ex2: task 3 report

gloom-rs/report/exercise2.md

@@ -4,8 +4,8 @@
 # Change them as you see fit.
 title: TDT4195 Exercise 2
 author:
-- Øyvind Nestvold
-- Fredrik Robertsen
+  - Øyvind Nestvold
+  - Fredrik Robertsen
 date: \today # This is a latex command, ignored for HTML output
 lang: en-US
 papersize: a4
@@ -35,8 +35,49 @@ OpenGL does a simple interpolation between the vertex colors, and assigns a colo
 
 ## Task 2)
 
-# a) Transparent triangles
+### a) Transparent triangles
 
 ![](./images/transparent_triangles.png)
 
 Here we have drawn three partially overlapping triangles drawn in the order "red - green - blue" with red having the highest z-index (furthest back), followed by green and finally blue. All triangles are rendered with alpha = 0.4
+
+### b) TODO
+
+## Task 3)
+
+### a-c)
+
+Given a matrix defined as
+
+$$
+\begin{bmatrix}
+a & b & 0 & c \\
+d & e & 0 & f \\
+0 & 0 & 1 & 0 \\
+0 & 0 & 0 & 1
+\end{bmatrix}
+$$
+
+we can adjust each variable $a, b, c, d, e$ and $f$ individually to observe all
+affine transformations except pure rotations. This can be intuitively understood
+from linear algebra, where a matrix represents a linear transformation of the
+basis vectors. For example, the upper left 2x2 matrix described by $a, b, d$ and
+$e$ tells us that we transform the $\mathbf{\hat{i}}$- and
+$\mathbf{\hat{j}}$-vectors to $[a, d]^\intercal$ and $[b, e]^\intercal$
+respectively. Thus can these four values alone perform shears and scaling on the
+$x y$-plane, since these transformations are simple linear transformations as
+such.
+
+- If $b$ and $d$ are both zero, we will perceive a scaling effect equal to $a$
+  in $x$-direction and $e$ in $y$-direction.
+- Similarily, if only one of $b$ and $d$ are zero, we can see a shear, as one of
+  the coordinate system vectors are only being scaled (it is being "held in
+  place"), while the other is linearly transformed freely.
+- Translation is encoded in $c$ and $f$, where they correspond to translation in
+  $x$- and $y$-directions respectively.
+
+If we want to perform a rotation, we have to modify all four values $a$, $b$,
+$d$ and $e$ in conjunction and in accordance with the 2D rotation matrix,
+otherwise we end up with a shear instead. Because we have to change these values
+together, it is impossible to observe rotational transformations from only
+changing one value at a time, starting from the identity matrix.