# Checking if $\beta \in \text{Isom}(\mathbb{R}^2)$

• MHB
• mathmari
In summary, the conversation discusses various maps in $\mathbb{R}^2$ and their properties as isometries. These maps include translations, rotations, and reflections. The conversation also mentions checking for the preservation of distances in order to determine if a map is an isometry. The goal is to prove that if a map is an isometry, then exactly one of the given statements about it is true.
mathmari
Gold Member
MHB
Hey! :giggle:

Let $\alpha\in \mathbb{R}$. We define the following maps:

(a) For $p\in \mathbb{R}^2$ let $\delta_{p,\alpha}=\tau_p\circ \delta_{\alpha}\circ\tau_p^{-1}$.
(b) For $p\in \mathbb{R}f_{\alpha}$ let $\sigma_{p,\alpha}=\tau_p\circ \sigma_{\alpha}\circ\tau_p^{-1}$.
(c) For $p\in \mathbb{R}f_{\alpha}$, $q\in \mathbb{R}e_{\alpha}$ let $\gamma_{q,p,\alpha}=\tau_q\circ \sigma_{p,\alpha}$.
Let $\beta\in \text{Isom}(\mathbb{R}^2)$. Then one of the following statements is true.

(i) $\beta=\text{id}$

(ii) $\beta=r_v$, $0\neq v\in \mathbb{R}^2$.

(iii) $\beta=\delta_{p,\alpha}$,$0<\alpha<2\pi$

(iv) $\beta=\sigma_{p,\alpha}$, $\alpha\in \mathbb{R}$, $p\in \mathbb{R}f_{\alpha}$

(v) $\beta=\gamma_{q,p,\alpha}$, $\alpha\in \mathbb{R}$, $p\in \mathbb{R}f_{\alpha}$ , $q\in \mathbb{R}e_{\alpha}$ , $0\neq q$
So we have to check which of these cases for $\beta$ we have an isometry, or not?

The identity function is in $\text{Isom}(\mathbb{R}^2)$, isn't it?

For the other ones do we have to checkif these maps are bijective? :unsure:

Hey mathmari!

What are $\mathbb Rf_\alpha$, $\mathbb Re_\alpha$, and $r_v$?

We would need to check if the $\beta$ are isometries. That is whether distances are preserved.

Btw, it looks as if all statements are true.
If $r_v$ is a typo and is actually $\tau_v$, then we have the list of all isometries in $\mathbb R^2$.

Klaas van Aarsen said:
What are $\mathbb Rf_\alpha$, $\mathbb Re_\alpha$, and $r_v$?

We would need to check if the $\beta$ are isometries. That is whether distances are preserved.

Btw, it looks as if all statements are true.
If $r_v$ is a typo and is actually $\tau_v$, then we have the list of all isometries in $\mathbb R^2$.

$\mathbb Re_\alpha$ is a multiple of the vector $\begin{pmatrix}\cos \left (\frac{\alpha}{2}\right ) \\ \sin \left (\frac{\alpha}{2}\right )\end{pmatrix}$ and $\mathbb Rf_\alpha$ is a a multiplie of the vector $\begin{pmatrix}-\sin \left (\frac{\alpha}{2}\right ) \\ \cos \left (\frac{\alpha}{2}\right )\end{pmatrix}$.

Yes it should be $\tau_v$, the translation, $\tau_v(x)=x+v$.

How do we see that all of the list are isometries in $\mathbb R^2$ ? How do we check of the distances are preserved? Do we calculate $\beta (x-y)$ ?

:unsure:

mathmari said:
How do we see that all of the list are isometries in $\mathbb R^2$ ?

Check out this section on wiki.

mathmari said:
How do we check of the distances are preserved? Do we calculate $\beta (x-y)$ ?
The definition of an isometry says that it's a map with $d(\beta (x),\beta (y))=d(x,y)$.

