notes/docs/lectures/graphics/04_transformations.md

Transformations

Translation

Translation is done by adding/subtracting the translation distance to either the x or y (or both) component

A translation of (3,2) done on vector u=\begin{pmatrix} u_0\\u_1\end{pmatrix}

u'=\begin{pmatrix} u_0+3\\u_1+2\end{pmatrix}

This can be applied to a triangle, where every vertex is translated by the same amount.

Rotation

Vector v can be rotated by angle \theta radians anticlockwise as follows

rot(v) = \begin{pmatrix}
cos\theta * v_0 - sin\theta * v_1 \\
sin\theta * v_0 - cos\theta * v_1
\end{pmatrix}

We can rotate triangles by rotating each vertex

Scale

The vector v can be scaled by scalar s in each dimension independently

v \cdot s = \begin{pmatrix} v_0 \cdot s_0 \\ v_1\cdot s_1 \end{pmatrix}

We can scale triangles by scaling each of its vertices

Transformation Matrix

Translation

This is where we represent a transformation in the form of a matrix

• The translation matrix, T, which translates by some vector t = (t_x, t_y, t_z)

T(t) = \begin{pmatrix}
1 \quad 0 \quad 0 \quad t_x \\
0 \quad 1 \quad 0 \quad t_y \\
0 \quad 0 \quad 1 \quad t_z \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}

Rotation

• The rotations around each axis by some angle \theta are represented as matrices R_x, R_y, R_z
• This is in 2 dimensions

Scale

S(t) = \begin{pmatrix}
s_x \quad 0 \quad 0 \quad 0 \\
0 \quad s_y \quad 0 \quad 0 \\
0 \quad 0 \quad s_z \quad 0 \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}

Homogeneous Coordinates

• Vector defines a direction or a position
• We can rotate directions and positions
• We can scale directions and positions
• We can only translate positions

A homogeneous vector is p=(p_x, p_y, p_z, p_w)

• Transforming a point p is done by multiplying the vector (with w set to 1) by the transformation matrix M
• To transform vectors the w must be set to 1

The translation matrix can be applied to point p by multiplying the point by the matrix

The scale matrix can be applied to a point p by multiplying the point by the matrix

The rotation around x matrix can be applied to a point p by multiplying the point by the matrix

y:

z:

Combining Transformations

For example if point p needs to be scaled by s=(2,1,1) and then translated by t=(1,0,0)

S = \begin{pmatrix}
2 \quad 0 \quad 0 \quad 0 \\
0 \quad 1 \quad 0 \quad 0 \\
0 \quad 0 \quad 1 \quad 0 \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}
\\\\
T = \begin{pmatrix}
1 \quad 0 \quad 0 \quad 1 \\
0 \quad 1 \quad 0 \quad 0 \\
0 \quad 0 \quad 1 \quad 0 \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}
\\\\
TS = T\times S =
\begin{pmatrix}
2 \quad 0 \quad 0 \quad 1 \\
0 \quad 1 \quad 0 \quad 0 \\
0 \quad 0 \quad 1 \quad 0 \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}

Matrix multiplication is read from right to left

The order of the transformations makes a difference and can change the resulting vector for example

ST = S\times T =
\begin{pmatrix}
2 \quad 0 \quad 0 \quad 2 \\
0 \quad 1 \quad 0 \quad 0 \\
0 \quad 0 \quad 1 \quad 0 \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}

The difference between translating and scaling vs scaling and translating vector v