Length of Arc of Small Circle
Theorem
Let $S$ be a sphere.
Let $\bigcirc FCD$ be a small circle on $S$.
Let $C$ and $D$ be the points on $\bigcirc FCD$ such that $CD$ is the arc of $\bigcirc FCD$ whose length is to be determined.
Construction
Let $P$ and $Q$ be the poles of $\bigcirc FCD$.
Let $\bigcirc PCQ$ and $\bigcirc PDQ$ be great circles on $S$.
Let $\bigcirc EAB$ be the great circle whose poles are $P$ and $Q$.
Let $A$ and $B$ be the points on $\bigcirc EAB$ which intersect $\bigcirc PCQ$ and $\bigcirc PDQ$.
The length of the arc $CD$ of $\bigcirc FCD$ is given by:
- $CD = AB \cos AC$
or:
- $CD = AB \sin PC$
Proof
Let $R$ denote the center of $\bigcirc FCD$.
Let $O$ denote the center of $S$, which is also the center of $\bigcirc EAB$.
We have:
- $CD = RC \times \angle CRD$
Similarly:
- $AB = OA \times \angle AOB$
By Circles with Same Poles are Parallel:
- $\bigcirc FCD \parallel \bigcirc EAB$
Hence $RC$ and $RD$ are parallel to $OA$ and $OB$ respectively.
Thus:
\(\ds \angle CRD\) | \(=\) | \(\ds \angle AOB\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds CD\) | \(=\) | \(\ds \dfrac {RC} {OA} AB\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {RC} {OC} AB\) | as $OA = OC$ are both radii of $S$ |
We also have that:
\(\ds RC\) | \(\perp\) | \(\ds OR\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds RC\) | \(=\) | \(\ds OC \cos \angle RCO\) |
and that:
\(\ds RC\) | \(\parallel\) | \(\ds OA\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \angle RCO\) | \(=\) | \(\ds \angle AOC\) |
We have that $\angle AOC$ is the (plane) angle subtended at $O$ by the arc $AC$ of $\bigcirc EAB$.
Thus:
\(\ds CD\) | \(=\) | \(\ds AB \cos AC\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds AB \, \map \cos {PA - PC}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds AB \sin PC\) | as $PA$ is a right angle, and Cosine of Complement equals Sine |
Hence the result.
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $3$. Length of a small circle arc.