Edge of Polyhedron has no Curvature
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Theorem
The edge of a polyhedron has zero curvature.
Proof
Let $X$ and $Y$ be two separate faces of a polyhedron separated by the edge $l$.
Let $P$ be a point on $X$ and let $Q$ be a point on $Y$.
The curvature inside an infinitesimal region $\delta a$ is given by the net angular displacement $\delta \theta$ a vector $v$ experiences as it is parallel transported along a closed path around $\delta a$.
The curvature is then given by:
- $R = \dfrac {\delta \theta} {\delta a}$
We must then prove that the vector $v$ experiences no net angular displacement as it is parallel transported from $P$ to $Q$ and back to $P$.
The two open curves $r$ and $s$ make a closed curve.
As the vector is parallel transported along the open curve $r$, it crosses the edge between the two faces $X$ and $Y$.
In doing so, it gains a finite angular displacement $\delta \theta_1$.
Then, when the vector is transported back along the open curve $s$, it gains another angular displacement $\delta \theta_2$.
Notice that because it is not being transported the other way (from $Y$ to $X$), the new angular displacement will be:
- $\delta \theta_2 = -\delta \theta_1$.
The curvature inside the region $\delta a$ is therefore given by:
\(\ds R\) | \(=\) | \(\ds \frac {\delta \theta_1 + \delta \theta_2} {\delta a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\delta \theta_1 - \delta \theta_1} {\delta a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 0 {\delta a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
The result follows.
$\blacksquare$