FundamentalForms

The fundamental forms are fundamental concepts in differential geometry and are used to define geometric properties of surfaces embedded in $\mathbb{R}^3$ or in a higher-dimensional Euclidean space.

1. First Fundamental Form (FFF): This form encodes information about the metric structure of the surface, including lengths, angles, and areas. It's given by the dot product of the tangent vectors, and can be represented by a symmetric $2 \times 2$ matrix given by: $$ I = \begin{bmatrix} E & F \ F & G \end{bmatrix} $$ where $$ \begin{align*} E &= \mathbf{X}_u \cdot \mathbf{X}_u \ F &= \mathbf{X}_u \cdot \mathbf{X}_v \ G &= \mathbf{X}_v \cdot \mathbf{X}_v \end{align*} $$ and $\mathbf{X}_u$ and $\mathbf{X}_v$ are the partial derivatives of the parameterization $\mathbf{X}(u,v)$ of the surface with respect to the parameters $u$ and $v$.

2. Second Fundamental Form (SFF): This form is related to the curvature of the surface and can be represented by a $2 \times 2$ matrix: $$ II = \begin{bmatrix} L & M \ M & N \end{bmatrix} $$ where $$ \begin{align*} L &= \mathbf{X}{uu} \cdot \mathbf{N} \ M &= \mathbf{X}{uv} \cdot \mathbf{N} \ N &= \mathbf{X}{vv} \cdot \mathbf{N} \end{align*} $$ Here, $\mathbf{X}{uu}, \mathbf{X}{uv}, \mathbf{X}{vv}$ are the second partial derivatives of the parameterization, and $\mathbf{N}$ is the unit normal vector to the surface.

The inputs to compute these forms would typically be a point on the surface and the parameterization of the surface in terms of two variables. You would need the derivatives of the parameterization with respect to these variables, as well as the normal vector, to compute the coefficients of the fundamental forms.