1. Introduction
The bi-Möbius transformations are functions
of the form:
(1)
where
and
. By denoting
and
,
, we have:
(2)
Let us notice that if
, then
, thus
is a Möbius transformation in
and if
, then
, thus
is a Möbius transformation in
, which justifies the name of bi-Möbius transformation we have given to
.
It is expected that the fixed points of
as a Möbius transformation in
depend on
. In reality, solving for
the equation
we obtain
. Since
, all the points
are fixed points of the Möbius transformation
. On the other hand, the equation
does not depend on
, therefore its solutions which are the fixed points
and
do not depend on
and are such that
and
. Similarly, the fixed points of
as a Möbius transformation in
do not depend on
. It can be easily checked that in fact they are the same, in other words,
for every
and
in
. In particular
, hence
and
. When
we have
, hence
, which has been excluded. When
, we have
, thus
defined by
has the double fixed point −1.
An easy computation shows that:
(3)
where
,
and
. Moreover,
is a Möbius transformation in each one of the variables, if the other variables do not take the values a or 1/a. Again, solving for
the equation
we obtain
, which indicates that
has the same fixed points
and
as the Möbius transformation in every variable when the other two
and
are such that
.
By using formula (3) repeatedly, we have computed in [1] several functions
. We provide here a list of them, which will be needed in Section 3 when dealing with the uniqueness of m-Möbius transformations. There is no harm to write
instead of
, where
.
When we checked for the fixed points of
as a Möbius transformation in
we got the equation:
where
are the symmetric sums in
, which shows that
has the same fixed points
and
as
and
. Due to the symmetry of
, this is true when it is considered as a Möbius transformation in any one of its variables if the other three are such that
. For
the equation is:
where
are the symmetric sums in
and again we find the same fixed points
and
when
is treated as a Möbius transformation in any one of its variables and
.
It is expected similar properties to be true for any m-Möbius transformation, yet for higher values of m, the computation becomes too tedious.
We notice that the coefficients of
, which are polynomials in
become more and more complicated as k increases. Yet, an interesting pattern should be noticed, namely that in every
the coefficient of
at the numerator is the same as the coefficient of
at the denominator. Also, if we compare the coefficients from
and from
we find another surprising pattern, which will be studied in detail in Section 3.
More generally, if
, then:
(4)
where
are symmetric sums of
and
are polynomials in
. Moreover,
is a Möbius transformation in every
if the other variables belong to
. This is an m-Möbius transformation, see [1] [2] [3] for more details about these transformations and their applications.
2. Images of Circles
The geometric properties of the m-Möbius transformations concern the way these mappings transform figures from each one of the planes (zk) into figures situated in the (w)-plane. As Möbius transformations, they are obviously conformal mappings, hence they will preserve the angles of those figures, except at singular points. They will transform circles (including straight lines, which can be considered circles of infinite radius) into circles. However, there are details which need to be examined.
A circle C centered at
and of radius r in the (w)-plane has the equation:
, or
.
It is convenient to write this equation under the form:
, (5)
where
and
. Indeed, we obtain (5) when we replace
by
and
by
into the equation
. Then, when
, this equation becomes that of a straight line. We will continue to call it circle (of infinite radius, or centered at infinity).
Theorem 1. The circle (5) is the image by
,
fixed, for
of circles:
(6)
from the (zk)-planes,
. More exactly, for every k, fixing
there is a unique circle (6) which is mapped bijectively by
onto the circle (5) from the (w)-plane.
Proof: Indeed, the coefficients
and
can be uniquely determined as functions of
and
by using
as follows. With fixed
, let us denote:
,
,
and:
.
Then:
hence, (5) becomes:
or:
which gives:
(7)
Obviously, different values of
will determine different circles (6) in the (zk)-plane which are mapped bijectively by
onto the circle (5). The bijective nature is assured by the fact that the Möbius transformations are bijective and
is a Möbius transformation of the (zk)-plane as long as every
remains fixed. Varying some of
, the coefficients
and
will all change and then
and
will be different, thus all circles (6) will change, despite of the fact that the circle (5) remains the same.
The concept of pre-image by
can be useful in order to describe this change. The pre-image by
of a point
is by definition
. The pre-image by
of a figure
is
. Since
depends only on the symmetric sums
the pre-image by
of any figure is invariant with respect to the permutations of the variables, i.e., it is a symmetric figure in
. In particular, the pre-image W of a single point w is a symmetric figure, which means that if
belongs to W, then so does any point obtained by a permutation of these coordinates.
