Let's consider the sphere with center at the origin \(O\) of the orthonormal frame of reference \(Oxyz\) and radius \(r>0\). We shall denote this sphere \(\mathbb{S}^{2}\).

Loxodrome whose trace has the shape of a spiral.

Let \(\ell_{\alpha}\) be a loxodrome whose angle of intersection with the meridians is \(\alpha\in\left[-\frac{\pi}{2},\,\frac{\pi}{2}\right]\), passing through point \(P\) with spherical coordinates \(\left(r,\theta_{P},\varphi_{P}\right)\,,\) with \(\theta_{P}\in[0\,,2\pi]\) and \(\varphi_{P}\in\ ]0\,,\pi[\). The parametrization of the loxodrome \(\ell_{\alpha}\) can be defined 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, its trace corresponds to a parallel with the same latitude as P; in particular, if \(\varphi_{P}=\frac{\pi}{2}\), \(\ell_{\alpha}\) we get a parametrization of the Equatorial line..

\[\begin{array}{ccll} \ell_{\alpha}: & ]0\,,\pi[ & \longrightarrow & \mathbb{S}^{2}\\ & \varphi & \mapsto & \left(r\cos\left(\theta_\alpha\left(\varphi\right)\right) \sin\varphi\,,\, r\sin\left(\theta_\alpha\left(\varphi\right)\right) \sin\varphi\,,\, r\cos\varphi\right)&,\end{array}\]

with \(\theta_\alpha(\varphi)=\theta_{P}+\tan\alpha\left[\ln\left(cotg \frac{\varphi}{2}\right)-\ln\left(cotg \frac{\varphi_{P}}{2}\right)\right]\).

Let's see some particular scenarios: