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 · y 2 + c · z 2 + 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

x21
 x22
 2
 x3
 2
 x4
x25
y12
y22
y32
y42
y52
z12
z22
z32
z42
z52
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x1 z1
x2 z2
x3 z3
x4 z4
x5 z5
 

 
a
y1 z1
0
b
  0
y2 z2 
 c  
   
y3 z3 
 · d = 0
 0
y4 z4  
e
y5 z5
0
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 twodimensional. One of these two degrees of freedom goes into the homogeneity of
10.1 Conic through five points
1
PSfrag replacements
165
4
PSfrag replacements
1
4
2
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 Plücker’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 Section 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: g1 := 1 × 3;
2: g2 := 2 × 4;
3: h1 := 1 × 3;
4: h2 := 2 × 4;
5: G := g1 g2T ;
6: H := h1 hT2 ;
7: M := q T HqG − q T 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
2
1
3
167
4
PSfrag replacements
p
q
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 determinants 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
(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. 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 C the value of (1, 2; 3, 4)p is invariant of the choice of p. Thus it
can be considered as a characteristic number that singles out the specific conic
C from all other conics through the four points. Thus we can take this number
as 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 use the 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 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 points
on 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 suitable
linear combinations αi ai + βi bi . If τ : Pl1 → Pl2 can be expressed as
10.3 Perspective generation of conics
PSfrag replacements
PSfrag replacements
1
2
3
4
q
p
1
2
3
4
q
p
169
Fig. 10.4. A point perspectivity and a line perspectivity.
' (
' (' ( ' (
α1
ab
α1
α2
τ(
)=
=
,
β1
cd
β1
β2
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 p1 and p2 be two distinct points in PR and o ∈ LR
not incident to p1 or p2 . Furthermore let Lp1 and Lp2 be the sets of lines
through the two points, respectively. The map τ : Lp1 → Lp2 defined 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 also
consider projective transformations τ : Lp1 → Lp2 in the corresponding dual
sense to point transformations. Again line-perspectivities are special projective transformations. Now, we will use Theorem 10.1 to prove the following
fact.
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 points
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. Consider 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 (l 1 , 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
2
3
4
q
p
1
2
3
4
q
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 C uniquely
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 C as well. Conversely, for
any point a on C{p, q} there is a line l that joins p and a. The intersection of
l 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 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 Lp and let it
sweep through the bundle. All intersections of l and τ (l) form the 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 coordinatisation (say we represent lines from the first bundle by λp l1 + µp l2 and points
from the second bundle
τ
1 + µ(
q m2 .) The projective transformation
' ( by'λq m
('
' (
λq
ab
λp
ab
can be written as
=
for a suitable matrix
. Thus the
µq
c d µp
cd
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 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 transformation τ : RP2 → 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 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
T
(T −1 p) A(T −1 p).
Thus we obtain the matrix of the transferred conic as
T
T −1 AT −1 .
Analogously the equation of the dual conic l T 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 C invariant. Such a transformation must map points on C to points on C. We here discuss the non-degenerate
case only and postpone the degenerate case 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 projective transformation τ : RP2 → RP2 that leaves C invariant induces a projective
transformations on C (considered as a projective line). For the following considerations we fix a non-degenearate conic C and identify it with the projective
line. As indicated in Section 10.2 we will speak of the cross ratio (1, 2; 3, 4)C
of four points on C which is (1, 2; 3, 4)p . for a suitably non-degenerate choice
of p.
Theorem 10.3. Let τ : RP2 → RP2 be a projective transformation that leaves
C 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 arbitrary point x on C is uniquely determined by the value of the cross ratio
(0, ∞; 1, x)C . Let τ : RP2 → RP2 be a projective transformation that leaves C
invariant. We will prove that the position of τ (x) is already defined by the
positions 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 are
well defined. Now p as well as τ (p) are on C. The other four image points are
also on C. Thus the above two cross ratios are the cross ratios are the cross
ratios (0, ∞; 1, x)C on C and (τ (0), τ (∞); τ (1), τ (x))C on C. Thus these two
cross ratios must be equal as claimed. This implies that the restriction of τ
to C must be a projective transformation.
+
*
The proof of the last theorem was algebraically simple but conceptually interesting. It relates a projective transformation on RP2 that leaves C invariant
to its action on C itself. With our concept of C representing the projective line
we see that in this world τ induces nothing else 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
C invariant by explicitly giving the images of four suitably chosen points on C.
10.4 Transformations and conics
d
173
τ (d)
τ (c)
PSfrag replacements
a
c
PSfrag replacements
τ
→
a
b
c
d
τ (b)
τ (a)
b
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 C such that (a, b; c, d)C = (a" , b" ; c" , d" )C . Then
there 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 points
a, b, c, d and the image points a" , b" , c" , d" . Thus we only have to show that τ
indeed leaves C invariant. Since a non-degenerate conic is uniquely determined
by five points on it it suffices to prove that there exists one more point p on
