<p><em>Note: I have answered before the edit from trichoplax and I thought you were searching for other transformations other than the one you mentioned. The informations below are still useful so I will keep the answer here, but it does not directly answer your question.</em></p>

<hr>

<p>Affine transformations (surprise!) map affine spaces to affine spaces. An affine space is substantially a vector space where you can establish an origin and define points as tuples of their coordinates. This is far from a formal definition, if you are interested in one I can edit the answer. </p>

<p>Now an affine transformation $T$ transform points into points and must preserve affine combinations: </p>

<p>$T(\lambda_1 P_1 + \lambda_2 P_2 + ...  +  \lambda_n P_n) = \lambda_1 T(P_1) + \lambda_2 T(P_2) + ... + \lambda_n T(P_n)  $   </p>

<p>Given $\sum_{i}^{n} \lambda_i = 1 $</p>

<p>However these transformations can't be arbitrary as the following must be preserved: </p>

<ul>
<li><p><strong><em>Parallelism is preserved</em></strong>. This means that if you transform parallel lines they remain parallel after the affine transformation.</p></li>
<li><p><strong><em>Relative ratios are preserved</em></strong>. This means that if you have $R = (1-\beta) P + \beta Q$  then $T(R) = (1 - \beta) T(P) + \beta T(Q)$ </p></li>
<li><p><strong><em>Addition between vector and points are preserved</em></strong>. Meaning that $T( P + \vec{v} ) = T(P) + T(\vec{v})$</p></li>
</ul>

<p>With these properties in mind you can come up with a very big number of affine transforms yourself. A couple of obvious ones other the ones you mentioned are: </p>

<p><strong>Translation</strong> </p>

<p>$$T_vp = 
\begin{bmatrix}
1&amp;0&amp;0&amp;v_x\\0&amp;1&amp;0&amp;v_y\\0&amp;0&amp;1&amp;v_z\\0&amp;0&amp;0&amp;1\end{bmatrix}
\begin{bmatrix}
p_x+v_x\\p_x+v_y\\p_z+v_z\\1\end{bmatrix}=p+v$$</p>

<p>That moves a point into a specific direction by a specific amount specified by a displacement vector. </p>

<p><strong>Shearing</strong> </p>

<p><a href="https://i.sstatic.net/fpg2D.png" rel="noreferrer"><img src="https://i.sstatic.net/fpg2D.png" alt="enter image description here"></a>
<em>image from wikipedia</em></p>

<p>That is a transform that displace all points in a given direction by an amount that is proportional to their perpendicular distance to a line parallel to that direction. </p>

<p>For example the transform matrix for an horizontal shear in a 2D space is given by: </p>

<p>$$
\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix}x+my\\y\end{bmatrix} = 
\begin{bmatrix}1&amp;m\\0&amp;1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}$$</p>

<hr>

<p>Again, these two are just example, the important information you should really keep is the definition (and properties) of an affine transform; with that in mind it shouldn't be too hard to recognize an affine transform.  Also note that combining affine transforms will give you an affine transform! </p>

CC BY-SA 3.0
2015-11-13T09:35:22.210Z
Note that I might have answered a question that is slightly different from yours, but I did that on purpose as yours is clearly an homework question and I preferred to give you the tools to come up with an answer by yourself rather than giving you one directly :)
cifz
CC BY-SA 3.0
2015-11-13T09:32:27.343Z
2015-11-13T13:08:52.040Z
2015-11-13T13:08:52.040Z
cifz
<p>Please help me tackle the below question:</p>

<blockquote>
  <p>Explain THREE forms of affine transformations using relevant examples for each case.</p>
</blockquote>

<p>The answers I want to see are scaling, rotation and reflection.</p>

<vector-graphics><affine-transformations>
What are forms of affine transformations?
BONIE
# What are forms of affine transformations?

(Asked by: username_0 on 2015-11-13)

Please help me tackle the below question:

> Explain THREE forms of affine transformations using relevant examples for each case.

The answers I want to see are scaling, rotation and reflection.

**Tags:** vector-graphics, affine-transformations


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## 1 answers

### Answer by username_1 on 2015-11-13. Score: 6

*Note: I have answered before the edit from trichoplax and I thought you were searching for other transformations other than the one you mentioned. The informations below are still useful so I will keep the answer here, but it does not directly answer your question.*

---

Affine transformations (surprise!) map affine spaces to affine spaces. An affine space is substantially a vector space where you can establish an origin and define points as tuples of their coordinates. This is far from a formal definition, if you are interested in one I can edit the answer.

Now an affine transformation $T$ transform points into points and must preserve affine combinations:

$T(\lambda\_1 P\_1 + \lambda\_2 P\_2 + ... + \lambda\_n P\_n) = \lambda\_1 T(P\_1) + \lambda\_2 T(P\_2) + ... + \lambda\_n T(P\_n) $

Given $\sum\_{i}^{n} \lambda\_i = 1 $

However these transformations can't be arbitrary as the following must be preserved:

* ***Parallelism is preserved***. This means that if you transform parallel lines they remain parallel after the affine transformation.
* ***Relative ratios are preserved***. This means that if you have $R = (1-\beta) P + \beta Q$ then $T(R) = (1 - \beta) T(P) + \beta T(Q)$
* ***Addition between vector and points are preserved***. Meaning that $T( P + \vec{v} ) = T(P) + T(\vec{v})$

With these properties in mind you can come up with a very big number of affine transforms yourself. A couple of obvious ones other the ones you mentioned are:

**Translation**

$$T\_vp =
\begin{bmatrix}
1&0&0&v\_x\\0&1&0&v\_y\\0&0&1&v\_z\\0&0&0&1\end{bmatrix}
\begin{bmatrix}
p\_x+v\_x\\p\_x+v\_y\\p\_z+v\_z\\1\end{bmatrix}=p+v$$

That moves a point into a specific direction by a specific amount specified by a displacement vector.

**Shearing**

[![enter image description here](https://i.sstatic.net/fpg2D.png)](https://i.sstatic.net/fpg2D.png)
*image from wikipedia*

That is a transform that displace all points in a given direction by an amount that is proportional to their perpendicular distance to a line parallel to that direction.

For example the transform matrix for an horizontal shear in a 2D space is given by:

$$
\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix}x+my\\y\end{bmatrix} =
\begin{bmatrix}1&m\\0&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}$$

---

Again, these two are just example, the important information you should really keep is the definition (and properties) of an affine transform; with that in mind it shouldn't be too hard to recognize an affine transform. Also note that combining affine transforms will give you an affine transform!

> **Comments:**
> **username_1**: Note that I might have answered a question that is slightly different from yours, but I did that on purpose as yours is clearly an homework question and I preferred to give you the tools to come up with an answer by yourself rather than giving you one directly :)  

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