Parametrization of the loxodrome

There are several different ways to define a curve in a mathematical context. For example, we can think of a curve as a set of points that verify a certain condition, like, for instance, a circumference. However, it will be more useful to adopt a different type of definition: the one of the curve as a parametrisation of a set (this parametrisation can have origin, for example, in the description of the movement of a particle).

In this work, we define a curve as a function \(\gamma:\,I\rightarrow\mathbb{R}^{3}\), where \(I\subseteq\mathbb{R}\) is an interval, that associates a point \(\gamma\left(t\right)\in\mathbb{R}^{3}\) to each value of a parameter \(t\in I\) . We assume also that the curves are smooth, that is, they have derivatives of any order.

The codomain of the curve is called trace of the curve. Notice that curves with different parametrisations can have the same trace.

The parametrisations of loxodromes used in this work will depend on colatitude or longitude, that is, the position of a point will only depend on its colatitude or its longitude.

The loxodrome is a curve on the sphere whose characteristic is making a constant angle with all the meridians it intersects. Two distinct cases can arise: the loxodrome is orthogonal to the meridians, and in this case, its parametrisation is given as a function of longitude (\(\theta\)); on the remaining cases, the parametrisation witl be given as a function of colatitude (\(\varphi\)).

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

Let \(\ell_{\alpha}\) denote a loxodrome whose angle of intersection with the meridians is \(\alpha\). Let \(P\) denote an arbitrary, but fixed, point on the trace of \(\ell_{\alpha}\), with spherical coordinates \((r,\theta_{P},\varphi_{P})\).

Notice that, if any of the poles belonged to \(\ell_{\alpha}\), this curve would intersect all meridians at that pole. In order for the loxodrome to make a constant angle with the meridians, the curve would have to have zero velocity at the poles and the angle with the meridians would not be defined. Therefore, we always assume that the poles do not belong to the curve. Therefore, we shall assume \(\varphi_{P}\in\ ]0\,,\pi[\).

Let \(m\) denote the meridian that intersects the curve \(\ell_{\alpha}\) at point \(P\). A parametrisation of the meridian \(m\) passing through \(P\) is the following:

\[m(\varphi)=\left(r\cos\left(\theta_{P}\right)\sin\varphi\,,\, r\sin\left(\theta_{P}\right)\sin\varphi\,,\, r\cos\varphi\right),\] com \(\varphi\in]0\,,\pi[\).

The meridian \(m\) and the curve \(\ell_{\alpha}\) intersect at \(P\) making an angle with amplitude \(\alpha\). Let's consider two different cases: in the first, we assume \(\alpha=\frac{\pi}{2}+n\pi\), for some \(n\in\mathbb{Z}\), that is, the curve \(\ell_{\alpha}\) is orthogonal to the meridians; in the second case, we assume \(\alpha\neq\frac{\pi}{2}+n\pi\), for all \(n\in\mathbb{Z}\).

1. \(\alpha=\frac{\pi}{2}+n\pi\), for some \(n\in\mathbb{Z}\)

Since the loxodrome is orthogonal to the meridians, its parametrisation is defined as a function of longitude, that is,

\[\ell_{\alpha}\left(\theta\right)=\left(r\cos\theta\sin\left(\varphi(\theta)\right)\,,\,r\sin\theta\sin\left(\varphi(\theta)\right)\,,\, r\cos\left(\varphi(\theta)\right)\right),\]

where the colatitude \(\varphi\) is a function of the parameter \(\theta\), \(\varphi:\,\left[0\,,2\pi\right]\rightarrow\ ]0\,,\pi[\).

The angle between both curves on the sphere can be determined by calculating the angle between the tangent vectors to both curves, at the intersection point (which can be calculated using the inner product of both vectors).

Since \(\ell_{\alpha}\) and \(m\) are orthogonal at \(P\), then \(\ell_{\alpha}^{\prime}\left(\theta_{P}\right)\,|\, m'\left(\varphi_{P}\right)=0\). Notice that \(\varphi(\theta_{P})=\varphi_{P}\).

It follows:

\[\begin{array}{rccl} \ell_{\alpha}^{\prime}(\theta_{P}) & = & & r\,\varphi'\left(\theta_{P}\right)\left(\cos\left(\varphi_{P}\right)\cos\left(\theta_{P}\right)\,,\cos\left(\varphi_{P}\right)\sin\left(\theta_{P}\right)\,,\,-\sin\left(\varphi_{P}\right)\right)\\ & & + & r\left(-\sin\left(\theta_{P}\right)\sin\left(\varphi_{P}\right)\,,\,\cos\left(\theta_{P}\right)\sin\left(\varphi_{P}\right),0\right) \end{array}\]


A straightforward calculation leads to \(\varphi'(\theta_{P})=0\).

Since \(P\) is an arbitrary point on the trace of \(\ell_{\alpha}\), it follows that \(\varphi(\theta)\) is constant and, changing the variable \(\varphi(\theta_{P})=\varphi_{P}\), we get \(\varphi\left(\theta\right)=\varphi_{P}\).

Therefore, a parametrisation \(\alpha=\frac{\pi}{2}+n\pi\) 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 trace \(\ell_{\alpha}\) corresponds to a parallel with colatitude equal to \(\varphi_{P}\).

Loxodrome whose trace corresponds to a parallel.

If \(\varphi_{P}=\frac{\pi}{2}\), then the trace of the curve is the Equatorial line.

