Merged assignments 1 and 2

_layouts/default.html

@@ -29,8 +29,7 @@ <h2 class="project-tagline">{{ page.description | default: site.description | de
       <a href="{% link index.md %}" class="btn">Start</a>
       <a href="{% link assignments/Assignment_1.md %}" class="btn">Week 1</a>
       <a href="{% link assignments/Assignment_2.md %}" class="btn">Week 2</a>
-      <a href="{% link assignments/Assignment_3.md %}" class="btn">Week 3</a>
-      <a href="{% link assignments/Assignment_4_practice.md %}" class="btn">Week 4</a>
+      <a href="{% link assignments/Assignment_3_practice.md %}" class="btn">Week 3</a>
       <a href="{% link assignments/FinalProject_1.md %}" class="btn">Final 1</a>
       <a href="{% link assignments/FinalProject_2.md %}" class="btn">Final 2</a>
       <a href="{% link assignments/FinalProject_3_optional.md %}" class="btn">Final Extra</a>

assignments/Assignment_1.md

@@ -4,8 +4,166 @@ title: BK7084 - Week 1
 ---
 
 # Assignment 1
+In this assignment, we will dive into transformations. It’s
+important that you get how to apply transformations to objects in 3D
+scenes and how to compose transformations, as you will use these
+operations a lot in the final assignment of this course. In this
+assignment, we slowly build up complexity: first, you will create
+transformation matrices from scratch to control a virtual car. Then you
+will compose transformation matrices to construct a lamp. Finally, you
+can apply these lessons to build an animated solar system. Follow the
+comments in *ex01.py*, *ex02.py*, and *ex03.py* to finish the
+assignments. You can use this document as a reference. There are
+questions in this document. Don’t skip over them, but think about them.
+If you can’t answer them, feel free to ask one of the TAs.
 
-Your first assignment is to write a program for a ray-triangle
+# Translation matrices
+
+You have learned about translation matrices and how they can be applied
+in order to move points in space. Below you can see how a point is moved
+by multiplying with a translation matrix
+
+$$
+\begin{bmatrix}
+ 1 & 0 & 0 & t_x \\
+ 0 & 1 & 0 & t_y \\
+ 0 & 0 & 1 & t_z \\
+ 0 & 0 & 0 & 1 \\
+\end{bmatrix}
+\begin{bmatrix}
+ v_x \\
+ v_y \\
+ v_z \\
+ 1
+\end{bmatrix} =
+\begin{bmatrix}
+ v_x + t_x \\
+ v_y + t_y \\
+ v_z + t_z\\
+ 1
+\end{bmatrix}.
+$$
+
+\$\rightarrow\$ Why are we using four dimensions in the matrix and
+coordinates of the point?  
+\$\rightarrow\$ Can you compute the result yourself?  
+\$\rightarrow\$ What would happen to the result if the last entry in the
+matrix was a 4?
+
+In the *translate* function you see that we create a matrix *mat* which
+will become your translation matrix. You must fill in the rows and
+columns of this matrix appropriately to form such a matrix. It starts
+off as the identity matrix
+
+$$
+\begin{bmatrix}
+ 1 & 0 & 0 & 0 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 0 \\
+ 0 & 0 & 0 & 1 \\
+\end{bmatrix}
+$$
+
+# Rotation matrices
+
+We use a rotation matrix to rotate points. In 3D, a point can be rotated
+around three possible axes (the x, y and z axes). Therefore, we need
+three rotation matrices to support all these rotations.
+
+$$
+R_x(\theta) = \begin{bmatrix}
+ 1 & 0 & 0 \\
+ 0 & \cos{\theta} & -\sin{\theta} \\
+ 0 & \sin{\theta} & \cos{\theta} \\
+\end{bmatrix}
+$$
+
+$$
+R_y(\theta) = \begin{bmatrix}
+ \cos{\theta} & 0 & \sin{\theta} \\
+ 0 & 1 & 0 \\
+ -\sin{\theta} & 0 & \cos{\theta} \\
+\end{bmatrix}
+$$
+
+$$
+R_z(\theta) = \begin{bmatrix}
+ \cos{\theta} & -\sin{\theta} & 0 \\
+ \sin{\theta} & \cos{\theta} & 0 \\
+ 0 & 0 & 1 \\
+\end{bmatrix}
+$$
+
+# Composing transformations
+
+Let’s say you want to rotate a point and then translate it. You have
+created the rotation matrix $\mathbf{R}$ and translation matrix
+$\mathbf{T}$. The point you want to transform is called
+$\mathbf{p}$. You could first compute the rotation
+\\[\mathbf{p}' = \mathbf{Rp}\\] and then the translation
+\\[\mathbf{p}'' = \mathbf{Tp}'.\\] Another way to write this is
+\\[\mathbf{p}' = \\mathbf{TRp}.\] You could also first compute the product
+of the matrices $\mathbf{T}$ and $\mathbf{R}$ and then multiply the
+result with $\mathbf{p}$.
+
+$$
+\begin{aligned}
+    \mathbf{M} = \mathbf{TR} \\
+    \mathbf{p}' = \mathbf{Mp}.
+\end{aligned}
+$$
+
+This is possible, because matrix products are *associative*, a fancy
+word to say that $(\mathbf{AB})\mathbf{C} = \mathbf{A}(\mathbf{BC})$.
+You know this property from regular old numbers (we call them
+*scalars*). For example, you know that
+$(3 \times 4) \times 5 = 3 \times (4 \times 5)$. This is probably so
+obvious to you that you never thought about it, but we’ll see that not
+all properties of scalars hold for matrices. The associativity property
+is nice, because it means we can compose all our transformations into
+*one* 4x4 matrix, which is then multiplied with all the points in our
+scene.
+
+We already mentioned that matrix multiplications don’t share all
+properties of scalars. One such property is *commutativity*.
+Commutativity means that you can swap the order of operations and the
+result will be the same: $a \times b = b \times a$. Try it out with
+numbers: $3 \times 4 = 4 \times 3$.
+
+$\rightarrow$ Why is this true? Hint: Can you think of a picture for
+multiplication?
+
+**Matrix multiplications are not commutative**. That means you cannot
+change the order of matrices and expect the same result. Try it out
+yourself:
+
+$$
+\begin{aligned}
+    \begin{bmatrix}
+     1 & 3  \\
+     4 & 2  \\
+    \end{bmatrix} 
+    \begin{bmatrix}
+     20 & 10  \\
+     40 & 60  \\
+    \end{bmatrix} &= ? \\\\
+    \begin{bmatrix}
+     20 & 10  \\
+     40 & 60  \\
+    \end{bmatrix}
+    \begin{bmatrix}
+     1 & 3  \\
+     4 & 2  \\
+    \end{bmatrix} 
+    &= ?\end{aligned}
+$$
+
+$\rightarrow$ What is the difference between first rotating and then
+translating and first translating and then rotating?
+
+# Extra
+
+This is an extra assignment in case you are done early with the first assignment. The task is to write a program for a ray-triangle
 intersection. This is one of the core tests for many light simulation
 algorithms, especially ray-tracing. You can think of the ray as being a
 ray of light and the triangle is part of some surface. You want to know