Let us deal for simplicity with the case of
given by (2). We can chose arbitrarily
. Solving for
the equation:
we get:
(8)
so, with this value of
we have
, i.e.,
. We notice that
, as a function of
is a Möbius transformation and therefore a bijective mapping of
onto
, which means that the pre-image by
of a single point
is a subset of
in one-to-one correspondence with
. Given
in the (
)-plane there is a unique
in the (
)-plane, namely that given by (8) such that
. Due to the symmetry of
. this happens if and only if
, which means that W is a symmetric figure in
.
When
, then:
(8')
hence every point
, where
is given by (8') is carried by
into the origin. Due to the symmetry of
, the same is true for every point
, where
is arbitrary and
.
Now, suppose that w belongs to the circle C of equation (5). Let us denote by
the pre-image by
of the circle C, i.e.,
. The formula (8) tells us that for every
, we can pick up arbitrarily
and if
is given by (8), then
. But, for
fixed (8) is a Möbius transformation in w and it maps the circle C onto a circle
into the (
)-plane. We will call it the projection onto the (
)-plane of the section of
by
. Analogously, a projection
onto the (
)-plane can be defined of the section of
by
. The circles
and
are mapped bijectively by
onto the circle C when we keep
respectively
fixed in
. An easy computation shows that in terms of
and
the equation (8) is
, where
,
and
. When
varies, all these parameters vary and we obtain different formulas (8) and therefore different circles
. Hence, there are infinitely many circles into the planes (
) and (
) which are mapped bijectively by
onto the circle C, one for every
, respectively
. They are all projections onto the two planes of sections of
by
, respectively
. However, we can prove:
Theorem 2. There is a unique Möbius transformation of the (
)-plane into the (
)-plane which maps bijectively every circle
onto a circle
.
Proof. For a given
to every circle C in the (w)-plane corresponds uniquely a circle
which is the image of C by the Möbius transformation
of the (w)-plane into the (
)-plane and for every given
a unique circle
exists, which is the image of C by the Möbius transformation
of the
(w)-plane into the (
)-plane. The function
is a Möbius transformation of the (
)-plane into the (
)-plane which carries the circle
into the circle
. The affirmation of the theorem is not trivial since it asserts the uniqueness of such a Möbius transformation, while it is known that there are infinitely many Möbius transformations which map a given circle onto another given circle. They differ by rotations around the center of any one of these circles. The theorem states that the mapping
is uniquely determined by the circles
and
. Indeed, the two circles are in turn uniquely determined by the circle C and the two fixed values of
and
, which define uniquely the functions
and
.
The general case can be treated similarly. We choose
arbitrarily in
and let:
(9)
such that
. Due to the symmetry of
, this happens if and only if
where
is an arbitrary permutation of
. The formula (9) represents a Möbius transformation in w which maps a circle C of equation (5) onto a circle
into the (
)-plane when all the other variables are kept constant in
. The theorem 2 in the general form states that for every k and j,
there is a unique Möbius transformation of the (
)-plane into the (
)-plane which carries every circle
into a circle
.
Let us deal now with the symmetry with respect to a circle (see [4], page 80) of Equation (5). As we have seen, that equation represents a proper circle when
or a line when
. When that line is the real axis, we say that the points w and
are symmetric with respect to it. Yet, for any line it is known what symmetric points with respect to that line mean, namely z and
are symmetric points with respect to the line L if and only if L is the bisecting normal of the segment determined by z and
. This concept can be extended to the case when
. In that case the Equation (5) is
, where
is the center of the circle and r is its radius. As shown in [4], page 81 by using the tool of cross ratios, w and
are symmetric with respect to this circle if and only if:
. (10)
The symmetry principle states that if a Möbius transformation carries a circle
into a circle
, then it transforms any pair of symmetric points with respect to
into a pair of symmetric points with respect to
. Here circle means proper circle or line. This principle can be extended to m-Möbius transformations in the following way.
Theorem 3 (The Main Theorem). Let w and
be symmetric points with respect to the circle (5) and let W and
be the pre-images by
of w and respectively
. Then the projection onto the (
)-plane of any section of W and
obtained by keeping
fixed,
are points symmetric with respect to the circle (6) corresponding to that section.
