Polar Coordinates

Assuming Cartesian coordinates as known, this section introduces polar coordinates and, in particular, presents the relations linking Cartesian and polar coordinates.

Figure 1.1: Correspondence between polar and Cartesian coordinates.

For a point in two-dimensional space, the relation linking these two coordinate systems is written as:

$$\begin{array}{rl} x &= \rho \cos \vartheta \\ y &= \rho \sin \vartheta \\ \end{array}$$ (1.52)

The inverse transformation, from Cartesian to polar coordinates, is

$$\begin{array}{rl} \rho &= \sqrt{x^2+y^2} \\ \vartheta &= \operatorname{atan2} (y,x) \\ \end{array}$$ (1.53)

A point on a sphere, however, does not have a unique representation: for the same reason, as will be emphasized several times in the appendix, there are infinitely many representations of a rotation in three-dimensional space.

A common choice is spherical polar coordinates (spherical coordinate system).

Figure 1.2: Polar coordinates in 3 dimensions: spherical coordinates.

With this convention, the relation between Cartesian and polar coordinates is written as

$$\begin{array}{rl} x &= \rho \sin \vartheta \cos \varphi \\ ... ...heta \sin \varphi \\ z &= \rho \cos \vartheta \\ \end{array}$$ (1.54)

where $\vartheta$ is defined as the zenith, while $\varphi$ is called the azimuth.

The inverse transformation, from Cartesian to polar coordinates, is obtained as

$$\begin{array}{rl} \rho &= \sqrt{x^2+y^2+z^2} \\ \varphi&=... ...torname{atan2}(\sqrt{x^2+y^2},z) = \arccos(z/\rho); \end{array}$$ (1.55)