Polar Coordinates

Assuming Cartesian coordinates, this section introduces polar coordinates and in particular shows the relationships that connect Cartesian coordinates to polar ones.

Figure 1.1: Correspondence between polar and Cartesian coordinates.

For a point in two-dimensional space, the relationship that links these two coordinate systems is written as:

$$\begin{array}{rl} x &= \rho \cos \vartheta \\ y &= \rho \sin \vartheta \\ \end{array}$$ (1.16)

The inverse transformation, from Cartesian to polar coordinates, is

$$\begin{array}{rl} \rho &= \sqrt{x^2+y^2} \\ \vartheta &= \operatorname{atan2} (y,x) \\ \end{array}$$ (1.17)

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

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

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

With this convention, the relationship between Cartesian and polar coordinates can be written as

$$\begin{array}{rl} x &= \rho \sin \vartheta \cos \varphi \\ ... ...heta \sin \varphi \\ z &= \rho \cos \vartheta \\ \end{array}$$ (1.18)

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

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

$$\begin{array}{rl} \rho &= \sqrt{x^2+y^2+z^2} \\ \varphi&=... ...torname{atan2}(\sqrt{x^2+y^2},z) = \arccos(z/\rho); \end{array}$$ (1.19)