Mappings - Similarity transformations

[mathmari]

Hey!

I want to check whether the following statements are correct. At each statement I wrote my idea/question:

1. A similitude mapping with exactly one fixed point is a scaling.
   
   A similitude mapping is a scaling, or a composition of scaling and rotation or a composition of scaling and glide reflection, right? The composition of scaling and rotation has also a fixed point (which is the same fixed point for the rotation and the scaling). So, the statement is wrong.
2. Similitude mappings $\neq id$ with more than one fixed point are reflections.
   
   For this one I don't have an idea.
3. The composition of two rotations with rotation angle $a$ and $b$ is a rotation iff $a+b=k\cdot 2\pi, k\in \mathbb{Z}$.
   
   I have shown that $R(a)R(b)=R(a+b)$. When we have that $a+b=k\cdot 2\pi$ do we not get again at the same point as at the beginning. So, is it then the identity?
4. For each line $g$ and each similitude mapping $\kappa$ it holds that $\kappa\circ\sigma\circ\kappa^{-1}=\sigma_{\kappa(g)}$.
   
   Unfortunately, also for this one I don't have an idea.

 

[I like Serena]

Hey!

I want to check whether the following statements are correct. At each statement I wrote my idea/question:

1. A similitude mapping with exactly one fixed point is a scaling.

A similitude mapping is a scaling, or a composition of scaling and rotation or a composition of scaling and glide reflection, right? The composition of scaling and rotation has also a fixed point (which is the same fixed point for the rotation and the scaling). So, the statement is wrong.

What's a similitude mapping?
Googling for it, I can't really find anything.
Is it perchance a congruence transformation or isometry?
Oh wait! Is it an affinity transformation or similarity transformation? (Wondering)

Anyway, indeed, a rotation has 1 fixed point, so the statement is wrong.

2. Similitude mappings $\neq id$ with more than one fixed point are reflections.

For this one I don't have an idea.

What are the fixed points of each of the different types of mappings? (Wondering)

3. The composition of two rotations with rotation angle $a$ and $b$ is a rotation iff $a+b=k\cdot 2\pi, k\in \mathbb{Z}$.

I have shown that $R(a)R(b)=R(a+b)$. When we have that $a+b=k\cdot 2\pi$ do we not get again at the same point as at the beginning. So, is it then the identity?

Yep.
Therefore it is not a rotation.

4. For each line $g$ and each similitude mapping $\kappa$ it holds that $\kappa\circ\sigma\circ\kappa^{-1}=\sigma_{\kappa(g)}$.

Unfortunately, also for this one I don't have an idea.

Let's first establish what a similitude mapping is supposed to represent. (Thinking)

 

[mathmari]

What's a similitude mapping?
Googling for it, I can't really find anything.
Is it perchance a congruence transformation or isometry?
Oh wait! Is it an affinity transformation or similarity transformation? (Wondering)

Ah it is similarity transformation.

What are the fixed points of each of the different types of mappings? (Wondering)

If there are more fixed points, does it mean that a whole line is fixed? (Wondering)

Therefore it is not a rotation.

Can we not consider it as a rotation with angle $0^{\circ}$ ? (Wondering)

Let's first establish what a similitude mapping is supposed to represent. (Thinking)

By the similarity mapping of a point do we get the same point? (Wondering)

 

[I like Serena]

If there are more fixed points, does it mean that a whole line is fixed?

I think so yes... unless the whole plane is fixed.

Can we not consider it as a rotation with angle $0^{\circ}$ ?

Sure we can. What do your notes say? (Wondering)

Note that if we do that, then we should also consider for instance a reflection a glide reflection (with a translation of 0).
And identity would be ambiguous, since it could be either a rotation (with angle 0), or a translation (with vector 0), or just identity.
So I'm used to making the distinction, and treating identity as distinct from a rotation, and also distinct from a translation.

By the similitude mapping of a point do we get the same point? (Wondering)

Not generally.

As for the statement at hand, suppose G is a point on the line g.
Then $\sigma(G)=G$, isn't it?
It would be a fixed point after all.

Would $\kappa(G)$ be a fixed point of $\kappa\circ\sigma\circ\kappa^{-1}$? (Wondering)
Oh, and if you don't mind, I'll rename the thread to 'Mappings - Similarity transformations'. (Malthe)

 

[mathmari]

As for the statement at hand, suppose G is a point on the line g.
Then $\sigma(G)=G$, isn't it?
It would be a fixed point after all.

