The loxodrome and two projections of the sphere Resize


Spherical coordinates

While a point in Earth's surface can be represented through its geographical coordinates, a point in the space \(\mathbb{R}^{3}\) can be characterised by its spherical coordinates. The location of a point \(P\) in \(\mathbb{R}^{3}\) on the orthonormal reference frame \(Oxyz\) can be given by its spherical coordinates \(\left(r,\theta,\varphi\right)\), where:

  • \(r\) is the distance from \(P\) to \(O\);
  • \(\theta\) is the angle made between the projection of \(\overrightarrow{OP}\) on the plane \(z = 0\) and the semi-positive axis \(Ox\), varying between \(0\) and \(2\pi rad\);*
  • \(\varphi\) is the angle that \(\overrightarrow{OP}\) makes with the positive semi-axis \(Oz\), varying between \(0\) and \(\pi rad\).

Spherical coordinates \(\left(r,\theta,\varphi\right)\) of a point \(P\).

Given that \(P\) lies on a sphere with center \(O\) and radius \(r\), let's \(\theta\) denote its longitude and \(\varphi\) its colatitude. On Earth's surface, the set of points with the same longitude is a meridian and the set of point with the same colatitude is a parallel. The Equatorial line is the set of points with colatitude equal to \(\frac{\pi}{2} rad\).

Therefore, the relation between the cartesian coordinates \(\left(x,y,z\right)\) and the spherical coordinates \(\left(r,\theta,\varphi\right)\) of a point in the sphere of radius \(r\) and center at the origin of the given reference frame is :

\(\begin{cases} x=r\cos\theta\sin\varphi\\ y=r\sin\theta\sin\varphi\\ z=r\cos\varphi
\end{cases}\), \(\theta\in[0\,,2\pi]\)  e  \(\varphi\in[0\,,\pi]\).

*We shall use angles measured in radians because it is more convenient.