ATRACTOR

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 \[ \begin{equation}
\tan\beta=\frac{\ln\left(\cot\frac{\varphi_{Q}}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)}{\theta_\alpha(\varphi_Q)-\theta_\alpha\left(\varphi_P\right)}\,.\;\;\;(1)
\end{equation}\] 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.