Length of loxodrome

The function arc length of a curve \(\gamma:\, I\rightarrow\mathbb{R}^{n}\), where I is a real interval and \(t_{0}\in I\) is fixed, is a function \(c:\, I\rightarrow\mathbb{R}\) defined by $$c(t)=\int_{t_{0}}^{t}\Vert\gamma'(x)\Vert\, dx\,.$$ The arc length of the curve \(\gamma:\,[a\,,b]\rightarrow\mathbb{R}^{n}\) is given by $$\int_{a}^{b}\Vert\gamma'(x)\Vert\, dx\,.$$ Let \(\ell_{\alpha}\) denote a loxodrome making an angle of amplitude \(\alpha\) with the meridians it intersects and whose trace contains the point P with spherical coordinates \(\left(r,\theta_{P},\varphi_{P}\right)\), with \(\theta_{P}\in[0\,,2\pi]\), \(\varphi_{P}\in\ ]0\,,\pi[\) and \(r\) equal to the radius of the sphere. Because of the parametrisation of \(\ell_{\alpha}\), we have to consider two different cases: \(\alpha=\frac{\pi}{2}+n\pi\), for some \(n\in\mathbb{Z}\), and \(\alpha\neq\frac{\pi}{2}+n\pi\), with \(n\in\mathbb{Z}\).

1. \(\alpha=\frac{\pi}{2}+n\pi\), for some \(n\in\mathbb{Z}\)

In red is displayed a loxodrome whose trace coincides with a parallel. In blue, the shortest path between the two points is shown (smallest great circle arc to which the points belong).

A parametrisation of \(\ell_{\alpha}\) is given by: \[\begin{array}{ccll} \ell_{\alpha}: & [0\,,2\pi] & \longrightarrow & \mathbb{S}^{2}\\ & \theta & \mapsto & \left(r\cos\theta\sin\left(\varphi_{P}\right)\,,\, r\sin\theta\sin\left(\varphi_{P}\right)\,,\, r\cos\left(\varphi_{P}\right)\right)&. \end{array}\] In this case, the arc length \(\ell_{\alpha}\) is given by \[\int_{0}^{2\pi}\Vert\ell_{\alpha}^{\prime}(\theta)\Vert\, d\theta\,.\] It follows that \[\ell_{\alpha}^{\prime}(\theta)=r\left(- \sin\theta\sin\left(\varphi_{P}\right)\,,\,\cos\theta\sin\left(\varphi_{P}\right)\,,\,0\right)\,.\] Calculating \(\Vert\ell_{\alpha}^{\prime}(\theta)\Vert^{2}\) one gets \[\Vert\ell_{\alpha}^{\prime}(\theta)\Vert^{2}=r^{2}\sin^{2}\left(\varphi_{P}\right)\,.\] Since \(\sin\left(\varphi_{P}\right) > 0\), pois \(\varphi_{P}\in\,]0,\pi[\), it finally follows that \(\Vert\ell_{\alpha}^{\prime}(\theta)\Vert=r\sin\left(\varphi_{P}\right)\).

Therefore, the arc length of \(\ell_{\alpha}\) is given by \[\int_{0}^{2\pi}r\sin\left(\varphi_{P}\right)d\theta=2\pi r\sin\left(\varphi_{P}\right)\,.\]

Notice that, since in this case the trace of the curve \(\ell_{\alpha}\) corresponds to a parallel with co-latitude equal to \(\varphi_{P}\), one could have calculated its length determining the perimeter of the circumference of radius \(r\sin\left(\varphi_{P}\right)\).

in particular, the arc length of \(\ell_{\alpha}\) between any two points of the curve \(Q\,\left(r,\theta_{Q},\varphi_{P}\right)\) and \(R\,\left(r,\theta_{R},\varphi_{P}\right)\) is equal to the length of the corresponding circumference arc of radius \(r\sin\left(\varphi_{P}\right)\), with value \(\left|\theta_{Q}-\theta_{R}\right|r\sin\left(\varphi_{P}\right)\).

