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plots/demo-plot-1/README.md

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+# Parametric Surface Plotter
+
+This project provides a simple way to **define and plot parametric 3D surfaces** (or curves) using Python.
+
+* A **surface** is defined with **two parameters**:
+
+  $$
+  x = x(t, u), \quad y = y(t, u), \quad z = z(t, u)
+  $$
+
+* A **curve** is defined with a **single parameter**:
+
+  $$
+  x = x(t), \quad y = y(t), \quad z = z(t)
+  $$
+
+The script will:
+
+* Generate a mesh of points (for surfaces) or a set of points (for curves)
+* Plot the result in 3D with Matplotlib
+* Save the plot as a high-definition `.svg` file, named after the parent directory of the script
+
+---
+
+## Requirements
+
+Install dependencies:
+
+```bash
+pip install -r requirements.txt
+```
+
+---
+
+## Usage
+
+Run the script:
+
+```bash
+python plot_surface.py
+```
+
+A window with the 3D plot will open, and an `.svg` will be saved in the same directory.
+
+---
+
+## Defining Your Own Geometry
+
+### 1. Surfaces (two parameters: t, u)
+
+Surfaces require two parameters because they are 2D objects.
+You define three functions in `plot_surface.py`:
+
+```python
+def x_func(t, u): return ...
+def y_func(t, u): return ...
+def z_func(t, u): return ...
+```
+
+Examples:
+
+* **Cone**
+
+  $$
+  x = 4t\cos(u), \quad y = 4t\sin(u), \quad z = 10t
+  $$
+
+  ```python
+  def x_func(t, u): return 4 * t * np.cos(u)
+  def y_func(t, u): return 4 * t * np.sin(u)
+  def z_func(t, u): return 10 * t
+  ```
+
+* **Sphere** (radius 1)
+
+  $$
+  x = \cos(u)\sin(t), \quad y = \sin(u)\sin(t), \quad z = \cos(t)
+  $$
+
+  ```python
+  def x_func(t, u): return np.cos(u) * np.sin(t)
+  def y_func(t, u): return np.sin(u) * np.sin(t)
+  def z_func(t, u): return np.cos(t)
+  ```
+
+You can adjust parameter ranges in `generate_mesh()`:
+
+```python
+def generate_mesh(t_range=(0, 5), u_range=(0, 2*np.pi), n_t=50, n_u=100):
+    ...
+```
+
+* Sphere → `t_range=(0, np.pi)`, `u_range=(0, 2*np.pi)`
+* Cone → `t_range=(0, 5)`, `u_range=(0, 2*np.pi)`
+
+---
+
+### 2. Curves (one parameter: t)
+
+If your object is not a surface but a **curve**, you only need one parameter $t$.
+
+Example: **Helix**
+
+$$
+x = \cos(t), \quad y = \sin(t), \quad z = t
+$$
+
+```python
+def x_func(t): return np.cos(t)
+def y_func(t): return np.sin(t)
+def z_func(t): return t
+```
+
+Here you don’t need `u` at all. The code can be simplified to generate a 1D array of points and plot them as a line in 3D.
+
+---
+
+## Which Should I Use?
+
+* **Two parameters (t, u):** Use this for **surfaces** (sphere, torus, cone, paraboloid, etc.).
+* **One parameter (t):** Use this for **curves** (helix, circle, line, parametric trajectory).
+
+👉 A quick rule:
+
+* If your equation is like $z = f(x, y)$, or implicit in 3 variables, you usually need **two parameters**.
+* If your equation is like $x = f(t), y = g(t), z = h(t)$, then **one parameter** is enough.
+
+---
+
+## Notes
+
+* Always use `numpy` functions (`np.sin`, `np.cos`, etc.), not Python’s built-ins, since the parameters are arrays.
+* Some surfaces (like cones, hyperboloids) have two branches; you can add an additional `z_func_neg` if needed.
+* The SVG is automatically named after the script’s parent directory.
+
+---
+
+✅ With this, you can parametrize and visualize **both surfaces (2D)** and **curves (1D)** by editing only a few functions.
+
+