Klaas van Aarsen said:
The definition of an isometry says that it's a map with $d(\beta (x),\beta (y))=d(x,y)$.

(i) For the identity we have that $\beta (x)=x$ and $\beta (y)=y$, so $d(\beta (x),\beta (y))=d(x,y)$ is satisfied.

(ii) If $\beta=\tau_v$ then $\beta(x)=\tau_v(x)=x+v$, and $\beta(y)=\tau_v(y)=y+v$. Then $d(\beta (x),\beta (y))=d(x+v,y+v)$ this is equal to $d(x,y)$, right?

(iii) $\beta (x)= \tau_p\circ \delta_{\alpha}\circ\tau_p^{-1} (x)= \tau_p( \delta_{\alpha}(x-p))=\tau_p( d_{\alpha}(x-p))=d_{\alpha}(x-p)+p$, right?

(iv) $\beta (x)= \tau_p\circ \sigma_{\alpha}\circ\tau_p^{-1} (x)= \tau_p( \sigma_{\alpha}(x-p))=\tau_p( s_{\alpha}(x-p))=s_{\alpha}(x-p)+p$, right?

(v) $\beta (x)= \tau_q\circ \sigma_{p,\alpha} (x)=\tau_q\circ \tau_p\circ \sigma_{\alpha}\circ\tau_p^{-1} (x)=s_{\alpha}(x-p)+p+q$, right? :unsure:

mathmari said:
(i) For the identity we have that $\beta (x)=x$ and $\beta (y)=y$, so $d(\beta (x),\beta (y))=d(x,y)$ is satisfied.

(ii) If $\beta=\tau_v$ then $\beta(x)=\tau_v(x)=x+v$, and $\beta(y)=\tau_v(y)=y+v$. Then $d(\beta (x),\beta (y))=d(x+v,y+v)$ this is equal to $d(x,y)$, right?

The standard Euclidean metric is $d(x,y)=\|x-y\|$.
So $d(\beta (x),\beta (y))=d(x+v,y+v)=\|(x+v)-(y+v)\|=\|x-y\|=d(x,y)$. So yes. (Nod)

mathmari said:
(iii) $\beta (x)= \tau_p\circ \delta_{\alpha}\circ\tau_p^{-1} (x)= \tau_p( \delta_{\alpha}(x-p))=\tau_p( d_{\alpha}(x-p))=d_{\alpha}(x-p)+p$, right?

(iv) $\beta (x)= \tau_p\circ \sigma_{\alpha}\circ\tau_p^{-1} (x)= \tau_p( \sigma_{\alpha}(x-p))=\tau_p( s_{\alpha}(x-p))=s_{\alpha}(x-p)+p$, right?

(v) $\beta (x)= \tau_q\circ \sigma_{p,\alpha} (x)=\tau_q\circ \tau_p\circ \sigma_{\alpha}\circ\tau_p^{-1} (x)=s_{\alpha}(x-p)+p+q$, right?
Yep.

So if we have that $\|d_\alpha y\|=\|y\|$ and $\|s_\alpha y\|=\|y\|$, then each of these will follow.

Klaas van Aarsen said:
The standard Euclidean metric is $d(x,y)=\|x-y\|$.
So $d(\beta (x),\beta (y))=d(x+v,y+v)=\|(x+v)-(y+v)\|=\|x-y\|=d(x,y)$. So yes. (Nod)Yep.

So if we have that $\|d_\alpha y\|=\|y\|$ and $\|s_\alpha y\|=\|y\|$, then each of these will follow.
I got it now! One question... At the question statement is it meant that $\beta$ can be ONLY one of the given maps or that these maps have the property of $\beta$, i.e . that they are isometries? :unsure:

mathmari said:
One question... At the question statement is it meant that $\beta$ can be ONLY one of the given maps or that these maps have the property of $\beta$, i.e . that they are isometries?
Makes sense yes.
I think we're supposed to prove that if $\beta$ is an isometry, that exactly 1 of the statements is true.

