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Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as an interpretation of $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$).

The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those. In particular, I don't see how to characterize Lorentz transformations using linear conditions.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ similar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \langle a,b\rangle = a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a$, $b$ intersect orthogonally if and only if $\langle a,b\rangle = 0$ since

$$\langle a,b\rangle = 2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\langle p,p\rangle = 0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.

Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as an interpretation of $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$).

The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those. In particular, I don't see how to characterize Lorentz transformations using linear conditions.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ similar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \langle a,b\rangle = a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a$, $b$ intersect orthogonally if and only if $\langle a,b\rangle = 0$ since

$$\langle a,b\rangle = 2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\langle p,p\rangle = 0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.

Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as an interpretation of $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$).

The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those. In particular, I don't see how to characterize Lorentz transformations using linear conditions.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ simiar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \left<a,b\right>=a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a,b$ intersect orthogonally iff $\left<a,b\right>=0$ since

$$\left<a,b\right>=2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\left<p,p\right>=0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.

Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as an interpretation of $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$).

The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those. In particular, I don't see how to characterize Lorentz transformations using linear conditions.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ similar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \langle a,b\rangle = a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a$, $b$ intersect orthogonally if and only if $\langle a,b\rangle = 0$ since

$$\langle a,b\rangle = 2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\langle p,p\rangle = 0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.

Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as geometry in $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$). The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ simiar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \left<a,b\right>=a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a,b$ intersect orthogonally iff $\left<a,b\right>=0$ since

$$\left<a,b\right>=2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\left<p,p\right>=0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.

Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as an interpretation of $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$). 

The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those. In particular, I don't see how to characterize Lorentz transformations using linear conditions.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ simiar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \left<a,b\right>=a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a,b$ intersect orthogonally iff $\left<a,b\right>=0$ since

$$\left<a,b\right>=2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\left<p,p\right>=0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.