Cartesian to spherical coordinates translation - how to differentiate x/y signs

Question

I am using this Wikipedia page translation formulae when writing my Cartesian2sherical function (sorry I don't have enough reputaton to post in-line image, but here is the link, and I will rewrite it too): link_to_wiki_formulae_image

$$r = \sqrt{x^2 + y^2 +z^2}$$ $$\theta = \arctan(\frac{\sqrt{x^2+y^2}}{z})$$ $$\phi = \arctan(\frac{y}{x}) =\arccos(\frac{x}{\sqrt{x^2+y^2}}) = \arcsin(\frac{y}{\sqrt{x^2+y^2}})$$

However, the above formulae do not handle the case where x or y is negative: the first two formulae squares x and y so the signs do not matter. Formula 3 falls apart when x or y are negative. For example, when x = -1 and y = -1, the 3 equalities in equation 3 don't even hold:

$$\arctan(\frac{y}{x}) = \arctan(1) = 0.7853982$$ $$\arccos(\frac{x}{\sqrt{x^2+y^2}}) = \arccos(-0.7071068) = 2.356194$$ $$\arcsin(\frac{y}{\sqrt{x^2+y^2}}) = \arcsin(-0.7071068) = -0.7853982$$

Does that mean the Wikipedia formulation is lacking some pre-assumptions of the x and y value? What would I change the formulae if I wanted to handel the case where x and y could have arbitrary signs? Thanks!

Answer 1

If you follow the link from that page to the main article on spherical coordinates, in particular, to the Cartesian Coordinates section of that article, you will find the following caveat:

The inverse tangent denoted in $\varphi = \arctan\frac yx$ must be suitably defined, taking into account the correct quadrant of $(x,y)$.

This is a common gotcha when working with the inverse tangent function as part of a coordinate transformation.

Answer 2

Inverse trig functions are multivalued. You just have to decide in which quadrant you want the answer.

Notice that $2.35619-\pi=-0.78539$