maths.first_fundamental_form
============================

.. py:module:: maths.first_fundamental_form

.. autoapi-nested-parse::

   Calculates the First Fundamental Form of a parametric surface.

   The first fundamental form allows the measurement of lengths, angles,
   and areas on a surface defined by parametric equations r(u, v).

   Reference:
   - https://en.wikipedia.org/wiki/First_fundamental_form

   Author: Chahat Sandhu
   GitHub: https://github.com/singhc7



Functions
---------

.. autoapisummary::

   maths.first_fundamental_form.first_fundamental_form


Module Contents
---------------

.. py:function:: first_fundamental_form(x_expr: str, y_expr: str, z_expr: str) -> tuple[sympy.Expr, sympy.Expr, sympy.Expr]

   Calculate the First Fundamental Form coefficients (E, F, G) for a surface
   defined by parametric equations x(u,v), y(u,v), z(u,v).

   Args:
       x_expr: A string representing the x component in terms of u and v.
       y_expr: A string representing the y component in terms of u and v.
       z_expr: A string representing the z component in terms of u and v.

   Returns:
       A tuple containing the sympy expressions for E, F, and G.

   Examples:
       >>> # Example 1: A simple plane r(u, v) = <u, v, 0>
       >>> E, F, G = first_fundamental_form("u", "v", "0")
       >>> print(f"E: {E}, F: {F}, G: {G}")
       E: 1, F: 0, G: 1

       >>> # Example 2: A paraboloid r(u, v) = <u, v, u**2 + v**2>
       >>> E, F, G = first_fundamental_form("u", "v", "u**2 + v**2")
       >>> print(f"E: {E}, F: {F}, G: {G}")
       E: 4*u**2 + 1, F: 4*u*v, G: 4*v**2 + 1

       >>> # Example 3: A cylinder r(u, v) = <cos(u), sin(u), v>
       >>> E, F, G = first_fundamental_form("cos(u)", "sin(u)", "v")
       >>> print(f"E: {sp.simplify(E)}, F: {F}, G: {G}")
       E: 1, F: 0, G: 1