# 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.