs rather thansets to cover also the situation of a double point (p, p).

If lines are represented by pairs of points what is the corresponding rep-resentation of a point of RP

2? In Hesse’s transfer principle points would berepresented by projective transformations on the projective line that are fur-thermore involutions (i.e. τ2 = id). Such a transformation is derived in the fol-lowing way. For a point p not on C we take two arbitrary distinct lines l and mthrough p that intersect C and consider the pairs of points HC(l) = (a1, a2) andHC(m) = (b1, b2). These four points are distinct and since they lie on a non-degenerate conic no three of them are collinear. Thus there is a unique projec-tive transformation τ : RP

2 → RP2 that simultaneously interchanges a1 with

a2 and b1 with b2. In particular this transformation leaves l, m and p invari-ant. Furthermore we have (by Theorem 4.2) (a1, a2; b1, b2)C = (a2, a1; b2, b1)C .This in turn implies by Theorem 10.3 that τ leaves the conic C invariant.Such a projective transformation induces by Theorem 10.2 a correspondingtransformation τp on C considered as RP

1. This is the object to which p istranslated. The crucial fact on the definition of τp is that it only depends onthe choice of p but it is independent of the particular choice of l and m. Wewill not do this here.

The reason for this is that we want to bypass a certain technical problemrelated to expressing a point p by a projective transformation τp. If the pointp is on the conic C then the above construction does not lead to a properprojective transformation, since a1 and b1 (or b2) become identical. Insteadof introducing a concrete object that represents a point we will characterizeconcurrence of lines k, l, m directly by a relation of the corresponding pointpairs HC(k), HC(l)and HC(m). This characterization also covers the degener-ate cases in which the coincident point lies on C.

Theorem 10.5. Let C be a conic and let k, l, m be lines of RP2. To exclude

the complex case we assume that they intersect or touch the conic. If k, l, mare concurrent then (HC(k);HC(l);HC(m)) form a quadrilateral set.

We will prove this theorem by restriction to a remarkable special case bya suitable projective transformation. This special case was communicated by


p1

p2

p

l1

H(l1) = (p1, p2)

l2

H(l1) = (p, p)

l3

H(l1) = (q, q)

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

Yuri Matiyasevich (private communication) who discovered this remarkableconfiguration as a high-school student. Matiyasevich’s configuration is a kindof geometric gadget for performing multiplications. He used this gadget togive a geometric construction for the prime numbers. We formulate it in thereal 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 threepoints are collinear.

det

−x y 0x2 y2 xy1 1 1

= −x · y2 + y · xy − (−x) · xy − y · x2 = 0.

*+

Figure 10.9 gives an impression of how the parabola-multiplication-deviceworks. For our purposes we must also cover the degenerate case y = −x. Thenthe join becomes a tangent and we obtain:

Lemma 10.2. Let x be a real number. Then the tangent at (−x, x2) to theparabola y = x2 crosses the y-axis at the point (0,−x2).

Proof. Also this can easily checked by direct calculation. The tangent hasslope −2x Hence the tangent has the equation f(t) = a− 2x.t resolving for agives −x2 = a − 2x(−x). Thus a must be −x2. *+


1

10

9

8

7

6

5

3

2

1

5 4 3 2 0 654321

4

11

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 verticallyprojected onto the x-axis. Thus the x-axis is the representation of RP

1 thatis isomorphic to the points on the conic (the unique infinite point of theparabola corresponds to the point at infinity of the x-axis). The line shownin 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 andblue point on the x axis corresponds to the Hesse-pair that represents theline. Now we are ready to prove Theorem 10.5 (which is essentially Hesse’stransfer principle) as a simple Application of Matiyasevich’s construction.

