Surface Normals and Orientability

Normal Vector

Another finding is that \(\partial_u\sigma(u_0, v_0)\times \partial_v\sigma(u_0, v_0)\) is perpendicular to \(T_p\Sigma\) since the cross product is perpendicular to the spanning vectors of the plane. Therefore, define the unit normal vector of \(\Sigma\) at \(p\) as

\[\hat N(p) = \frac{\partial_u\sigma(u_0, v_0)\times \partial_v\sigma(u_0, v_0)}{\|\partial_u\sigma(u_0, v_0)\times \partial_v\sigma(u_0, v_0)\|}\]

Note that the tangent plane is independent of the choice of the path. Therefore, there are only 2 possible normal vectors. Say \(\hat N(p) = \frac{\partial_u\sigma(u_0, v_0)\times \partial_v\sigma(u_0, v_0)}{\|\partial_u\sigma(u_0, v_0)\times \partial_v\sigma(u_0, v_0)\|}\), then another possible reparameterization of the surface patch will result in \(\tilde N(p) = \frac{\partial_v\sigma(u_0, v_0)\times \partial_u\sigma(u_0, v_0)}{\|\partial_v\sigma(u_0, v_0)\times \partial_u\sigma(u_0, v_0)\|}=-\hat N(p)\)

Reparameterization of Surface Patches

Note that a corollary of the regularity condition: Consider the intersection of the 2 surface patch of a regular surface \(\sigma_1: U_1\rightarrow V_1, \sigma_2: U_2\rightarrow V_2. V_1\cap V_2\neq\emptyset\). Then, let \(f:U_1\rightarrow U_2\) be the map from one plane to another. Then, \(f\) is invertible, and \(f\) and \(f^{-1}\) are both differentiable.

Claim If \(\sigma: U\rightarrow V\) is a regular coordinate patch of a surface and \(f: \tilde U\rightarrow U\) is a reparameterization. Then \(\sigma\circ f: \tilde U\rightarrow V\) is also a regular coordinate patch.

Another interesting finding is that the normal vector of the reparameterized tangent plane.

\[\begin{align*} \tilde N &= (\frac{\partial \sigma}{\partial u} \frac{\partial u}{\partial \tilde u} + \frac{\partial \sigma}{\partial v} \frac{\partial v}{\partial \tilde u})\times (\frac{\partial \sigma}{\partial u} \frac{\partial u}{\partial \tilde v} + \frac{\partial \sigma}{\partial v} \frac{\partial v}{\partial \tilde v})\\ &= \frac{\partial \sigma}{\partial u} \times \frac{\partial \sigma}{\partial v} (\frac{\partial u}{\partial \tilde u} \frac{\partial v}{\partial \tilde v} - \frac{\partial v}{\partial \tilde u} \frac{\partial u}{\partial \tilde v})\\ &= \det(Df)(\frac{\partial \sigma}{\partial u} \times \frac{\partial \sigma}{\partial v}) \end{align*}\]

Therefore, the unit normal will be

\[\hat{\tilde N} = \frac{\tilde N}{\|\tilde N\|} = \frac{\det(Df)}{|\det(Df)|}\hat N = \text{sign}(\det(Df))\hat N\]

Orientable Surface

From the above findings, we can define an surface's orientability through its atlas A surface \(\Sigma\) is orientable if exists an atlas with the property that, if \(f\) is the transition map between any two surface patches, then \(\det(Df) > 0\). In other words, for any two overlapping surface patches \(\sigma_1: U_1\rightarrow V_1, \sigma_2: U_2\rightarrow V_2. V_1\cap V_2\neq \emptyset\) the transition map \(f:= \sigma_1^{-1}\circ \sigma_2:W_2\rightarrow W_1\) where \(W_i = \sigma_i^{-1}(V_1\cap V_2)\subseteq U_i\) has a positive Jacobian determinant.

Equivalently, there exists a unit normal function \(\hat N_\Sigma: \Sigma\rightarrow C = \{(x,y,z): x^2+y^2+z^2=1\}\) where \(\hat N_\Sigma\) is smooth and continuous. In other words, the change is unit normal vector along any path on the surface must be continuous.

Example: Mobius Band (Non-orientable)

The Mobius band is the surface obtained by rotating a straight line segment \(l\) around its midpoint \(p\) at the same time as \(p\) moves around a circle \(C\).

Source code

import plotly.graph_objects as go
import numpy as np

t = np.arange(0, 2*np.pi, 0.05 * np.pi)
p = np.array((np.cos(t), np.sin(t), np.zeros_like(t)))
orient = np.array((np.sin(t/2), np.zeros_like(t), np.cos(t/2))) / 2.
top = p + orient
bot = p - orient
surface = np.empty((3, top.shape[1] * 2))
surface[:, ::2] = top
surface[:, 1::2] = bot

fig = go.Figure(
    data=[
        go.Scatter3d(x=top[0], y=top[1], z=top[2], mode="lines", name="'top'"),
        go.Scatter3d(x=bot[0], y=bot[1], z=bot[2], mode="lines", name="'bottom'"),
        go.Scatter3d(x=surface[0, ::3], y=surface[1, ::3], z=surface[2, ::3], mode="lines", name="normal")
    ],
    )
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0), height=480)
with open("../assets/mobius.json", "w") as f:
    f.write(fig.to_json())

As shown in the example, let \(C\) be the unit circle, and \(l\) be the line segment connecting \((0,0, -1/2), (0,0, 1/2)\).

The Mobius band can be parameterized by 2 surface patches

\[\sigma(t, \theta) = (\cos\theta(1-t\sin\frac{\theta}{2}), \sin\theta(1-t\sin\frac{\theta}{2}), t\cos\frac{\theta}{2})\]

where \(U_1 = (-1/2, 1/2) \times (0,2\pi), U_2 = (-1/2, 1/2)\times (-\pi, \pi)\).

Consider the unit normal

\[\begin{align*} \partial_t\sigma|{(0,\theta_0)} &= (-\sin\frac{\theta_0}{2}\cos\theta_0, \sin\theta_0\sin\frac{\theta_0}{2}, \cos\frac{\theta_0}{2})\\ \partial_\theta\sigma|_{(0, \theta_0)} &= (-\sin\theta_0, \cos\theta_0, 0)\\ \partial_t\sigma\times \partial_\theta\sigma &= (-\cos\frac{\theta}{2}\cos\theta, -\cos\frac{\theta}{2}\sin\theta, -\sin\frac{\theta}{2}\cos^2\theta -\sin\frac{\theta}{2}\sin^2\theta)\\ N&= (-\cos\frac{\theta}{2}\cos\theta, -\cos\frac{\theta}{2}\sin\theta, -\sin\frac{\theta}{2}) \end{align*}\]

Note that the unit normal vector \(\hat N = N\) since \(\|N\| = 1\) Then, note that

\[\lim_{\theta\rightarrow0+} N(0,\theta) =(-1, 0, 0) \neq (1, 0, 0) = \lim_{\theta\rightarrow 2\pi-}N(0, \theta)\]

contradict with the orientability assumption.

Furthermore, a surface is non-orientable if it contains a sub surface diffeomorphic to the Mobius band.