What are forms of affine transformations?

Question

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.

Answer 1

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.

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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

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:

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

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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!