Proof of Theorem 10.5: Since three tangents of a conic C never intersect inone point at least one of the lines must meet the conic in two points. Aftera suitable projective transformation we may assume that the conic p is theparabola 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 ∞ withthe RP 1 associated to the conic. The corresponding mapping goes via verticalprojection. Thus k is mapped to HC(k) = (0,∞). Now assume that the othertwo lines l and m intersect the y-axis at the same point as required by thetheorem. Let the corresponding point pairs on the x-axis be HC(l) = (lx, ly)and HC(m) = (mx, my). Since the two lines in the theorem intersect the y-axisin the same point we can consider them as an two instances of Matiyasevich’sconstruction and we get:

(−lx) · (ly) = (−mx) · (my).

This expression can be used to prove the corresponding quadset relation. Forthis we introduce homogeneous coordinates

(lx1

),

(ly1

),

(mx

1

),

(my

1

),

(01

),

(10

)


Fig. 10.10. Hesse’s transfer principle as incidence theorem.

and calculate the characteristic quadset equation of Section 8.2. For the sixpoints (lx, ly; mx, my; 0,∞) being a quadset we must show

[lx,∞][mx, ly][0, my] = [lx, my][mx,∞][0, ly].

This expands to∣∣∣∣lx 11 0

∣∣∣∣

∣∣∣∣mx ly1 1

∣∣∣∣

∣∣∣∣0 my

1 1

∣∣∣∣ =

∣∣∣∣lx my

1 1

∣∣∣∣

∣∣∣∣mx 11 0

∣∣∣∣

∣∣∣∣0 ly1 1

∣∣∣∣.

Expanding the determinants yields:

(−1)(mx − ly)(−my) = (lx − my)(−1)(−ly)

which reduces to−mxmy + lymy = −lxly + myly.

Subtracting lymy on both sides leaves us exactly with the identity proved byMatiyasevich’s equation. *+

With Theorem 10.5 we reduced the essence of Hesse’s transfer principle toan incidence theorem in the projective plane. Lines are represented by pairsof points. Three lines intersect if the corresponding three pairs of points forma quadrilateral set. Figure 10.10 summarizes the essence of Hesse’s transferprinciple as an incidence theorem. The green lines are the three lines thatintersect. The six points of intersection seen from one point on the boundaryof the conic generate a line bundle that must form a quadrilateral set. Thered part of the figure certifies the quadset relation by the construction givenin Figure 8.2.


1

5 3

4

26

X

Y

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’sTheorem. This theorem was discovered already in 1640 by the famous BlaisePascal 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 intersec-tions 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 ofHesse’s transfer principle. We may assume that the three intersection pointsare distinct, since otherwise they are trivially collinear. For the labeling refer toFigure 10.11. In order to apply the transfer principle we will simply express thethree 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 (applyingthe transfer principle) with points in RP

1. We will also need two more pointson the conic, namely the intersections X and Y with the central conclusionline. 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 ) formsa 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 theequation:

[3Y ][14][X6] = [36][1Y ][X4].

Multiplying both left and right sides and canceling brackets that appear onboth (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 theo-rem. *+

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 (morethan 150 years after Pascal’s Theorem!).

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

Pascal’s Theorem also holds in limit cases in which one upto three consecu-tive points of the hexagon (1, . . . , 6) coincide. The join of two such consecutivepoints then becomes a tangent to the conic. We refer the reader to Section 1.4for 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 showthat it is extremely simple to construct a harmonic point on a non-degenerateconic. For this again we identify the conic C with the projective line. If threepoints a, b, c on C are given we want to construct a fourth point d such that(a, b; c, d)C = −1 holds. The construction us shown in Figure 10.13 and justconsists of two tangents at a and b and a join of their intersection to c. ByHesses 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


a

b

c

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]

[ad][cb]= −1.

Which is after a slight reordering of the letters easily recognized as the con-dition for (a, b; c, d) being harmonic.

It is an amazing fact that the construction of a harmonic point on a conicturns out to be even simpler than the corresponding task on a line. Thisreflects on the one hand the fundamental importance of conics and on the otherhand the fact that conics are closely related to involutions and involutions areclosely related to harmonic sets. In particular if we fix a and b and considerthe construction of point d as a function τ : RP

1 → RP1 with τ(c) = d, then

this map τ turns out to be a projective involution with fixpoints a and b.

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10 Conics and perspectivities