Notice that, contrary to what is suggested by the notation, the curve \(\ell_{\alpha}\) depends not only on \(\alpha\) but also on \(\varphi_{P}\). We chose this notation for the sake of simplicity.

2. \(\alpha\neq\frac{\pi}{2}+n\pi\), \(n\in\mathbb{Z}\)

In this case, the parametrisation of \(\ell_{\alpha}\) is given as a function of colatitude* \(\varphi\in\ ]0\,,\pi[\):

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

with the longitude \(\theta_\alpha\) depending on the parameter \(\varphi\), \(\theta_\alpha:\ ]0\,,\pi[\rightarrow\mathbb{R}\).

As it has been mentioned in the previous point, the angle between two curves on the sphere corresponds to the angle between the vectors, tangent to the respective curves, at the intersection point. Therefore,


It holds:

\ell_{\alpha}^{\prime}(\varphi_{P}) & = & & r\,\theta'_\alpha\left(\varphi_{P}\right)\left(-\sin\left(\theta_{P}\right)\sin\left(\varphi_{P}\right)\,,\,\cos\left(\theta_{P}\right)\sin\left(\varphi_{P}\right),0\right)\\ & & + & r\left(\cos\left(\varphi_{P}\right)\cos\left(\theta_{P}\right)\,,\,\cos\left(\varphi_{P}\right)\,\sin\left(\theta_{P}\right)\,,\,-\sin\left(\varphi_{P}\right)\right)

A further calculation leads to

\[\cos\alpha=\frac{1}{\sqrt{1+\left[\theta'_\alpha\left(\varphi_{P}\right)\right]^{2}\sin^{2}\left(\varphi_{P}\right)}}\;.\] Sicne \(\cos\alpha\neq0\) and \(\varphi_{P}\in\ ]0\,,\pi[\), it follows that \[\begin{array}{rcl}
\left(\theta'_\alpha(\varphi_{P})\right)^{2}\sin^{2}\left(\varphi_{P}\right) & = & \frac{1}{\cos^{2}\left(\alpha\right)}-1\Leftrightarrow\\ \theta'_\alpha(\varphi_{P}) & = & \pm\frac{\tan\alpha}{\sin\left(\varphi_{P}\right)}.

If \(\alpha\) is an integer multiple of \(\pi\), then \(\theta'_\alpha(\varphi)=0\) for all \( \varphi \). If \(\alpha\) is not an integer multiple of \(\pi\), \(|\theta'_\alpha(\varphi_{P})|\geq|\tan\alpha|>0\), and \(\theta'_\alpha\) has constant sign. Due to geometric reasoning, we choose \(\theta'_\alpha(\varphi)=-\frac{\tan\alpha}{\sin\varphi}\) for \( \varphi \in ]0\,,\pi[\).

Integrating with respect to \(\varphi\), it follows \(\theta_\alpha(\varphi)=\tan\alpha\ln\left(\cot\frac{\varphi}{2}\right)+k\), for some constant \(k\in\mathbb{R}\). Since \(\theta_\alpha(\varphi_{P})=\theta_{P}\), we get \(k=\theta_{P}-\tan\alpha\ln\left(\cot\frac{\varphi_{P}}{2}\right)\).

Therefore, if \(\alpha\neq\frac{\pi}{2}+n\pi\), with \(n\in\mathbb{Z}\), then a parametrisation of \(\ell_{\alpha}\) is given by:

\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)

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

Notice that, when \(\alpha=0\), the trace of the curve coincides with a meridian (without the poles) because, in this case, \( \theta_\alpha\) is constant, \(\theta_\alpha(\varphi)=\theta_{P}\).

In the remaining cases (\(\alpha\neq\frac{\pi}{2}+n\pi\), with \(n\in\mathbb{Z}\)), the trace of the loxodrome take the shape of a spiral around the poles because, when \(\varphi\rightarrow0^{+}\) or when \(\varphi\rightarrow\pi^{-}\), \(\theta_\alpha(\varphi)\rightarrow\infty\), which shows that the curve infinitely circles around the z-axis.

Loxodrome whose trace has the shape of a spiral.

Although we have already seen that, if the curve is smooth (that is, if its velocity is always nonzero), the poles cannot belong to the curve's trace (assuming that the property of the curve making a constant angle with the meridians holds), it is possible to extend the loxodrome in order to contain the poles.

In fact, if \(\alpha\neq\frac{\pi}{2}n\), \(n\in\mathbb{Z}\), despite \(\lim_{\varphi\rightarrow0^{+}} \theta_\alpha\left(\varphi\right)=\pm\infty\) and \(\underset{\varphi\rightarrow\pi^{-}}{\lim}\theta_\alpha\left(\varphi\right)=\pm\infty\), \(\underset{\varphi\rightarrow0^{+}}{\lim}\ell_{\alpha}\left(\varphi\right)=\left(0,0,r\right)\) and \(\underset{\varphi\rightarrow\pi^{-}}{\lim}\ell_{\alpha}\left(\varphi\right)=\left(0,0,-r\right)\).

Therefore, it is possible to extend \(\ell_{\alpha}\) to \(\left[0\,,\pi\right]\) continuously. However, this new curve does not have the property of making a constant angle with the meridians it intersects because of the points with \(\varphi=0\) and \(\varphi=\pi\), corresponding to the poles.

*Notice that, if \(\alpha\neq\frac{\pi}{2}+n\pi\), for all \(n\in\mathbb{Z}\), then no two points on the curve have the same colatitude.