maths.first_fundamental_form

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

first_fundamental_form(→ tuple[sympy.Expr, sympy.Expr, ...)

Calculate the First Fundamental Form coefficients (E, F, G) for a surface

Module Contents

maths.first_fundamental_form.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