Proof: By the Theorem 1, the circle (5) is the image by Möbius transformations of every circle (6) from the (
)-plane when
are kept fixed. Then the symmetry principle applied to these circles is exactly this theorem.
This theorem describes in fact a phenomenon happening in
related to m-Möbius transformations. If we consider
as a generalized circle in
and W as the equivalent of a point from
, then it makes sense to say that
is the symmetric of W with respect to
, since this is true for the projections on every (
)-plane of any section of them obtained by keeping
fixed,
. The projection of the respective section of
is
and that of the sections of W and
is
and
. The symmetric of
with respect to
. The theorem states that a m-Möbius transformation carries symmetric points with respect to
into symmetric points with respect to C.
There is a one-to-one mapping of W onto
assigning to every
the point
, where
is the symmetric of
with respect to
. It can be called reflection in
, extending to
a concept pertinent to
(see [4], page 81). The reflection in a circle is an involution and the composition of two reflections with respect to different circles result in a Möbius transformation. More generally, the composition of an odd numbers of reflections with respect to different circles is again a reflection and the composition of an even number of reflections is a Möbius transformation. Moreover, as shown in [4], page 172, every Möbius transformation can be obtained as the composition of two or four reflections in circles.
The symmetry principle states that if a Möbius transformation
carries the circle
into the circle
, then M composed with reflection in
is reflection in
. In other words
, where
on the left hand side means reflection in
and on the right hand side reflection in
.
We notice that (10) does not represent a Möbius transformation since it is anticonformal, while Möbius transformations are conformal mappings. Yet, in terms of mapping properties there are some similarities between the reflections in circles and Möbius transformations. One of them is that of preservation of circles ( [5], page 126). Even more can be said: if a line L does not pass through the center
of the circle C, then its reflection in C is a circle
passing through
and vice-versa, if a circle
passes through
then its reflection in C is a line L not passing through
. If the proper circle
does not pass through
then its reflection in C is another proper circle not passing through
.
Next we will investigate some similar properties of the reflections with respect to
and m-Möbius transformations of
.
Theorem 4. Let
be the pre-image by a m-Möbius transformation
of the circle
. Then
composed with reflection in
is reflection in C. If L is a line not passing through
, then
, where
and
are the pre-images of L and
. Moreover, if
is the image of
by reflection in
, then
. Reciprocally, if
is a proper circle which passes through
then
and
.
Proof: By the formula (9) and the symmetry principle we have
for every
. Then:
and the first term shows reflection in C, while the last one is
composed with reflection in
. The second affirmation is true since the pre-image of the intersection of two sets is equal to the intersection of the pre-images of those sets. Next,
moves
into the reflection of L in C which passes through
, hence the pre-image of
is included in
. Finally, if a proper circle
passes through
, then
should pass through
, which is the reflection in C of
. Thus
is a line not passing through
and then
.
Theorem 5. Given a circle
in the (
)-plane, for every
there are infinitely many circles
in every (
)-plane such that
and
have the same image by
when all the other variables are kept fixed.
Proof: Suppose that a circle C of Equation (5) is given in the (w)-plane. We are looking for a circle (6) which is mapped bijectively by
onto the circle C when
are kept all fixed. With the notations of Theorem 1, we have that if
is on
and w is on C where
, then
, as in Theorem 1, thus
, and the equation of:
becomes:
, or
which is:
,
where:
(11)
Hence the circle
is mapped bijectively by
onto the circle C when
and
are kept fixed. On the other hand, the projection onto the (
)-plane of any section of the pre-image
of C by
is a circle
which is mapped by
bijectively onto C when all the variables except
are kept fixed in
, therefore there are infinitely many circles
in every (
)-plane such that
and
have the same image by
.
The theory of Apollonius circles and of Steiner nets ( [4], page 85) can be extended word by word to m-Möbius transformations. Let us deal first with the case
. With the notation
,
and
the formula (1) becomes
, which shows that
corresponds to
and
corresponds to
, thus straight lines through the origin of the (w)-plane are images by
of circles
through the limit points
and
from the (
)-plane for every
. On the other hand, the concentric circles about the origin
, are the images by
of circles with equation
for every
. These are the Apollonius circles
with the limit poins
and
. Together with the
circles they form the Steiner net. Every
circle is orthogonal to all
circles and every
circle is orthogonal to all
circles.There is exactly one
circle and one
circle from the Seiner net defined by
passing through every point of the (
)-plane except the points
and
. Reflection in a
circle switch
and
and transforms every
circle into itself and every
circle into another
circle.
