S3D: Assignment #2

The code for this assignment can be found here.

For this assignment, a simple scene was prepared in Unity. It is composed of:

• A terrain (from Unity);
• Water (from Unity’s Standard Assets);
• Procedural rocks (from Assignment 1);
• Procedural vases (from Assignment 1);
• And procedural trees, that grow over time (based on the Bushes from Assignment 1).

Rocks

The same algorithm was used, with the only addition of the UV mapping, which was simply done by mapping a triangle with vertices \(v_0\), \(v_1\), \(v_2\) to \((0, 0)\), \((1, 0)\), \((1, 1)\).

Vases

The same algorithm was used, and the UV mapping was simply \((\theta, t)\) for every generated vertex.

Growing Trees

The tree is based on the Bushes from Assignment 1, which are generated using a Stochastic L-System on an alphabet with turtle graphics interpretation. The one used there was

\[\begin{array}{cclcl} \omega & : & \mathtt{A} \\ p_0' & : & \mathtt{A} & \xrightarrow{0.34} & \mathtt{[\&FLA]/////[\&FLA]///////[FBA]} \\ p_0'' & : & \mathtt{A} & \xrightarrow{0.33} & \mathtt{[\&LA]/////[\&FLA]///////[FBA]} \\ p_0''' & : & \mathtt{A} & \xrightarrow{0.33} & \mathtt{[\&FLA]/////[\&LA]} \\ p_1 & : & \mathtt{F} & \xrightarrow{} & \mathtt{S/////F} \\ p_2 & : & \mathtt{S} & \xrightarrow{} & \mathtt{FL} \\ p_3 & : & \mathtt{L} & \xrightarrow{} & \mathtt{[ \wedge \wedge \{-f+f+f-|-f+f+f \}]} \\ p_4 & : & \mathtt{B} & \xrightarrow{} & \mathtt{[-Fb]} \end{array}\]

With the alphabet

Since the flower (the b instruction above) is simply a static mesh, we can’t animate it growing. Also, since we want more of a tree than a bush, we add some forward instructions before the A symbol in the axiom. The resulting L-System is

\[\begin{array}{cclcl} \omega & : & \mathtt{0$SA} \\ p_0' & : & \mathtt{A} & \xrightarrow{0.34} & \mathtt{[\&F\%LA]/////[\&F\%LA]///////[F\%A]} \\ p_0'' & : & \mathtt{A} & \xrightarrow{0.33} & \mathtt{[\&LA]/////[\&F\%LA]///////[F\%A]} \\ p_0''' & : & \mathtt{A} & \xrightarrow{0.33} & \mathtt{[\&F\%LA]/////[\&LA]} \\ p_1 & : & \mathtt{F} & \xrightarrow{} & \mathtt{S/////F} \\ p_2 & : & \mathtt{S} & \xrightarrow{} & \mathtt{FL} \\ p_3 & : & \mathtt{L} & \xrightarrow{} & \mathtt{[1 \wedge \wedge \{-f+f+f-|-f+f+f \}]} \\ \end{array}\]

With the alphabet

We start by generating a list of instructions for our turtle, as if we were making a static mesh. The growing animation is accomplished by recreating a mesh from the instructions for each frame, but stopping each time after a certain amount of increasing steps.

The parameters necessary are:

• Number of steps per “Forward” instruction;
• Number of steps to start at (usually 0)
• Maximum number of steps we want to progress at the time;

Then, to generate a mesh for some frame where we have already progressed \(i\) frames, we do as normal (starting from our origin), with the following exceptions:

• Forward: the amount to step, instead of simply being \(\delta\) (step size), becomes \((\delta / n) \min \{m - i, n\},\) with \(m\) being the maximum number of steps to progress, \(i\) the number of steps already progressed and \(n\) the number of steps per “Forward” instruction. Finally, we step \(\min \{m - i, n\}\).
• Rotations: the shoulder (to not leave a gap into the mesh) should not be inserted if \(i \geq m\) (where \(i\) is the number of steps progressed so far and \(m\) is the maximum number of steps to progress).
• Starting Polygons: (This is used to generate the leaves in the trees) When we start the polygon, we calculate a step size for it: \(\delta_0 = (\delta / n) \min \{m - i, n\}\). Whenever we use an instruction that will add an edge to the polygon, we use \(\delta_0\) as our step size.

Finally, to apply textures, we need an UV mapping to be applied to our mesh’s faces. This way, the rendering engine knows which coordinate of the texture image to apply to that position in the mesh’s surface.

If the tree was static, the mapping would be quite simple; for each face, its vertices \(v_0\), \(v_1\), \(v_2\) and \(v_3\) have UVs \((0, 0)\), \((1, 0)\), \((1, 1)\) and \((0, 1)\), respectively. However, it is growing; if we were to use the same mapping, the textures would move along with the branches, which would look quite strange. Instead, we use the mapping \((0, 0)\), \(((1/n) \min \{m - i, n\}, 0)\), \(((1/n) \min \{m - i, n\}, 1)\), \((0, 1)\).

Screenshots