When does a reflection around a point result in a reflection along a line?

[mathmari]

Hey!

I am looking the following:

For each point $Z$ let $\delta_Z$ be a reflection around the point $Z$. For which points $Z$ is $\sigma_g\circ\delta_Z$ a refection along a line, where $\sigma_g$ is the reflection along the line $2x+3y+4=0$?
I have done the following:

A reflection around a point is the composition of two reflection along a line, $\sigma_a$ and $\sigma_b$, where $a$ is perpendicular to $b$.

So, we have that $\sigma_g\circ\delta_Z=\sigma_g\circ\sigma_a\circ\sigma_b$.

So, we have a composition of three reflections along a line.

We have that the composition of two parallel reflections is a translation and that the composition of a reflection and a translation is a glide reflection.
So, when $g$ is not parallel to $a$ and $b$ then the composition is a reflection. Is this correct? (Wondering)

 

[I like Serena]

Hey mathmari! (Smile)

When we write a point reflection as the composition of 2 perpendicular reflections, we have the freedom to pick the angle of one line.
So we can always pick one of the lines to be parallel to $g$, yielding a translation.
When would the combination of a reflection and a translation still be a reflection (and not a glide reflection)? (Wondering)

 

[mathmari]

When we write a point reflection as the composition of 2 perpendicular reflections, we have the freedom to pick the angle of one line.

Isn't a point reflection around $Z$ the composition of 2 perpendicular reflections, and the reflection lines interset in $Z$ ?

When would the combination of a reflection and a translation still be a reflection (and not a glide reflection)? (Wondering)

To have a glide translation the translation must be parallel to the reflection line, or not? So, to have again a reflection the translation must not be parallel to the reflection line, right? (Wondering)

 

[I like Serena]

Isn't a point reflection around $Z$ the composition of 2 perpendicular reflections, and the reflection lines interset in $Z$ ?

Yes... any 2 perpendicular reflections. So we can choose the angle of one of them, after which the other is fixed. (Thinking)

To have a glide translation the translation must be parallel to the reflection line, or not? So, to have again a reflection the translation must not be parallel to the reflection line, right?

We can write each glide reflection with a parallel translation vector.
But if the translation vector is at an angle, won't we still have a glide reflection?

At least if the translation vector is zero, we will have a reflection.
What if the translation vector is perpendicular to the reflection line? (Wondering)

 

[mathmari]

Yes... any 2 perpendicular reflections. So we can choose the angle of one of them, after which the other is fixed. (Thinking)

Ah ok (Nod)

We can write each glide reflection with a parallel translation vector.
But if the translation vector is at an angle, won't we still have a glide reflection?

At least if the translation vector is zero, we will have a reflection.
What if the translation vector is perpendicular to the reflection line? (Wondering)

If the translation vector is perpendicular to the reflection line we get a reflection, or not? (Wondering)

 

[I like Serena]

If the translation vector is perpendicular to the reflection line we get a reflection, or not?

Suppose we have the reflection in the x-axis given by $(x,y)\mapsto (x,-y)$.
And suppose we translate by $(0,2)$ after the reflection.
Which reflection or glide reflection do we get? (Wondering)

What if we translate by $(2,2)$?
And what if we translate by $(2,0)$?

Note that we can rotate, and shift this example anywhere we want, meaning what we find applies to all reflections. (Nerd)

 

[mathmari]

Suppose we have the reflection in the x-axis given by $(x,y)\mapsto (x,-y)$.
And suppose we translate by $(0,2)$ after the reflection.
Which reflection or glide reflection do we get? (Wondering)

What if we translate by $(2,2)$?
And what if we translate by $(2,0)$?

Note that we can rotate, and shift this example anywhere we want, meaning what we find applies to all reflections. (Nerd)

With the translation by $(0,2)$ we have a reflection.
With the translations by $(2,2)$ and $(2,0)$ we have a glide reflection.

Is this correct? (Wondering)

 

[I like Serena]

With the translation by $(0,2)$ we have a reflection.
With the translations by $(2,2)$ and $(2,0)$ we have a glide reflection.

Is this correct?

Yep.
Just to improve our understanding, which (glide) reflections? (Wondering)

Anyway, so we have a (non-glide) reflection if either the translation vector is 0, or if it is perpendicular to the line of reflection.
Can we tell which points $Z$ we're looking for now? (Wondering)

 

[mathmari]

Yep.
Just to improve our understanding, which (glide) reflections? (Wondering)

Reflection to $g$ and glide reflections the reflection to $g$ and a parallel translation, or not? (Wondering)

Anyway, so we have a (non-glide) reflection if either the translation vector is 0, or if it is perpendicular to the line of reflection.
Can we tell which points $Z$ we're looking for now? (Wondering)

If the translation vector is 0 then the points are on line $g$, right? But what if it is perpendicular to the line of reflection? (Wondering)

 

[I like Serena]

Reflection to $g$ and glide reflections the reflection to $g$ and a parallel translation, or not?

I don't think so. (Worried)

If the translation vector is 0 then the points are on line $g$, right? But what if it is perpendicular to the line of reflection?