2. \(\alpha\neq\frac{\pi}{2}+n\pi\), \(n\in\mathbb{Z}\)

In red it is shown a loxodrome whose trace has the shape of a spiral. In blue, the shortest path between the two points is displayed.

A parametrisation of \(\ell_{\alpha}\) is given by:\[\begin{array}{ccll} \ell_{\alpha}: & ]0\,,\pi[ & \longrightarrow & \mathbb{S}^{2}\\ & \varphi & \mapsto & \left(r\cos\left(\theta_\alpha\left(\varphi\right)\right) \sin\varphi\,,\, r\sin\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi, r\cos\varphi\right)&,\end{array}\] com \(\theta_\alpha(\varphi)=\theta_{P}+\tan\alpha\left[\ln\left(\cot\frac{\varphi}{2}\right)- \ln\left(\cot\frac{\varphi_{P}}{2}\right)\right]\,.\) As we have already seen, this curve can be extended continuously to \([0\,,\pi]\), and consequently the arc length of \(\ell_{\alpha}\) is given by the integral \(\int_{0}^{\pi}\Vert\ell_{\alpha}^{\prime}(\varphi)\Vert\, d\varphi\). It holds that $$\begin{array}{rccl} \ell_{\alpha}^{\prime}(\varphi) & = & & r\,\theta'_\alpha\left(\varphi\right)\left(-\sin\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi\,,\,\cos\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi\,,\,0\right)\\ & & + & r\left(\cos\varphi\cos\left(\theta_\alpha\left(\varphi\right)\right)\,,\,\cos\varphi\sin\left(\theta_\alpha\left(\varphi\right)\right)\,,\,-\sin\varphi\right)\,. \end{array}$$ Calculating \(\Vert\ell_{\alpha}^{\prime}(\varphi)\Vert\) one obtains $$\Vert\ell_{\alpha}^{\prime}(\varphi)\Vert=r\sqrt{\left[\theta'_\alpha(\varphi)\right]^{2}\sin^{2}\varphi+1}\,.$$ Since \(\theta'_\alpha\left(\varphi\right)=-\frac{\tan\alpha}{\sin\varphi}\), it follows that \(\Vert\ell_{\alpha}^{\prime}(\varphi)\Vert=r\sqrt{\tan{}^{2}\alpha+1}=\frac{r}{\left|\cos\alpha\right|}\,.\)

Therefore, if \(\alpha\neq\frac{\pi}{2}+n\pi\), with \(n\in\mathbb{Z}\), the arc length of the curve \(\ell_{\alpha}\) is \[\int_{0}^{\pi}\frac{r}{\left|\cos\alpha\right|}\,d\varphi=\frac{\pi}{\left|\cos\alpha\right|}r.\]

Loxodrome that coincides with a meridian. In this case, the arc length of the curve between two points corresponds to the shortest distance between them.

When \(\alpha=0\), the trace of the curve coincides with a meridian. Therefore, its arc length is equal to half of the perimeter of a circumference of radius \(r\), \(\pi r\).

Notice also that, in the remainning cases (\(\alpha\neq\frac{\pi}{2}+n\pi\)), although the trace of the loxodrome takes the form of an "infinite spiral", its arc length is finite.

If we want to calculate the arc length of the curve \(\ell_{\alpha}\) between any two given points on the curve \(Q\,\left(r,\theta_{Q},\varphi_{Q}\right)\) e \(R\,\left(r,\theta_{R},\varphi_{R}\right)\) we have to determine the value of \(\left|\int_{\varphi_{R}}^{\varphi_{Q}}\frac{r}{\left|\cos\alpha\right|}\, d\varphi\right|\), which is equal to \(\frac{\left|\varphi_{Q}-\varphi_{R}\right|}{\left|\cos\alpha\right|}r\).

One can show that this arc length is bigger or equal to the length of the smallest great circle arc defined by both points, that is, the route following a loxodrome is not, in general, the shortest path between two points.