 First fundamental form

The first fundamental form of a surface $S\subseteq {\mathbb{R}}^3$ is the restriction of the ambient inner product to each tangent space $T_{p}S$. For a parametrized surface $X(u,v)$, it is

The First Fundamental Form is a mathematical concept in differential geometry, which provides a way to measure distances and angles on a surface. It essentially encodes the geometric properties of a surface in terms of its intrinsic metrics. For a surface described by a parametric representation, the First Fundamental Form can be constructed from the parameters of that representation.