The pre-image by
of any Steiner net from the (w)-plane is an object in
whose sections by
and
are Steiner nets in the (
)-plane, respectively the (
)-plane. Indeed, this is true due to the formula (8) and the preservation of circles by Möbius transformations. The Theorem 2 implies the following:
Corollary 1. There is a unique Möbius transformation of the (
)-plane into the (
)-plane which carries the Steiner net determined by
into the Steiner net determined by
.
Proof: Indeed, by the Theorem 2 there is a unique Möbius transformation
which carries an Apollonius circle
into another Apollonius circle
. Then, by the symmetry principle every
circle is transformed into a
circle. Yet the family of these circles determines uniquely the family of the
circles, which are all the orthogonal circles to them. Finally, the whole Steiner net from the (
)-plane is mapped by
into the Steiner net determined by
.
The pre-image by
of a Steiner net from the (w)-plane is an object in
whose sections obtained by keeping all
fixed for
is a Steiner net in the (
)-plane, due to the formula (9) and the preservation of circles by Möbius transformations. Every point of
, except the pre-image of the limit points of the net, belongs to the pre-image of both families of circles belonging to the Steiner net from the (w)-plane.
Given a Steiner net in the (w)-plane, its pre-image by
into
projects into Steiner nets in every (
)-plane, the image of which by
is the original Steiner net from the (w)-plane. By Theorem 2, there is a unique Möbius transformation of any (
)-plane into the (
)-plane carying one such Steiner net into the other.
The question arises: what is the pre-image by
of a triangle, rectangle, or in general of an arbitrary polygon? We cannot describe these pre-images, yet we can imagine their sections when keeping constant all the variables except one. Since such a section of the pre-image of a line segment is an arc of a circle or a half line, when one of the ends is sent to infinity, the answer to that question is the following. The projection onto the (
)-plane of the sections as previously defined of the pre-image by
of a triangle is a curvilinear triangle having the same angles as the original one. Some of these triangles can be infinite, in the sense that one side is an arc of a circle and the other two are half-lines. An analogous situation appears for the pre-image of an arbitrary polygon.
3. Uniqueness of m-Möbius Transformations
It is known that there is a unique Möbius transformation in the plane moving three distinct points into other three distinct points. In what follows, we will study similar properties of m-Möbius transformations.
Theorem 6. There is a unique m-Möbius transformation
moving a given point
into a given point
.
Proof: Indeed, such transformations have the form see ( [1] )
, respectively
. Then, in the first case,
implies
, hence
is uniquely determined by
. Analogously, in the second case,
implies
, thus
is uniquely determined by
.
This theorem is a particular case of the following:
Theorem 7. For
and
,
the equation
has the degree k in
and therefore, with the exception of multiple roots, it has k solutions, which means that with those exceptions there are k different m-Möbius transformations
moving a given point
into a given point
.
Proof: We need a pretty elaborate induction argument. It can be made more obvious if we write the m-Möbius transformations as matrices
whose
entries are the polynomials in
appearing at the numerator and at the denominator of
and arrange the coefficients of these polynomials also as matrices. The examples of
which follow can be found in [1].
For
we have the matrix expression:
For
the matrix is:
We notice that starting with
every matrix
is built around the matrix
by adding a first and a last column as follows:
Let us notice that a simplification with
occurs in every
such that
and
have the same degree as rational functions of
. We need to prove this affirmation thoroughly by induction. It is clear that the induction hypothesis should be:
where
and
are polynomials of degree k and
and
are polynomials of degree
. Moreover, it can be easily checked that for every
we have
,
,
and
and these equalities should be a part of the induction hypothesis.
The proof consists in showing that the formula:
produces matrices obtained from the last two by replacing k with
, i.e.:
We have:
which shows that indeed, the matrix corresponding to
is:
The computation for the second matrix is similar. This proof provides more information than that about the degree of the equation
, namely it shows the structure of
.
