GPT is reall good at generating comments and fixing spelling error

math.md

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+# Silk Particle Simulation: Mathematical Formulation
+
+## 1. Stream Function
+
+The velocity field is derived from a scalar stream function $\psi(x, y, t)$:
+
+$$
+\psi(x, y, t) = \sum_{i=1}^{4} A_i \Psi_i(x, y, t)
+$$
+
+where:
+
+$$
+\begin{align}
+\Psi_1 &= \sin(\omega_1 x + \phi_1(t)) \sin(\omega_1 y) \\
+\Psi_2 &= \sin(\omega_2 x) \sin(\omega_2 y + \phi_2(t)) \\
+\Psi_3 &= \sin(\omega_3 r + \phi_3(t)), \quad r = \sqrt{(x - 0.5)^2 + (y - 0.5)^2} \\
+\Psi_4 &= (x - 0.5)(y - 0.5) \sin(\omega_4 t)
+\end{align}
+$$
+
+Suggested parameters: $A = (0.3, 0.25, 0.2, 0.15)$, $\omega = (2\pi, 3\pi, 4\pi, 0.5)$, $\phi_i(t) = c_i t$ with $c = (0.5, 0.3, 0.4)$
+
+---
+
+## 2. Velocity Field
+
+Incompressible flow via curl of stream function:
+
+$$
+\mathbf{v} = \nabla \times \psi = \left(\frac{\partial \psi}{\partial y}, -\frac{\partial \psi}{\partial x}\right)
+$$
+
+This ensures $\nabla \cdot \mathbf{v} = 0$ (divergence-free).
+
+---
+
+## 3. Vorticity (for color)
+
+Scalar vorticity equals the negative Laplacian of the stream function:
+
+$$
+\omega = \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} = -\nabla^2 \psi = -\left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right)
+$$
+
+---
+
+## 4. Boundary Confinement
+
+Soft quartic potential keeps particles on screen:
+
+$$
+V_{\text{boundary}}(x, y) = k \left[(x - 0.5)^4 + (y - 0.5)^4\right]
+$$
+
+Boundary force:
+
+$$
+\mathbf{F}_{\text{boundary}} = -\nabla V_{\text{boundary}} = -4k\left[(x - 0.5)^3, (y - 0.5)^3\right]
+$$
+
+---
+
+## 5. Equations of Motion
+
+Particle position $\mathbf{q} = (x, y)$ evolves as:
+
+$$
+\frac{d\mathbf{q}}{dt} = (1 - \alpha)\mathbf{v}(\mathbf{q}, t) + \beta \mathbf{F}_{\text{boundary}}(\mathbf{q})
+$$
+
+where $\alpha \approx 0.1$ (damping), $\beta \approx 0.05$ (boundary strength).
+
+---
+
+## 6. RK4 Integration
+
+Fourth-order Runge-Kutta for state $\mathbf{s}$ with derivative $\mathbf{f}(\mathbf{s}, t)$:
+
+$$
+\begin{align}
+\mathbf{k}_1 &= \mathbf{f}(\mathbf{s}, t) \\
+\mathbf{k}_2 &= \mathbf{f}(\mathbf{s} + \tfrac{\Delta t}{2} \mathbf{k}_1, t + \tfrac{\Delta t}{2}) \\
+\mathbf{k}_3 &= \mathbf{f}(\mathbf{s} + \tfrac{\Delta t}{2} \mathbf{k}_2, t + \tfrac{\Delta t}{2}) \\
+\mathbf{k}_4 &= \mathbf{f}(\mathbf{s} + \Delta t \mathbf{k}_3, t + \Delta t) \\
+\mathbf{s}_{\text{next}} &= \mathbf{s} + \tfrac{\Delta t}{6} (\mathbf{k}_1 + 2\mathbf{k}_2 + 2\mathbf{k}_3 + \mathbf{k}_4)
+\end{align}
+$$
+
+---
+
+## 7. Color Mapping
+
+Map vorticity to HSV hue:
+
+$$
+H = \text{mod}\left(\frac{\omega - \omega_{\min}}{\omega_{\max} - \omega_{\min}} \cdot 360°, 360°\right), \quad S = 0.8, \quad V = 0.9
+$$
+
+Positive $\omega$ (CCW) → warm colors; negative $\omega$ (CW) → cool colors.