]> 3.6 Polar Coordinates

## 3.6 Polar Coordinates

A 2-vector $( x , y )$ can be described by two numbers that are not coefficients in a sum: its length, and the angle its vector makes with the $x$ axis.
The first of these is usually written as $r$ , the second as $θ$ .

These parameters obey $r 2 = x 2 + y 2$ and $tan ⁡ θ = y x$ ;
the inverse relations are $x = r cos ⁡ θ , y = r sin ⁡ θ$ .
$r$ and $θ$ are called polar coordinates.

Calculating the angle $θ$ in polar coordinates is a bit tricky; the obvious thing to try is $atan ⁡ ( y , x )$ but that is defined only between $− π 2$ and $π 2$ , while $θ$ has a domain of size $2 π$ .

Here is something that works: $acos ⁡ ( x x 2 + y 2 ) * if ( y < 0 , − 1 , 1 )$ .

This gives theta in the range $− π$ to $π$ . If you want it to have range 0 to $2 π$ you can add $if ( y < 0 , 8 * atan ⁡ ( 1 ) , 0 )$ to it.