Is the composition of the isometries a rotation?

In summary, the conversation discusses the geometric descriptions of isometries involving rotations and reflections. It is shown that the transformation $\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ involves a translation back by $v$, a rotation around $\alpha$, and then another translation by $v$. Similarly, $\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$ involves a translation back by $v$, a reflection about a line through the origin with angle $\frac{\alpha}{2}$ with the $x$-axis, and then another translation by $v$. It is also mentioned that these transformations can be seen as rotations around a point and reflections in a line through a point
  • #1
mathmari
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Hey! 😊

Let $\delta_a:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the rotation around the origin with angle $\alpha$ and let $\sigma_{\alpha}:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the reflection about a line through the origin that has angle $\frac{\alpha}{2}$ with the $x$-axis.

Let $v\in V$ and $\alpha\in \mathbb{R}$.

I want to determine the geometric description of the isometries $\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ and $\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$.

$\tau_v$ is the translation about $v$, i.e. $\tau_v(x)=x+v$.

After that I want to show that for $a\in O_2$ and $v\in V$ it holds that $\phi_a\circ \tau_v\circ\phi_a^{-1}=\tau_{av}$.
Let's consider the isometry $\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$:
$$(\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1})(x)=\tau_v(\delta_{\alpha}(\tau_v^{-1}(x)))=\tau_v(\delta_{\alpha}(x-v))=\delta_{\alpha}(x-v)+v$$ Is the total result a rotation? :unsure:

Let's consider now the isometry $\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$:
$$(\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1})(x)=\tau_v(\sigma_{\alpha}(\tau_v^{-1}(x)))=\tau_v(\sigma_{\alpha}(x-v))=\sigma_{\alpha}(x-v)+v$$ Is the total result again a rotation? :unsure:For the other question: We have that $\phi_a:V\rightarrow V, \ v\mapsto av$.

$O_2$ is the set of orthogonal $2\times 2$ matrices.

We have that \begin{align*}(\phi_a\circ \tau_v\circ\phi_a^{-1})(x)&=\phi_a( \tau_v(\phi_a^{-1}(x)))=\phi_a( \tau_v(a^{-1}x))=\phi_a(a^{-1}x+v) \\ & =a(a^{-1}x+v)=x+av=\tau_{av}(x)\end{align*} So it is done. Is this correct? Where did we use here that $a\in O_2$ ? :unsure:
 
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  • #2
Hey mathmari!

Those are not geometric descriptions are they?
A geometric description is for instance that we have rotation with a specific angle around a specific point. 🧐
 
  • #3
$\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ is a translation back by $v$, a rotation aroung $\alpha$ and then again a translation by $v$.

$\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$ is a translation back by $v$, a refection about a line throught origin that has angle $\frac{\alpha}{2}$ with the $x$-axis and then again a translation by $v$.

Are these the geometric descriptions ? :unsure:
 
  • #4
mathmari said:
$\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ is a translation back by $v$, a rotation aroung $\alpha$ and then again a translation by $v$.

$\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$ is a translation back by $v$, a refection about a line throught origin that has angle $\frac{\alpha}{2}$ with the $x$-axis and then again a translation by $v$.

Are these the geometric descriptions ?

More or less, but I think we can do a bit better.
One step further from your description, we can say that the transformation $\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ converts an absolute vector to a vector relative to $v$, which is then rotated by an angle $\alpha$, and then converted back into an absolute vector, can't we? 🤔

The proper geometric description would be that $\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ is a rotation by an angle $\alpha$ around the point $v$.
How does that sound? 🤔
 
  • #5
What do you mean by "absolute vector" ?

And at the second case we have the same but instead of rotation we have reflection?

:unsure:
 
  • #6
mathmari said:
What do you mean by "absolute vector" ?

I've learned that an "absolute vector" is simply a vector relative to the origin that identifies a point.
As opposed to a "relative vector" that indicates that it is relative to some point.
Now that I look it up, I see that it is terminology that seems to be specific for Computer Graphics. 🧐
Either way, that is where a main application of isometries is.

mathmari said:
And at the second case we have the same but instead of rotation we have reflection?

Yes, but with a different angle, and it is not around a point, is it? 🤔
 
  • #7
We convert the vector by $v$ then we reflect it by the line and then we convert the vector back by v. Or not? But which is the angle?
 
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  • #8
mathmari said:
We convert the vector by $v$ then we reflect it by the line and then we convert the vector back by v. Or not? But which is the angle?
Doesn't the problem statement say: "reflection about a line through the origin that has angle $\frac α2$ with the x-axis"? 🤔

I'd say: $\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$ is a reflection in a line through point $v$ that has angle $\frac α2$ with the x-axis. 🧐
 

Related to Is the composition of the isometries a rotation?

1. What is an isometry?

An isometry is a transformation in geometry that preserves the shape and size of an object. It is also known as a rigid motion or congruence transformation.

2. What is the composition of isometries?

The composition of isometries refers to combining two or more isometries to create a new transformation. This can be done by applying the individual isometries in succession.

3. How is a rotation defined in geometry?

In geometry, a rotation is a transformation that moves points around a fixed point, known as the center of rotation, by a certain angle in a specific direction. It preserves the shape and size of the object being rotated.

4. Is the composition of isometries always a rotation?

No, the composition of isometries can result in various transformations, not just rotations. It depends on the specific isometries being combined and their properties.

5. How can we determine if the composition of isometries is a rotation?

To determine if the composition of isometries is a rotation, we can analyze the properties of the individual isometries being combined. For example, if all the isometries involved are rotations around the same center of rotation, then the composition will also be a rotation.

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