The general model for simulating grass blade behavior is as follows:

Now let's look at each of these parameters separately.

As mentioned earlier, the grass blade consists of segments. Let blade height be $H$, and number of segments be $N$. Then the height of each segment is $L=\frac{H}{N}$. We also need the concept of relative segment height $S=\frac{i}{N} \in [0, 1]$, where $i$ is the segment index. One more required parameter in this model is $\alpha_{i}$, the angle that defines how far segment $i$ is tilted from its vertical position. This parameter will later be determined by wind and influence functions.

The key idea here is that wind is a function $\omega(t)\in[-1,1]$ that defines how strongly the wind blows to the right or left at time $t$. This function can be defined in different ways, for example by a sine function. However, to get closer to realistic behavior, we will use a smoothed deterministic noise function.

Assume wind changes every $\Delta t$ seconds. Let the time step be $j$. That means wind changes at time $j\cdot\Delta t$. For example, if $\Delta t=0.5$, then wind changes at times $0.5, 1.0, 1.5, 2.0, ...$

Now let's see how to get wind values at each step. For this we need a generator that returns a deterministic value in the range $[-1, 1]$ for each $j$. There are many possible implementations. Here is one of them:

The coefficients $127.1$ and $43758.5453$ were chosen arbitrarily.

Now the wind generator is defined, but if we change wind values abruptly, the blade will instantly change its position. To avoid that, we need to smooth transitions between steps. For this we need a smoothing function. As with the generator, there are several options. We will use one of the most common ones: ease-in/ease-out interpolation.

Here, $u$ is the normalized time offset between step $j-1$ and step $j$, so $u \in [0, 1]$.

To keep this function between $Hash(j)$ and $Hash(j-1)$, we interpolate it: take $Smooth(u)$ as the interpolation parameter and smoothly blend values $Hash(j-1)$ and $Hash(j)$.

So we can define the resulting smoothed deterministic function at time $t$:

As a final step to make behavior more realistic, we combine slow and fast wind oscillations. For this, we take values $F(t,\Delta t)$ with different $\Delta t$ and corresponding coefficients. As a result, we get the wind function we need:

Thus, $F(t,0.8)$ gives slow smooth oscillations with higher weight, while $F(t,0.15)$ gives faster oscillations with lower weight, adding realism.

To make the blade behave naturally, its root should stay firmly attached to the ground, while the tip is freer and affected the most. In other words, wind should contribute differently at different heights. This introduces the function $Influence(S), S \in [0, 1]$. Here, $S=0$ is the root and $S=1$ is the tip. From this behavior we can conclude that the function should be increasing, but not too steep, otherwise the blade will simply be pressed to the ground. In general, there is room to experiment here. The simplest and most straightforward option is a linear increasing function:

In this case, wind influence increases uniformly from root to tip. The behavior is reasonably realistic. You can also use a parabolic function:

where $p$ is a coefficient chosen manually, for example $p=2$. With this function, influence near the tip becomes much stronger than near the root. You can increase this effect even more:

Overall, there is room for tuning and experimentation.

After everything above, only one step remains: compute the final coordinates of each segment. For this we calculate the deviation angle. It cannot exceed 90 degrees or $\frac{\pi}{2}$, so the blade does not bend below the ground. The final deviation angle for segment $i$ is:

And coordinates of segment $i$:

At the same time, $x_0(t), y_0(t)$ are always fixed, because the blade is rigidly attached to the ground.

Now that we have the formula for computing coordinates of each segment, take this challenge and implement this logic. Good luck.