## The loxodrome and two projections of the sphere

### 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.