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docs/lectures/graphics/04_transformations.md

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+# 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
+
+$$
+rot(v) = \begin{pmatrix}
+cos\theta * v_0 - sin\theta * v_1 \\
+sin\theta * v_0 - cos\theta * v_1
+\end{pmatrix}
+$$
+
+![1646415138.png](img/1646415138.png)
+
+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
+
+![1646415811.png](img/1646415811.png)
+
+#### 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
+
+![1646416149.png](img/1646416149.png)
+
+The translation matrix can be applied to point $p$ by multiplying the point by the matrix
+
+![1646416282.png](img/1646416282.png)
+
+The scale matrix can be applied to a point $p$ by multiplying the point by the matrix
+
+![1646416316.png](img/1646416316.png)
+
+The rotation around x matrix can be applied to a point $p$ by multiplying the point by the matrix
+
+![1646416447.png](img/1646416447.png)
+
+y:
+
+![1646416456.png](img/1646416456.png)
+
+z:
+
+![1646416468.png](img/1646416468.png)
+
+#### 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
+
+$$
+ST = S\times T =
+\begin{pmatrix}
+2 \quad 0 \quad 0 \quad 2 \\
+0 \quad 1 \quad 0 \quad 0 \\
+0 \quad 0 \quad 1 \quad 0 \\
+0 \quad 0 \quad 0 \quad 1
+\end{pmatrix}
+$$
+
+![1646417286.png](img/1646417286.png)
+
+The difference between translating and scaling vs scaling and translating vector $v$