Would $\kappa(G)$ be a fixed point of $\kappa\circ\sigma\circ\kappa^{-1}$? (Wondering)

Yes, $\kappa(G)$ be a fixed point of $\kappa\circ\sigma\circ\kappa^{-1}$, since:
$$\kappa\circ\sigma\circ\kappa^{-1}(\kappa(G))=\kappa\circ\sigma\circ(\kappa^{-1}\kappa(G))=\kappa\circ\sigma (G)=\kappa (G)$$

Oh, and if you don't mind, I'll rename the thread to 'Mappings - Similarity transformations'. (Malthe)

Sure! (Yes)

 

[mathmari]

$\sigma_{\kappa(g)}$ is the reflection along $\kappa(g)$.

$\kappa(g)$ is the similar mapping the line $g$.

$\kappa\circ\sigma\circ\kappa^{-1}=\sigma_{\kappa(g)}$ means that every $\kappa (G)$ with $g\in G$ is a fixed point for $\kappa\circ\sigma\circ\kappa^{-1}$ since $\kappa (G)$ is a fixed point for that reflection, right? (Wondering)

2. Similitude mappings $\neq id$ with more than one fixed point are reflections.

So, if we have more than on fixed point, a whole line or plane is fixed.

But why can we have only reflections? (Wondering)

 

[I like Serena]

Yes, $\kappa(G)$ be a fixed point of $\kappa\circ\sigma\circ\kappa^{-1}$, since:
$$\kappa\circ\sigma\circ\kappa^{-1}(\kappa(G))=\kappa\circ\sigma\circ(\kappa^{-1}\kappa(G))=\kappa\circ\sigma (G)=\kappa (G)$$

$\sigma_{\kappa(g)}$ is the reflection along $\kappa(g)$.

$\kappa(g)$ is the similar mapping the line $g$.

$\kappa\circ\sigma\circ\kappa^{-1}=\sigma_{\kappa(g)}$ means that every $\kappa (G)$ with $g\in G$ is a fixed point for $\kappa\circ\sigma\circ\kappa^{-1}$ since $\kappa (G)$ is a fixed point for that reflection, right?

Yep. And similarities have the property that they map a line to a line.

So, if we have more than on fixed point, a whole line or plane is fixed.

But why can we have only reflections?

What else could it be?
Scalings, translations, rotations, and glide reflections all have 0 or 1 fixed point.
In theory there could a composition of them (including reflections) that also has a fixed line, but I don't think there is one. (Thinking)

 

[mathmari]

What else could it be?
Scalings, translations, rotations, and glide reflections all have 0 or 1 fixed point.
In theory there could a composition of them (including reflections) that also has a fixed line, but I don't think there is one. (Thinking)

Similarity transformations are the scalings, translations, rotations, glide reflections and rflections and their compositions, right? (Wondering)

What exactly is the difference between an affine transformation and a similarity transformation? (Wondering)

 

[I like Serena]

Similarity transformations are the scalings, translations, rotations, glide reflections and reflections and their compositions, right? (Wondering)

Yes. And I propose to include identity as a separate transformation (all points are fixed, which is not the case for 'real' rotations and translations).

What exactly is the difference between an affine transformation and a similarity transformation? (Wondering)

A similarity transformation preserves the shape of objects, but they can be rotated, translated, reflected, or scaled.
All similarity transformations are affine transformations.
However, affine transformations include also for instance shear transformations, that do change the shape of the object.

Mathematically, the similarity transformations are exactly those transformations that can be written as $\mathbf x \mapsto rA\mathbf x + \mathbf b$, where the scaling factor $r$ is any positive real number, $A$ is any orthogonal transformation, and $\mathbf b$ is any translation vector.
Affine transformations are $\mathbf x\mapsto M\mathbf x + \mathbf b$, where $M$ is any linear transformation, and $\mathbf b$ is any translation vector. (Nerd)

 

[mathmari]

Yes. And I propose to include identity as a separate transformation (all points are fixed, which is not the case for 'real' rotations and translations).
A similarity transformation preserves the shape of objects, but they can be rotated, translated, reflected, or scaled.
All similarity transformations are affine transformations.
However, affine transformations include also for instance shear transformations, that do change the shape of the object.

Mathematically, the similarity transformations are exactly those transformations that can be written as $\mathbf x \mapsto rA\mathbf x + \mathbf b$, where the scaling factor $r$ is any positive real number, $A$ is any orthogonal transformation, and $\mathbf b$ is any translation vector.
Affine transformations are $\mathbf x\mapsto M\mathbf x + \mathbf b$, where $M$ is any linear transformation, and $\mathbf b$ is any translation vector. (Nerd)

I understand! Thank you so much! (Yes)