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Lec 03 Transformation

Menu of this lecture

Viewing transformation

Viewing transformation: 3D -> 2D projection

Scale

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} =\begin{pmatrix} s_x & 0 \\ 0 & s_y \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

Multiplying a matrix on the left corresponds to performing a row operation.

Reflection

E.g. reflection about the y-axis:

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} =\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

Sheer

Illustration of sheer tranformation:
Sheer tranformation

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} =\begin{pmatrix} x+ay \\ y \end{pmatrix} =\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

Rotation

By default, the rotation is around the origin.
\(R_{45}\) refers to rotatiing 45 degrees counterclockwose about the origin.

\[ R_{\theta}=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \]

Homogeneous Coordinates

Translation is not linear transform, so it cannot be represented in matrix form. But we don't want it to be a special case, so homogeneous coordinates are used to represent all transformations.

Add a third coordinate (w-coordinate), to represent the translation characters of points or vectors.

Affine transformations:
affine map = linear map + translation map

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} =\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} +\begin{pmatrix} t_x \\ t_y \end{pmatrix} \]

2D version

  • 2D point: \(\begin{pmatrix}x , y , 1 \end{pmatrix}^T\)
  • 2D vector:\(\begin{pmatrix}x , y , 0 \end{pmatrix}^T\)

When \(w\neq 0\), 2D point \(\begin{pmatrix}x , y , w \end{pmatrix}^T\) means \(\begin{pmatrix}x/w , y/w , 1 \end{pmatrix}^T\)

(The w-coordinate of vectors are 0, which means vectors are translation invariant. By comparison, w-coordinate of points are 1)

Use homogeneous Coordinates to represent affine translations:

\[ \begin{pmatrix} x' \\ y' \\ w' \end{pmatrix} =\begin{pmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix} =\begin{pmatrix} x+t_x \\ y+t_y \\ 1 \end{pmatrix} \]

Properties:
1. The last row must be 0 0 1.
2. The top-left 2×2 matrix represents a linear transformation.
3. The rightmost column represents translation. 4. Relations between points and vectors: - vector + vector = vector - point - point = vector - point + vector = point - point + point = point

Scale:

\[ S(s_x, s_y)=\begin{pmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Rotation:

\[ R(\alpha)=\begin{pmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Translation:

\[ T(t_x, t_y)=\begin{pmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{pmatrix} \]

Inverse transformations: \(M^{-1}\)

Esp. matrix \(R_{\theta}\) for rotation is orthogonal, so \(R_{-\theta}=R^{-1}=R^T\).

3D version

  • 3D point: \(\begin{pmatrix}x , y , z, 1 \end{pmatrix}^T\)
  • 3D vector:\(\begin{pmatrix}x , y , z, 0 \end{pmatrix}^T\)

Use 4×4 matrix for affine transformations.

Composing Transforms

All matrices are left-multiplied to the original coordinates, and are composed in the order of transformations from right to left.

Examples

Transformations: A1 -> A2 -> ... -> An
Matrix: \(A_n\cdots A_2A_1\begin{pmatrix}x \\ y \\ 1\end{pmatrix}\)