First Fundamental Form

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 \(v\in T_p\Sigma\),

\[\langle v, v\rangle_{T_p\Sigma} = \|a\sigma_u + b\sigma_v\|^2 = Ea^2 + 2Fab + Gb^2\]

The First fundamentally form is the expression

\[\langle v, v\rangle_{T_p\Sigma} = Eu'^2 + 2Fu'v' + Gv'^2\]

Arc-length of curve on surface

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

\[\gamma(t) = (x(t), y(t), z(t)) = \sigma(u(t), v(t))\]

Since \(\sigma\) is a homeomorphism, such \(u,v\) exists and are smooth. Therefore, for some segment of the curve,

\[\begin{align*} L &= \int \|\gamma'(t)\| dt\\ &= \int \|\frac{\partial \sigma}{\partial u}\frac{du}{dt}+\frac{\partial \sigma}{\partial v}\frac{dv}{dt}\|dt\\ &= \int \sqrt{(\sigma_u\cdot\sigma_u) (u')^2+ 2(\sigma_u\sigma_v) u'v' + (\sigma_v\cdot\sigma_v) (v')^2}dt\\ &= \int \sqrt{E(u')^2 + 2F(u'v') + G(v')^2} dt \end{align*}\]

Area of Surface

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,

\[\begin{align*} A(\Omega) &= \iint_{U_1} \|\partial_u\times \partial_v\| dudv\\ &= \iint_{U_1} \sqrt{(\sigma_u\cdot \sigma_u)(\sigma_v\cdot \sigma_v) - (\sigma_u\cdot\sigma_v)^2} dudv\\ &= \int_{U_1} \sqrt{EG - F^2} dudv \end{align*}\]

Angle between Curves

Let \(\gamma_1: \mathbb R\rightarrow \Sigma, \gamma_2: \mathbb R\rightarrow\Sigma\) with \(\gamma_1(t_1) = \gamma_2(t_2) = p\in\Sigma\). Then, at point \(p, \gamma'_1, \gamma'_2 \in T_p\Sigma\).

Define angle two 2 curves on the surface as the angle on \(T_p\Sigma\) at intersection point \(p\).

Note that for each \(\gamma = \sigma(u(t), v(t))\) for \(u,v:\mathbb R\rightarrow\mathbb R\), its derivative is

\[\frac{d}{dt} (\sigma(u(t), v(t))) = \sigma_u\cdot u' + \sigma_v\cdot v'\]

Therefore, the angle is defined as

\[\begin{align*} \angle(\gamma_1,\gamma_2) &= \arccos(\frac{\gamma_1'}{\|\gamma_1'\|}\cdot \frac{\gamma_1'}{\|\gamma_1'\|})\\ &= \frac{(\sigma_{u}\cdot u_1' + \sigma_{v}\cdot v_1')\cdot(\sigma_{u}\cdot u_2' + \sigma_{v}\cdot {v_2}')}{\|\gamma_1'\|\|\gamma_2'\|}\\ &= \frac{(\sigma_u\cdot\sigma_u) (u_1'\cdot u_2') + (\sigma_u\cdot\sigma_v)(u_1'v_2'+u_2'v_1') + (\sigma_v\cdot\sigma_v) (v_1'\cdot v_2')}{(E(u'_1)^2 + 2F(u_1'\cdot v_1') + G(v_1')^2)(E(u'_2)^2 + 2F(u_2'\cdot v_2') + G(v_2')^2)}\\ &= \frac{E (u_1'\cdot u_2') +F(u_1'v_2'+u_2'v_1') + G (v_1'\cdot v_2')}{(E(u'_1)^2 + 2F(u_1'\cdot v_1') + G(v_1')^2)(E(u'_2)^2 + 2F(u_2'\cdot v_2') + G(v_2')^2)} \end{align*}\]

Example: Parameter Curve

The parameter curves on a surface patch \(\sigma(u,v)\) can be parametrized by

\[\gamma_1(t) = \sigma(u_0, t), \gamma_2(t) = \sigma(t, v_0)\]

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

import plotly.graph_objects as go
from plotly.subplots import make_subplots
import numpy as np
from numpy import sinh, cosh
from IPython.display import display, IFrame

theta, psi = np.mgrid[-.5 * np.pi:.5 * np.pi:10j, -np.pi: np.pi:10j]
u, v = np.mgrid[-.5 :.5 :10j, -1: 1:10j]
S1 = go.Surface(x=sinh(theta) * sinh(psi), y=sinh(theta) * cosh(psi), z=sinh(theta))
S2 = go.Surface(x=u - v, y=u + v, z=u * u + v * v)
S3 = go.Surface(x=cosh(theta), y=sinh(theta), z=v)
S4 = go.Surface(x=u, y=v, z=u * u + v * v)

fig = make_subplots(cols=2, rows=2, horizontal_spacing=0, vertical_spacing=0, 
                    specs=[[{"type": "scene"}, {"type": "scene"}], [{"type": "scene"}, {"type": "scene"}]])
fig.add_trace(S1, row=1, col=1)
fig.add_trace(S2, row=1, col=2)
fig.add_trace(S3, row=2, col=1)
fig.add_trace(S4, row=2, col=2)
fig.update_traces(showscale=False)
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0), height=720)
with open("../assets/fff.json", "w") as f:
    f.write(fig.to_json())

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

\[\frac{\partial u}{\partial t} = \frac{\partial u}{\partial \tilde u}\frac{\partial \tilde u}{\partial t}+\frac{\partial u}{\partial \tilde v}\frac{\partial \tilde v}{\partial t}, \frac{\partial v}{\partial t} = \frac{\partial v}{\partial \tilde u}\frac{\partial \tilde u}{\partial t}+\frac{\partial v}{\partial \tilde v}\frac{\partial \tilde v}{\partial t}\]

If we plug in this, we have that the first fundamental form of \(\tilde \sigma\) is

\[E(\partial_{\tilde u} u\tilde u' + \partial_{\tilde v} u\tilde v')^2 + 2F(\partial_{\tilde u} u\tilde u' + \partial_{\tilde v} u\tilde v')(\partial_{\tilde u} v\tilde u' + \partial_{\tilde v} v\tilde v') + G(\partial_{\tilde u} v\tilde u' + \partial_{\tilde v} v\tilde v')^2\]

Simplify the equation, we can have the formula as wanted

\[\begin{bmatrix}\tilde E &\tilde F\\\tilde F&\tilde G\end{bmatrix} = \begin{bmatrix}\partial_{\tilde u} u &\partial_{\tilde u} v\\\partial_{\tilde v} u&\partial_{\tilde v} v\end{bmatrix} \begin{bmatrix}E&F\\F&G\end{bmatrix} \begin{bmatrix}\partial_{\tilde u} u &\partial_{\tilde v} u\\\partial_{\tilde u} v&\partial_{\tilde v} v\end{bmatrix}\]

Example: Surface of Revolution

For surface pf revolution

\[\sigma(u,v) = (f(u)\cos v, f(u)\sin v, g(u))\]

with the assumption that \(\gamma(t) = (f(t), g(t))\) resides on the plane is unit-speed. Then, we have

\[\sigma_u = (f'\cos v, f'\sin v, g'), \sigma_v = (-f\sin v, f\cos v, 0)\]

\[E = f'^2+g'^2 = 1, F = 0. G = f^2\]

Therefore, the first fundamental form is

\[u'^2 + f(u)^2 v'^2\]

For sphere, we have \(f(u) = \cos u\), which implies the first fundamental form is

\[u'^2 + \cos^2 u v'^2\]