Transformations

• Demo for today.
• This is sort of Section 2.3
  • I think he is looking at non-math people.
• This is where linear algebra comes into play for me.
• Points
  • We will represent points as a 3 or 4 tuple.
    • For 2d: $[x, y, 1]$
    • For 3d: $[x, y, z, 1]$
  • This will make the next step work.
• Review matrix vector multiplication.
  • $\begin{bmatrix} a & d & g \\ b & e & h \\ c & f & i \end{bmatrix} \times \begin{bmatrix}x \\ y \\ z\end{bmatrix} = [ ax + dy + gz, bx + ey + hz, cx + fy + iz] $
• Matrix matrix multiplication
  • Remember $A \times B \neq B \times A $ in general.
• Translating in the plane
  • We would like to move a point in the plane by $(t_x, t_y)$
  • The simplest is to do $(x + t_x, y + t_y)$
  • But linear gives us a cool system so let's try
  • $\begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix}x \\ y \\ 1\end{bmatrix} $
• This is the base 2d translation matrix.
• Scaling in the plane
  • Again scale by $(s_x, s_y)$
  • Which would be $(x \times s_x, y \times s_y)$
  • $\begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix}x \\ y \\ 1\end{bmatrix} $
  • By the way, this scales about the origin.
• The neat part is that matrix multiplication preserves the order of the transformations.
  • Let's try scale then translate
    • $\begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} s_x & 0 & s_x t_x \\ 0 & s_y & s_y t_y \\ 0 & 0 & 1 \\ \end{bmatrix}$
  • Let's try translate then scale
    • $\begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} s_x & 0 & t_x \\ 0 & s_y & t_y \\ 0 & 0 & 1 \\ \end{bmatrix}$
  • In the first case, we translate the point, then scale it.
  • In the second we scale, then translate.
  • In the first, the translation is also scaled, but not so in the second.