Lecture 8: Coordinate Transformations

Basics

Multiplying a matrix by a vector allows us to produce a new, changed vectors.

$$A * v = \begin{bmatrix}a_{1,1} & a_{1,2}\\a_{2,1} & a_{2,2}\end{bmatrix} * \begin{bmatrix}x\\y\end{bmatrix}$$ $$= \begin{bmatrix}a_{1,1} * x + a_{1,2} * y\\a_{2,1} * x + a_{2,2} * y\end{bmatrix}$$

Such a transformation can be used to scale, shear, and rotate a vector or Cartesian coordinates.

Scale

Matrix formulation:

$$A = \begin{bmatrix}s_x & 0\\s_y & 0\end{bmatrix}$$

Shear

Matrix formulation:

$$A = \begin{bmatrix}1 & tan \theta\\0 & 1\end{bmatrix}$$

Rotation

Matrix formulation:

$$A = \begin{bmatrix}cos \theta & -sin \theta\\sin \theta & cos \theta\end{bmatrix}$$

Matrix derivation:

There is a requisite diagram for understanding this which is coming soon…

$$\vec a = (x_a, y_a)$$ $$\vec b = (x_b, y_b)$$

$\vec b$ is $\vec a$ rotated by $\theta$

$\vec a$ makes angle $\alpha$ with the x-axis and its length is $r = x_a^2 + y_a^2$

Therefore:

$$x_a = r * cos \alpha$$ $$y_a = r * sin \alpha$$

And:

$$x_b = r * cos(\alpha + \theta)$$ $$y_b = r * sin(\alpha + \theta)$$

Using trigonometric relationships:

$$sin(A + B) = sin(A) * cos(B) + sin(B) * cos(A)$$ $$cos(A + B) = cos(A) * cos(B) - sin(B) * sin(A)$$

Thus:

$$x_b = r * cos(\alpha) * cos(\theta) - r * sin(\alpha) * sin(\theta)$$ $$y_b = r * sin(\alpha) * cos(\theta) + r * cos(\alpha) * sin(\theta)$$

Substituting the equations for $x_a$ and $y_a$:

$$x_b = x_a * cos(\theta) - y_a * sin(\theta)$$ $$y_b = y_a * cos(\theta) + x_a * sin(\theta)$$

And the transformation matrix is revealed.