Affine transformations

An affine transformation is a type of geometric transformation which preserves collinearity (if a collection of points sits on a line before the transformation, they all sit on a line afterwards) and the ratios of distances between points on a line. Types of affine transformations include translation (moving a figure), scaling (increasing or decreasing the size of a figure), and rotation (turning a figure about a point).

Note that affine transformations can be done $\mathbb{R}^n$, for $n\geq 1$, although some of the transformations do not make sense for $n=1$. Matrix algebra will be used to unify the presentation. To get an unique affine transformation matrix, one more point is needed than the $n$ of the $\mathbb{R}^n$ space. It also is assumed that the points do not share a common $\mathbb{R}^{n-1}$ space. For example, in $\mathbb{R}^2$ space, 3 points are required and they must not all in in the same $\mathbb{R}^1$ space, that is, must not be collinear.

The examples and computations will be in $\mathbb{R}^2$. Hopefully the extension to $n\gt 2$ will be obvious.

In each example, the before is red and solid and the after is blue and dashed. The corners of the example triangle will be labeled as follows: the first will have a small disk, the second will have a small quadrilateral and the third vertex will have a small five-sided object. All of these labels will be in translucent gray. The original triangle will be (0,0), (1,0) and (0,1), in each example.

Translation

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Right by $\frac12$ and up by $\frac13$

Scaling

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By $\frac12$ vertically. By definition, scaling is in one direction only. Note, multiple scaling on more than axis is permissible and in on all axes, becomes a magnification or homothety..

Reflection

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A reflection about an infinite line passing through $(0,1)$ with a slope of $-\frac12$:

Rotation

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A rotation about the origin through an angle of $\frac23\pi$ radians:

Shearing or shear

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A shearing about the origin through an angle of $\text{arctan}(\frac12)$ radians:

Affine transformations become linear transformations in one dimension higher. By assigning a point a next coordinate of $1$, e.g., $(x,y)$ becomes $(x,y,1)$, these are called homogeneous coordinates. Since linear transformations are represented easily by matrices, the corresponding entity is an augmented matrix, where the $a$s provide all transformation except translation and those are represented by the $b$s. $$\bm{a}= \left( \begin{array}{ccc} a_{1,1} & a_{1,2} & b_1 \\ a_{2,1} & a_{2,2} & b_2 \\ 0 & 0 & 1 \\ \end{array} \right)$$

The example sets of points are the original set and the final set.

The original set are $(-2,2)$, $(0,1)$ and $(2,0)$,. In a $3\times 3$ matrix, that becomes: $$\bm{o}= \left( \begin{array}{ccc} -2 & 0 & 2 \\ 2 & -1 & 0 \\ 1 & 1 & 1 \\ \end{array} \right)$$

The desired final set is $(1,0)$, $(0,\frac{\sqrt{3}}{2})$ and $(0,\frac{\sqrt{3}}{2})$. In a $3\times 3$ matrix, that becomes: $$\bm{n} \left( \begin{array}{ccc} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ 1 & 1 & 1 \\ \end{array} \right)$$

Therefore, the affine transformation matrix is the $\bm{a}$ of $\bm{a}.\bm{o}=\bm{n}$.

That can be solved for by $\bm{a}=\bm{n}.\bm{o}^{-1}$. Note the following, because matrix multiplication is not commutative, i. e., the multiplicands may not swap sides, the inverse of $\bm{o}$ has to go to the side of $\bm{n}$ that $\bm{o}$ was on on $\bm{a}$ previously. Since, in this article, column vectors and left side matrix multiplication is being used by convention, that means that $\bm{o}^{-1}$ would go to the right side of $\bm{n}$. Also, note that if $\bm{o}$ will not invert, that means the original points are in a common subspace, e.g., are collinear in the case of 2D, coplanar in the case of 3D, etc.

$$\bm{n}.\bm{o}^{-1}= \left( \begin{array}{ccc} -\frac{3}{16} & \frac{3}{8} & -\frac{1}{8} \\ -\frac{1}{16} \left(5 \sqrt{3}\right) & -\frac{1}{8} \left(3 \sqrt{3}\right) & \frac{\sqrt{3}}{8} \\ 0 & 0 & 1 \\ \end{array} \right)$$

The reverse transformation is matrix inverse of the forward transformation: $$\left( \begin{array}{ccc} -2 & -\frac{2}{\sqrt{3}} & 0 \\ \frac{5}{3} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \\ 0 & 0 & 1 \\ \end{array} \right)$$

Each frame of the movie is $\frac{1}{180}$ of the total affine transformation from all to none and back again.

Affine Spaces.