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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.

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