4. Vectors in
The orthogonality of two vectors in
is expressed by the cancellation of their inner product (see [5], page 151). With the help of m-Möbius transformations we can say more, namely an angle of two arbitrary two vectors in
can be defined, such that the respective angle is
when their inner product is zero. For simplicity, we deal first with the case
. Let
be arbitrary points. Following a tradition (see [4], page 12), we will keep the same notation for their position vectors, i.e., vectors pointing from the origin to those points. The inner product of the vectors
and
is by definition
. We say that
and
are orthogonal if and only if
. Suppose that
and
are not those given by (8'), i.e., their images by
is not zero. We denote by
a point which is mapped by
into zero, i.e.,
, for an arbitrary
and
. Then
and
are vectors with the initial point
and the final points respectively
and
. Their images by
are position vectors u and v in the (w)-plane. They make an angle
which remains invariant to a conformal mapping.
Theorem 8. Let
be a bi-Möbius transformation of parameter
moving the points
and
into u and v. Then the projection of the section of the pre-image of u and v by
does not depend on
.
Proof: Indeed, let us deal with the mapping
obtained by solving for
the equation
. For
fixed this is a
conformal mapping of the (w)-plane onto the (
)-plane. For
we have
, which implies that
. Due to the symmetry of
we obtain a similar result for the projection onto the (
)-plane.
If u and v are orthogonal, so are
and
, respectively
and
. Then
and
, thus
, which means that
and
are orthogonal. We can put by definition
, which agrees with the the definition of orthogonality. These concepts generalize trivially to
.
5. The Cross-Ratio
The cross-ratio of four points used by Desargues in his studies of projective geometry (see [5], page 154), reappears in complex analysis as a means of dealing with Möbius transformations. It is known that there is a unique Möbius transformation which carries three arbitrary distinct points
into
. This is
and it is called the cross-ratio of the
four points
. In other words, the cross-ratio
is the image of z by the Möbius transformation which carries
into
. It is also known (see [4], page 79) that for any Möbius transformation
and for any four distinct points
we have
, where
,
, hence the cross-ratio of four points is an invariant with respect to Möbius transformations.
What can be said if
is a m-Möbius transformation? Obviously, the cross-ratio cannot be defined in
. Yet, we can use the pre-image by M of the four points
and the fact that the projections of the sections obtained by keeping
fixed,
of this pre-image onto any (
)-plane are complex numbers. More exactly, keeping fixed all
,
, the section of the pre-image of
is a unique point. Let us denote by
the projection onto the (
)-plane of the section of the pre-image of
obtained by keeping
fixed,
. We have:
(12)
as in Section 2. Thus, we can state:
Theorem 9. For every m-Möbius transformation
the cross-ratio of four points from every (
)-plane is preserved.
Proof: Let us take
four distinct points in the (
)-plane and for
arbitrary in the (
)-planes,
denote
,
. The projections onto the
(
)-plane of the sections of the pre-image by M of
obtained when we keep
fixed are exactly the points
. Since (9) is a Möbius transformation, we have
, which proves the theorem.
Theorem 10. The cross-ratio
of four points in any (
)-plane is real if and only if for arbitrary
, the points
, where
lie on a circle in the (w)-plane.
Proof: It is known that cross-ratio of four points in the complex plane is real if and only if the four points lie on a circle in that plane (see [4], page 79). Let the plane be a (
)-plane and let
lie on a circle. Then
is real and
. The images of that circle by
for different choices of
are circles in the (w)-plane which contain the points
. Reciprocally, if
lie on a circle in the (w)-plane, let us take the pre-image by
of that circle. Keeping
constant we obtain a section of that pre-image whose projection onto the (
)-plane is a circle containing the points
, therefore, the cross-ratio of these points must be real.
6. Conclusion
The purpose of this work was to extend to
the geometric concepts pertinent to Möbius transformations in the plane. The introduction of a function
, which is a Möbius transformation in each one of the variables, when keeping the others constant allowed us to perform this task. The most remarkable achievement was the extension to
of the symmetry principle. The concept of the angle of two vectors in
has been also dealt with. However, just a few properties have been visited, so the potential for other developments is obvious. In particular, visualization in the style done in [6] might be possible having in view the fact that
is a classic Möbius transformation in every one of its variables when others are kept fixed. For the same reason, characterizations as those made by Haruki and Rassias in [7] [8] [9] [10] are expected.
Acknowledgements
I thank Aneta Costin for her support with technical matters.