Klaas van Aarsen said:
Makes sense yes.
I think we're supposed to prove that if $\beta$ is an isometry, that exactly 1 of the statements is true.

But how can exactly one be true if all maps are isometries? I got stuck right now. :unsure:

mathmari said:
But how can exactly one be true if all maps are isometries? I got stuck right now.
An isometry cannot be both a reflection and a rotation can it?

Klaas van Aarsen said:
An isometry cannot be both a reflection and a rotation can it?

I got stuck right now, you mean that not all maps can be an isometry although the distances are preserved? :unsure:

mathmari said:
I got stuck right now, you mean that not all maps can be an isometry although the distances are preserved?
All those maps are isometries.
We need to prove that any given isometry must be one of them.
That is that we didn't 'miss' any isometries. And moreover that if an isometry matches one of them, that it won't simultaneously match another one as well.

Klaas van Aarsen said:
All those maps are isometries.
We need to prove that any given isometry must be one of them.
That is that we didn't 'miss' any isometries. And moreover that if an isometry matches one of them, that it won't simultaneously match another one as well.

Ah ok!

To show that we didn't 'miss' any isometries, do we suppose that we miss some and we get a contradiction, or how do we show that? :unsure:

mathmari said:
To show that we didn't 'miss' any isometries, do we suppose that we miss some and we get a contradiction, or how do we show that?
I think we can state the list from wiki as the list of the possible euclidean isometries.
Then we should probably just prove that the given statements correspond to them and that they are mutually exclusive.

Klaas van Aarsen said:
I think we can state the list from wiki as the list of the possible euclidean isometries.
Then we should probably just prove that the given statements correspond to them and that they are mutually exclusive.

Ok!

(i) is the identity
(ii) is the translation nas for $v$
(iii) is the rotation around the point $p$ and with angle $a$
(iv) is the reflection about aline through the point $p$ and with angle $a$
(v) is a glide reflection Is that correct? :unsure:

mathmari said:
(i) is the identity
(ii) is the translation nas for $v$
(iii) is the rotation around the point $p$ and with angle $a$
(iv) is the reflection about a line through the point $p$ and with angle $a$
(v) is a glide reflection
Is that correct?
Yep. That is, it is in one direction. (Nod)
I think we still need to verify that every rotation can be written with a $p$ in $\mathbb Rf_\alpha$.
And similarly for reflections and glide reflections.

And also that a $\gamma$ as we defined it, is really a glide reflection and not a regular reflection.

## 1. What does it mean for $\beta$ to be in $\text{Isom}(\mathbb{R}^2)$?

Being in $\text{Isom}(\mathbb{R}^2)$ means that $\beta$ is an isometry, or a transformation that preserves distance and angles, in the two-dimensional Euclidean plane.

## 2. How can I check if $\beta$ is in $\text{Isom}(\mathbb{R}^2)$?

To check if $\beta$ is in $\text{Isom}(\mathbb{R}^2)$, you can apply the definition of an isometry and see if it satisfies the criteria of preserving distance and angles.

## 3. What are some common examples of transformations in $\text{Isom}(\mathbb{R}^2)$?

Some common examples of transformations in $\text{Isom}(\mathbb{R}^2)$ include translations, rotations, reflections, and combinations of these.

## 4. Can a transformation be in $\text{Isom}(\mathbb{R}^2)$ if it does not preserve distance?

No, by definition, a transformation in $\text{Isom}(\mathbb{R}^2)$ must preserve distance. If a transformation changes the distance between points, it is not considered an isometry.

## 5. How is $\text{Isom}(\mathbb{R}^2)$ related to other mathematical concepts?

$\text{Isom}(\mathbb{R}^2)$ is closely related to other mathematical concepts such as symmetry, similarity, and congruence. It is also a fundamental concept in geometry and has applications in fields such as computer graphics and robotics.

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