152 lines
3.5 KiB
Markdown
152 lines
3.5 KiB
Markdown
# Transformations
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### Translation
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Translation is done by adding/subtracting the translation distance to either the x or y (or both) component
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A translation of $(3,2)$ done on vector $u=\begin{pmatrix} u_0\\u_1\end{pmatrix}$
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$u'=\begin{pmatrix} u_0+3\\u_1+2\end{pmatrix}$
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This can be applied to a triangle, where every vertex is translated by the same amount.
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### Rotation
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Vector $v$ can be rotated by angle $\theta$ radians anticlockwise as follows
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$$
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rot(v) = \begin{pmatrix}
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cos\theta * v_0 - sin\theta * v_1 \\
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sin\theta * v_0 - cos\theta * v_1
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\end{pmatrix}
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$$
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We can rotate triangles by rotating each vertex
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### Scale
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The vector $v$ can be scaled by scalar $s$ in each dimension independently
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$$
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v \cdot s = \begin{pmatrix} v_0 \cdot s_0 \\ v_1\cdot s_1 \end{pmatrix}
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$$
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We can scale triangles by scaling each of their vertices
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### Transformation Matrix
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#### Translation
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This is where we represent a transformation in the form of a matrix
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- The translation matrix, **T**, which translates by some vector **t** $= (t_x, t_y, t_z)$
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$$
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T(t) = \begin{pmatrix}
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1 \quad 0 \quad 0 \quad t_x \\
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0 \quad 1 \quad 0 \quad t_y \\
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0 \quad 0 \quad 1 \quad t_z \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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$$
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#### Rotation
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- The rotations around each axis by some angle $\theta$ are represented as matrices $R_x, R_y, R_z$
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- This is in 2 dimensions
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#### Scale
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$$
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S(t) = \begin{pmatrix}
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s_x \quad 0 \quad 0 \quad 0 \\
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0 \quad s_y \quad 0 \quad 0 \\
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0 \quad 0 \quad s_z \quad 0 \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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$$
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### Homogeneous Coordinates
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- Vector defines a **direction** or a **position**
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- We can rotate directions and positions
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- We can scale directions and positions
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- We can only translate positions
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A homogeneous vector is $p=(p_x, p_y, p_z, p_w)$
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- Transforming a point $p$ is done by multiplying the vector (with $w$ set to 1) by the transformation matrix $M$
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- To transform vectors the $w$ must be set to 1
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The translation matrix can be applied to point $p$ by multiplying the point by the matrix
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The scale matrix can be applied to a point $p$ by multiplying the point by the matrix
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The rotation around x matrix can be applied to a point $p$ by multiplying the point by the matrix
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y:
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z:
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#### Combining Transformations
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For example, if point $p$ needs to be scaled by $s=(2,1,1)$ and then translated by $t=(1,0,0)$
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$$
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S = \begin{pmatrix}
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2 \quad 0 \quad 0 \quad 0 \\
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0 \quad 1 \quad 0 \quad 0 \\
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0 \quad 0 \quad 1 \quad 0 \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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\\\\
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T = \begin{pmatrix}
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1 \quad 0 \quad 0 \quad 1 \\
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0 \quad 1 \quad 0 \quad 0 \\
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0 \quad 0 \quad 1 \quad 0 \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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\\\\
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TS = T\times S =
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\begin{pmatrix}
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2 \quad 0 \quad 0 \quad 1 \\
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0 \quad 1 \quad 0 \quad 0 \\
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0 \quad 0 \quad 1 \quad 0 \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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$$
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Matrix multiplication is read from right to left
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The order of the transformations makes a difference and can change the resulting vector for example
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$$
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ST = S\times T =
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\begin{pmatrix}
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2 \quad 0 \quad 0 \quad 2 \\
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0 \quad 1 \quad 0 \quad 0 \\
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0 \quad 0 \quad 1 \quad 0 \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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$$
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The difference between translating and scaling vs scaling and translating vector $v$
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