Let \(E = \|\sigma_u\|^2, F = \sigma_u\cdot\sigma_v, G = \|\sigma_v\|^2\), consideer the matrix \(M = \begin{bmatrix}E&F\\F&G\end{bmatrix}\), note that \(E, G\) are always positve, and since \(\det(M) = EG-F^2 > 0, M\) is always a postive definite symmetric matrix.
Consider the inner product equipped with dot product and restricted to the tangent plane at \(p\). Then, for any \(v\in T_p\Sigma\),
Let \(\gamma: \mathbb R\rightarrow \Sigma\) be some smooth curve on the surface \(\Sigma\), then for some parameterization \(\sigma\) (assuming the surface is covered by one patch) we have that
Assume doamin \(\Omega \subseteq \sigma(U)\), say \(\Omega = \sigma(U_1)\). The area of \(U_1\) is then integrated by \(\iint_{U_1}dudv\), i.e. integrate along each square \((u, v), (u+du, v+dv)\). For small \(du,dv\), since \(\sigma\) is a local diffeomorphism, it is approximated by the tangent plane \(\sigma_u, \sigma_v\). Therefore, the area of the mapped square is \(\|\partial_u\times \partial_v\|\), by change of variable,
so that \(D\gamma_1 = (1, 0), D\gamma_2 = (0, 1)\)\(\cos\theta = \frac{E(0) + F(1) + G(0)}{\sqrt{EG} } = \frac{F}{\sqrt{EG} }\)
Examples
\(\sigma = (\sinh u\sinh v, \sinh u\cosh v, sinh u)\)
\[\begin{align*} \sigma_u &= (\cosh u\sinh v, \cosh u\cosh v, \cosh u)\\ \sigma_v &= (\sinh u\cosh v, \sinh u\sinh v, 0)\\ E = \sigma_u\cdot\sigma_v &= \cosh^2 u\sinh^2 v + \cosh^2 u\cosh^2 v + \cosh^2 u \\ &= \cosh^2 u((\sinh^2 v + 1)+\cosh^2 v)\\ &= \cosh^2 u (\cosh^2 v + \cosh^2 v)\\ &= 2\cosh^2u \cosh^2 v\\ F = \sigma_u\cdot\sigma_v &=2\cosh u\cosh v\sinh u\sinh v\\ G = \sigma_v\cdot\sigma_v &=\sinh^2 u (\cosh^2 v + \sinh^2 v) = \sinh^2 u \cosh (2v)\\ \end{align*}\]
\(\sigma = (u-v, u+v, u^2+v^2)\)
\[\sigma_u = (1, 1, 2u)\sigma_v = (-1, 1, 2v)\]
\[(2 + 4u^2)du^2 + 8uv dudv + (2+4v^2)dv^2\]
\(\sigma = (\cosh u, \sinh u, v)\)
\[\sigma_u = (\sinh u, \cosh u, 0)\sigma_v = (0, 0, 1)\]
\[(\sinh^2 u +\cosh^2 u)du^2 + dv^2\]
\(\sigma = (u, v, u^2+v^2)\)
\[\sigma_u = (1, 0, 2u)\sigma_v = (0, 1, 2v)\]
\[(1 + 4u^2)du^2 + 8uv dudv + (1+4v^2)dv^2\]
Source code
First Fundamental Forms with Reparameterization
Theorem Let \(\tilde \sigma(\tilde u, \tilde v)\) be a reparameterization of some \(\sigma(u,v)\) with transformation map \(\Phi\). Then, the first fundamental form of the reparameterization is
\[\begin{bmatrix}\tilde E &\tilde F\\\tilde F&\tilde G\end{bmatrix} = J(\Phi)^T \begin{bmatrix}E&F\\F&G\end{bmatrix}J(\Phi)\]
where \(J\) is the Jacobian matrix of \(\Phi\).
proof. First, by the reparameterization, we have that \((u, v) = \Phi(\tilde u, \tilde v)\), therefore