The loxodrome and two projections of the sphere

Definition

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:

• If $$\alpha=\pm\frac{\pi}{2}$$,

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

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..

• If $$\alpha\in\left]-\frac{\pi}{2},\,\frac{\pi}{2}\right[$$,

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

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:

• when $$\alpha=0$$, the trace of the curve coincides with a meridian (without the poles);
• when $$\alpha=\pm\frac{\pi}{2}$$, the trace of the curve corresponds to a geographical parallel;
• in the remaining cases, the trace of the loxodrome takes the form of a spiral around the poles (but without containing them).