## The loxodrome and two projections of the sphere

### 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 $$\pi rad$$.

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.