Indeed what then?
Perhaps we should first take a look at what happens when we have a reflection in the x-axis with a perpendicular translation? (Wondering)

 

[mathmari]

Perhaps we should first take a look at what happens when we have a reflection in the x-axis with a perpendicular translation? (Wondering)

When we have the point $(x,y)$ after the reflection we get the point $(x,-y)$. After a perpendicular translation we get the point $(x,y')$, for some $y'$, or not? (Wondering)

 

[I like Serena]

When we have the point $(x,y)$ after the reflection we get the point $(x,-y)$. After a perpendicular translation we get the point $(x,y')$, for some $y'$, or not? (Wondering)

We have to pick a translation.
Let's pick $\tau: (x,y)\mapsto (x,y+2)$.
Then $\tau \circ \sigma_x: (x,y)\mapsto (x,-y+2)$.
We think that this is a reflection. If so, which one? What is the fixed line? (Wondering)

 

[mathmari]

We have to pick a translation.
Let's pick $\tau: (x,y)\mapsto (x,y+2)$.
Then $\tau \circ \sigma_x: (x,y)\mapsto (x,-y+2)$.
We think that this is a reflection. If so, which one? What is the fixed line? (Wondering)

The fixed line is $y=2$, right? So, this line depends on the translation. If we translate by $(0,b)$, then the line is $y=b$, or not? (Wondering)

 

[I like Serena]

The fixed line is $y=2$, right?

Let's see... we pick a point on that line, say $(x,2)$.
Its image is $(x,-2+2)=(x,0)$.
That's not the same point is it? (Wondering)

 

[mathmari]

Let's see... we pick a point on that line, say $(x,2)$.
Its image is $(x,-2+2)=(x,0)$.
That's not the same point is it? (Wondering)

I got stuck right now... Shouldn't a point after being reflected be the same why do we check the map according to the translation? The fixed line according to the translation is a line perpendicular to the $x$-axis, so in this case $x=2$, or not? (Wondering)

 

[I like Serena]

I got stuck right now... Shouldn't a point after being reflected be the same why do we check the map according to the translation? The fixed line according to the translation is a line perpendicular to the $x$-axis, so in this case $x=2$, or not? (Wondering)

Not sure what you mean.

We have the reflection in the x-axis given by $\sigma_x(x,y)=(x,-y)$ and the translation $\tau(x,y)=(x,y+2)$.
Its composition is $\tau\circ\sigma(x,y)=(x,-y+2)$.
Its fixed points are when $\tau\circ\sigma(x,y)=(x,y)$, so $y=-y+2\Rightarrow y=1$.
In other words, it's a reflection in the line $y=1$.

Remember that 2 reflections with angle $\phi$ make a rotation with angle $2\phi$?
Similarly 2 parallel reflections at distance $d$ make a translation along a perpendicular vector with length $2d$.
And we have just seen that a reflection followed by a perpendicular translation of length $2d$ results in a reflection with the line shifted by $d$. (Thinking)

 

[mathmari]

So, if the translation vector is 0 then the points $Z$ are on line $g$ and if it is perpendicular to the line of reflection then the points $Z$ are on $g$ but shifted by $d$ ? (Wondering)

 

[I like Serena]

So, if the translation vector is 0 then the points $Z$ are on line $g$

Yep. (Nod)

and if it is perpendicular to the line of reflection then the points $Z$ are on $g$ but shifted by $d$ ?

I think we'll need a closer look.
We were making a composition with 3 reflections.
Which composition was that exactly? (Wondering)

 

[mathmari]

We have the composition of three reflections, where the two reflection lines $a$ and $b$ are perpendicular.

If $g$ is parallel to one of the lines $a$ or $b$ we get a composition of a reflection and a translation. That is either a reflection or a glide reflection. We get a reflection either when the translation vector is the zero vector or when the translation vector is perpendicular to the reflection line.

If $g$ is not parallel to one of the lines $a$ or $b$, we always get a reflection.

Have I understood it correct so far? (Wondering)

 

[I like Serena]

We have the composition of three reflections, where the two reflection lines $a$ and $b$ are perpendicular.

If $g$ is parallel to one of the lines $a$ or $b$ we get a composition of a reflection and a translation. That is either a reflection or a glide reflection. We get a reflection either when the translation vector is the zero vector or when the translation vector is perpendicular to the reflection line.

Yes.
And which reflection line exactly are we talking about in this case?

If $g$ is not parallel to one of the lines $a$ or $b$, we always get a reflection.

We'll always get a reflection or a glide reflection (from 3 reflections).
If $g$ is not parallel to either $a$ or $b$, we can't tell which one.
We have:
$$\sigma_g\circ\sigma_a\circ\sigma_b$$
and we can choose $a$ to be parallel to $g$, meaning $b$ is perpendicular to both $g$ and $a$.
Now we can get a clue whether it's a reflection or a glide reflection. (Thinking)

 

[mathmari]

And which reflection line exactly are we talking about in this case?

In post #16 we had the translation by $d=2$ and the new reflection line is $y=\frac{d}{2}$. But in that we case we had that the first reflection line was the $x$-axis. If we had a general line, the result would different, right? Would the new line maybe be the old line shifted by $\frac{d}{2}$ ? (Wondering)