## The loxodrome and two projections of the sphere

### Stereographic projection

Given a fixed point $$C$$ on the sphere, which we shall call projection's origin, the stereographic projection of a point $$P\neq C$$ of the sphere will lie on the plane tangent to the sphere in the antipode (diametrically opposed) of $$C$$. This plane is called plane of projection. The projection of $$P$$ is the result of the intersection of the half line $$CP$$ with the plane of projection.

The stereographic projection now defined has the South Pole as projection origin and, therefore, the plane of projection is the plane tangent to the North Pole. Notice that the projection of the point corresponding to the South Pole is not defined.

Schematic representation of the Stereographic Projection with origin at the South Pole.

Given a point $$P$$ on the sphere with spherical coordinates $$\left(r,\theta,\varphi\right)$$ and $$P'$$, the stereographic projection defined above, the coordinates of $$P'$$ are given by $$2r\tan\left(\frac{\varphi}{2}\right)\left(\cos\theta\,,\,\sin\theta\right)$$.

However, for a better visualisation of the countries on the map produced by the stereographic projection chosen, we take the plane of the map obtained by the rotation of the projection plane by an angle of amplitude $$\frac{\pi}{2}$$ in the clockwise direction and center at the origin of the reference frame.

Therefore, the stereographic projection is the function $$\mathcal{E}$$ from the sphere of radius r onto the plane defined by: $\begin{array}{ccll} \mathcal{E}: & \mathbb{R}\,\times\,]0\,,\pi[ & \longrightarrow &\mathbb{R}^{2}\\& \left(\theta,\varphi\right) & \mapsto & 2r\tan\frac{\varphi}{2}\left(\cos\left(\theta-\frac{\pi}{2}\right)\,,\,\sin\left(\theta-\frac{\pi}{2}\right)\right) \end{array}\,.$

Any given point in the plane $$\mathbb{R}^{2}$$ can be characterised by its polar coordinates: $$\left(\rho,\theta\right)$$, where $$\rho$$ is the distance to the reference frame's origin O and $$\theta$$ is the amplitude of the angle that the vector $$\overrightarrow{OP}$$ makes with the positive x semi-axis. Therefore, its cartesian coordinates are given by $$\rho\left(\cos\theta\,,\,\sin\theta\right)$$.

1. We recall that if $$\alpha\neq\frac{\pi}{2}+n\pi$$, $$n\in\mathbb{Z}$$, a parametrisation of $$\ell_{\alpha}$$ passing through a point P with spherical coordinates $$\left(r,\theta_{P},\varphi_{P}\right)$$ is given byr:

$\ell_{\alpha}\left(\varphi\right)=\left(r\cos\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi\,,\,r\sin\left(\theta_\alpha\left(\varphi\right)\right)\sin\varphi\,,\,r\cos\varphi\right)\,,$ com $$\varphi\in\,]0\,,\pi[$$   e   $$\theta_\alpha(\varphi)=\theta_{P}+\tan\alpha\left[\ln\left(\cot\frac{\varphi}{2}\right)-\ln\left(\cot\frac{\varphi_{P}}{2}\right)\right]\,.$$

Therefore, the projection of the loxodrome is the function $$\mathcal{E}\circ\ell_{\alpha}$$ defined by: $\begin{array}{ccll} \mathcal{E\circ}\ell_{\alpha}: & ]0\,,\pi[ & \longrightarrow & \mathbb{R}^{2}\\ & \varphi & \mapsto & 2r\tan\frac{\varphi}{2}\left(\cos\left(\theta_\alpha\left(\varphi\right)-\frac{\pi}{2}\right)\,,\,\sin\left(\theta_\alpha\left(\varphi\right)-\frac{\pi}{2}\right)\right) \end{array}\,.$

Assuming additionally that $$\alpha\neq n\pi,\, n\in\mathbb{Z}$$, and defining $$\varphi$$ as a function of $$\theta$$, it follows that $$\varphi_\alpha\left(\theta\right)=2\acot\left(e^{\frac{\theta-k}{\tan\alpha}}\right)$$, com $$k=\theta_{P}-\tan\alpha\ln\left(\cot\frac{\varphi_{P}}{2}\right)$$.

We can now reparametrise the stereographic projection of the curve in the following way: $\begin{array}{ccll} \mathcal{E\circ}\ell_{\alpha}\circ\varphi_\alpha: & \mathbb{R} & \longrightarrow & \mathbb{R}^{2}\\ & \theta & \mapsto & 2r\, e^{\frac{k-\theta}{\tan\alpha}}\left(\cos\left(\theta-\frac{\pi}{2}\right)\,,\,\sin\left(\theta-\frac{\pi}{2}\right)\right) \end{array}\,,$ com $$k=\theta_{P}-\tan\alpha\ln\left(\cot\frac{\varphi_{P}}{2}\right)$$.

The trace of this curve is a spiral.

Loxodrome projected onto the plane by the Stereographic Projection. The latitude varies between -60º and 90º.

2. If $$\alpha=\frac{\pi}{2}+n\pi$$, for some $$n\in\mathbb{Z}$$, a parametrisation of $$\ell_{\alpha}$$ that passed through a point $$P$$ with spherical coordinates $$\left(r,\theta_{P},\varphi_{P}\right)$$ is given by:

$\ell_{\alpha}\left(\theta\right)=\left(r\cos\left(\theta\right)\sin\left(\varphi_{P}\right)\,,\, r\sin\left(\theta\right)\sin\left(\varphi_{P}\right)\,,\, r\cos\left(\varphi_{P}\right)\right)\,,$   com $$\theta\in\,[0,2\pi],$$

whose trace on the sphere is a parallel.

In this case, the projection of the loxodrome is the function $$\mathcal{E}\circ\ell_{\alpha}$$ defined by: $\begin{array}{ccll}\mathcal{E\circ}\ell_{\alpha}: & [0\,,2\pi] & \longrightarrow & \mathbb{R}^{2}\\& \theta & \mapsto & 2r\tan\frac{\varphi_{P}}{2}\left(\cos\left(\theta-\frac{\pi}{2}\right)\,,\,\sin\left(\theta-\frac{\pi}{2}\right)\right)\;. \end{array}$ Since $$2r\tan\frac{\varphi_{P}}{2}$$ is a positive constant, the function defines a circumference on the plane of radius $$2r\tan\frac{\varphi_{P}}{2}$$ and center at the origin of the reference frame.

Loxodrome orthogonal to the meridians projected onto the plane using the Stereographic Projection. The latitude varies between -60º and 90º.

3. If $$\alpha=n\pi$$, for some $$n\in\mathbb{Z}$$, a parametrisation of $$\ell_{\alpha}$$ passing through a point $$P$$ with spherical coordinates $$\left(r,\theta_{P},\varphi_{P}\right)$$ is given by:

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

whose trace on the sphere is a meridian.

In this case, the projection of the loxodrome is the function $$\mathcal{E}\circ\ell_{\alpha}$$ defined by: $\begin{array}{ccll}\mathcal{E\circ}\ell_{\alpha}: & ]0\,,\pi[ & \longrightarrow & \mathbb{R}^{2}\\& \varphi & \mapsto & 2r\tan\frac{\varphi}{2}\left(\cos\left(\theta_{P}-\frac{\pi}{2}\right)\,,\,\sin\left(\theta_{P}-\frac{\pi}{2}\right)\right)\;, \end{array}$ a half line with origin at $$(0,0)$$ and slope $$\tan\left(\theta_{P}-\frac{\pi}{2}\right)$$.