## The loxodrome and two projections of the sphere

### Mercator projection

We want to determine a projection of the sphere with the following properties: loxodromes must be projected onto straight lines and the projection must preserve angles, that is, it must be conformal. One projection satisfying these requirements is Mercator's projection.

If $$\alpha\neq\frac{\pi}{2}+n\pi$$, with $$n\in\mathbb{Z}$$, a parametrisation of the loxodrome $$\ell_{\alpha}$$ passing through $$P$$ with spherical coordinates $$\left(r,\theta_{P},\varphi_{P}\right)$$ is: $\begin{array}{ccll} \ell_{\alpha}: & ]0\,,\pi[ & \longrightarrow & \mathbb{S}^{2}\\& \varphi & \mapsto & \left(r\cos\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi\,,\,r\sin\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi\,,\, r\cos\varphi\right) \end{array}\,,$ with $\theta_\alpha(\varphi)=\theta_{P}+\tan\alpha\left[\ln\left(\cot\frac{\varphi}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)\right]\,.$ Let $$Q$$ be a point belonging to the trace of the curve $$\ell_{\alpha}$$. Let's assume it has spherical coordinates $$\left(r,\theta_{Q},\varphi_{Q}\right)$$, with $$\varphi_{Q}\in\ ]0\,,\pi[$$ and $$\theta_{Q}\in[0\,,2\pi]$$.

Since $$Q$$ belongs to the trace of the curve, then $\theta_\alpha(\varphi_{Q})=\theta_{P}+\tan\alpha\left[\ln\left(\cot\frac{\varphi_{Q}}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)\right].$ Since $$\theta_\alpha\left(\varphi_{P}\right) = \theta_{P}$$ it follows that $\tan\alpha=\frac{\theta_\alpha(\varphi_{Q}) -\theta_\alpha\left(\varphi_{P}\right)}{\ln\left(\cot\frac{\varphi_{Q}}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)}\,.$

If $$\beta=\frac{\pi}{2}-\alpha$$ and assuming that $$\alpha$$ is not an integer multiple of $$\pi$$, we get $\tan\beta=\frac{\ln\left(\cot\frac{\varphi_{Q}}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)}{\theta_\alpha(\varphi_Q)-\theta_\alpha\left(\varphi_P\right)}\,.\;\;\;(1)$ Observe that (1)  $$\tan\beta$$ is the slope of the line passing through points $$\left(\theta_\alpha\left(\varphi_{P}\right)\,,\ln\left(\cot\frac{\varphi_{P}}{2}\right)\right)$$ and $$\left(\theta_\alpha(\varphi_{Q})\,,\ln\left(\cot\frac{\varphi_{Q}}{2}\right)\right)$$.

Therefore, we define Mercator's projection as the function $$\mathcal{M}$$ of the sphere onto the plane that maps each point in the sphere with spherical coordinates $$\left(r,\theta,\varphi\right)$$ to a point in the plane with cartesian coordinates $$\left(\theta_\alpha\left(\varphi\right)\,,\ln\left(\cot\frac{\varphi}{2}\right)\right)$$. However, the longitude $$\theta_\alpha\left(\varphi\right)$$ of a point on the loxodrome can take any real value. Therefore, we have to identify points with respect to their longitude. We want the map to represent longitudes between $$-\pi$$ and $$\pi$$. Notice also that, given a point $$Q$$ on the loxodrome $$\ell_{\alpha}$$, $$\theta_\alpha(\varphi_Q)$$ and $$\theta_Q$$ differ by an integer multiple of $$2\pi$$.

Therefore, we define $$\begin{array}{ccll} \mathcal{M}: & \mathbb{R}\,\times\,]0\,,\pi[ & \longrightarrow &\mathbb{R}^{2}\\ & \left(\theta,\varphi\right) & \mapsto & \left(\theta-2\pi\,m\left(\theta\right)\,,\ln\left(\cot\frac{\varphi}{2}\right)\right) \end{array}\,,$$ with $$m\left(\theta\right)=\left\lfloor \frac{\theta+\pi}{2\pi}\right\rfloor$$ the biggest integer smaller or equal than $$\frac{\theta+\pi}{2\pi}$$.

1. If $$\alpha\neq\frac{\pi}{2}+n\pi$$, $$n\in\mathbb{Z}$$:

The projection of the loxodrome is the curve on the plane with parametrisation $$\mathcal{M}\circ\ell_{\alpha}$$: $\begin{array}{ccll} \mathcal{M}\circ\ell_{\alpha}: & ]0\,,\pi[ & \longrightarrow & \mathbb{R}^{2}\\& \varphi & \mapsto & \left(\theta_\alpha\left(\varphi\right)-2\pi\,m\left(\theta_\alpha\left(\varphi\right)\right)\,,\ln\left(\cot\frac{\varphi}{2}\right)\right) \end{array}\,,$ with

$\theta_\alpha(\varphi)=\theta_{P}+\tan\alpha\left[\ln\left(\cot\frac{\varphi}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)\right]$
and
$m\left(\theta_\alpha\left(\varphi\right)\right)=\left\lfloor\frac{\theta_\alpha\left(\varphi\right)+\pi}{2\pi}\right\rfloor.$

It holds $\mathcal{M}\circ\ell_{\alpha}\left(\varphi\right)=\left(-2\pi\,m\left(\theta_\alpha\left(\varphi\right)\right)\,,0\right)+\left(\theta_{P}-\tan\alpha\ln\left(\cot\frac{\varphi_{P}}{2}\right)\,,0\right)+\ln\left(\cot\frac{\varphi}{2}\right)\left(\tan\alpha,1\right)\,.$ Therefore, the trace of $$\mathcal{M}\circ\ell_{\alpha}$$ is a subset of the union of parallel line segments with slope $$\frac{1}{\tan\alpha}=\tan\left(\frac{\pi}{2}-\alpha\right)$$.

Loxodrome projected to the plane by Mercator's Projection. The latitude varies between -80º and 80º.

2. If $$\alpha=\frac{\pi}{2}+n\pi$$, for some $$n\in\mathbb{Z}$$:

The projection of the loxodrome is parametrised by: $\begin{array}{ccll} \mathcal{M}\circ\ell_{\alpha}: & \left[0\,,2\pi\right] & \longrightarrow & \mathbb{R}^{2}\\ & \theta & \mapsto & \left(\theta-2\pi\,m\left(\theta\right)\,,\ln\left(\cot\frac{\varphi_{P}}{2}\right)\right) \end{array}\,,$ with $$m\left(\theta\right)=\left\lfloor \frac{\theta+\pi}{2\pi}\right\rfloor$$ the biggest integer smaller or equal than $$\frac{\theta+\pi}{2\pi}$$.

In this case, the trace of the projected curve is the horizontal line segment with equation $$y=\ln\left(\cot\frac{\varphi_{P}}{2}\right)$$, with $$x\in\left[-\pi\,,\pi\right[$$.

Loxodrome orthogonal to the meridians, projected to the plane by Mercator's Projection. The latitude varies between -80º and 80º.

Notice that, by definition, $$\mathcal{M}$$ preserves angles between loxodromes and meridians. In fact, $$\mathcal{M}$$ is conformal.