C whose image τ (p) is also on C. For this let p be an arbitrary point distinct
from 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 fact
that (a" , b" ; c" , d" )C = (a" , b" ; c" , d" )τ (p) shows that τ (p) also must lie on the
conic C.
+
*
Figure 10.6 shows a circle before and after a projective transformation that
leaves 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 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 .
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.
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 C to a line in
Lp : Each line is associated to its intersection with C different from p. There is
one line in the bundle that has to be treated separately. The tangent through
p is associated to p itself. (This reflects the limit situation when the point on
C 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 the
projective 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 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 Lp . It point p were outside the conic not all lines of the bundle would
intersect 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 C
and the lines in the bundle.
Let us close this section with a remark on the group structure of the set of
those transformations that leave C invariant. Theorem 10.3 can be interpreted
in the following way: The group of all projective transformations that leaves a
non-degenerate conic invariant is isomorphic to the group of transformations
of RP1 .
10.5 Hesses “Übertragungsprinzip”
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
RP2 by suitable objects of the projective line. This allows us to represent
statements in the two-dimensional world of RP2 by corresponding statements
10.5 Hesses “Übertragungsprinzip”
φ"p (c)
φp (d)
φp (f )
φp (e)
φp (h)
e
f
φp (g)
φp (h)
φp (i)
175
φp (j)
#
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 “Übertragungsprinzip”. 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.
In his original work Hesse related Points in RP2 to solutions of onedimensional 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
C as an image of a projective line. Now to a line l in RP2 we associate its
two points of intersection with C. A word of caution is necessary. First of
all not all lines will have two intersections with C. This corresponds to the
situation that Hesse studied solutions of arbitrary 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
talk about CP1 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 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 C in two (real or complex) points p1 , and
p2 is represented by the pair HC (l) := (p1, p2). If l is tangent to the conic at
point p we represent it by the pair HC (l) := (p, p). In all our considerations
related to Hesse’s transfer principle the 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 representation of a point of RP2 ? 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 C we take two arbitrary distinct lines l and m
through p that intersect C and consider the pairs of points HC (l) = (a1 , a2 ) and
HC (m) = (b1 , b2 ). These four points are distinct and since they lie on a nondegenerate conic no three of them are collinear. Thus there is a unique projective transformation τ : RP2 → RP2 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 )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 corresponding
transformation τp on C considered as RP1 . This is the object to which p is
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 C then the above construction does not lead to a proper
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 HC (k), HC (l)and HC (m). This characterization also covers the degenerate 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, m
are concurrent then (HC (k); HC (l); HC (m)) form a quadrilateral set.
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 “Übertragungsprinzip”
177
l3
H(l1 ) = (q, q)
l1
H(l1 ) = (p1 , p2 )
PSfrag replacements
p2
l2
p1
H(l1 ) = (p, p)
p
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, y 2 ) 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 y 2 xy  = −x · y 2 + 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 .
+
*
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10 Conics and perspectivities
11
10
9
8
7
6
5
4
3
2
1
PSfrag replacements
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
projected onto the x-axis. Thus the x-axis is the representation of RP1 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 C never intersect in
one point at least one of the lines must meet the conic in 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 HC (k) = (0, ∞). Now assume that the other
two lines l and m intersect the y-axis at the same point as required by the
theorem. 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-axis
in the same point we can consider them as an two instances of Matiyasevich’s
construction and we get:
(−lx ) · (ly ) = (−mx ) · (my ).
This expression can be used to prove the corresponding quadset relation. For
this we introduce homogeneous coordinates
' ( ' ( ' ( ' ( ' ( ' (
0
1
my
mx
l
lx
,
,
,
, y ,
1
1
0
1
1
1
10.5 Hesses “Übertragungsprinzip”
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 (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 1 )) mx ly )) 0 my
))
))
)
) 1 0 )) 1 1 )) 1 1
Expanding the determinants yields:
))
) )
))
) ) lx my )) mx 1 )) 0 ly
))
)=)
))
) ) 1 1 )) 1 0 )) 1 1
)
)
).
)
(−1)(mx − ly )(−my ) = (lx − my )(−1)(−ly)
which reduces to
−mx my + ly my = −lx ly + my ly.
Subtracting ly my 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.
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10 Conics and perspectivities
5
PSfrag replacements
3
1
X
Y
4
2
6
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. Figure 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
the transfer principle) with points in RP1 . 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 C with the projective line. If three
points 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 just
consists of 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
a
c
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
the 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.