First fundamental form

The first fundamental form is the inner product restricted to tangent vectors. Let be two tangent vectors $\vec{v},\vec{w}\in\overrightarrow{T_{x_0}S}$:

$$\mathbb{I}(\vec{v}, \vec{w}) = \vec{v}\cdot\vec{w}$$

The first fundamental form is independent from a particular surface representation, thus invariant under parameter transformation. Another point of view for a parameterised surface is to use $2D$ coordinates $\vec{\tau} = (\tau_u, \tau_v)$ to describe tangent vectors. Let us note $x_u = \frac{\delta\xi}{\delta u}(u_0,v_0)$, $x_v = \frac{\delta\xi}{\delta u}(u_0,v_0)$. In the basis $(x_u, x_v)$, the first fundamental form applied to $(\tau_u, \tau_v)$ is entirely defined by the numbers:

$$\begin{array}{rclll} E &=& x_u^2\\ F &=& x_ux_v\\ G &=& x_v^2\\ \end{array}$$

Thus if $\vec{\tau}\in \overrightarrow{T_xS}$ has components $(\tau_u, \tau_v)$ in $(x_u, x_v)$, the first fundamental form defines a positive definite quadratic form in the tangent plane:

$$\begin{array}{rcl} \mathbb{I}(\vec{\tau},\vec{\tau}) &=& E\... ...n{array}{c} \tau_u \\ \tau_v \end{array}\!\!\right) \end{array}$$

where $E$, $F$ and $G$ are the first fundamental coefficients, and are not invariant under a parameter transformation. We will refer to the matrix of these coefficients as the first fundamental matrix.