Reword spherical coords in book 3, chap 4, para 3

books/RayTracingTheRestOfYourLife.html

@@ -1389,12 +1389,15 @@
 space is called _the rejection method_, and is found all over the literature. The method covered
 in the last chapter is referred to as _the inversion method_ because we invert a PDF.
 
-Every direction in 3D space has an associated point on the unit sphere and can be generated by
-solving for the vector that travels from the origin to that associated point. You can think of
-choosing a random direction as choosing a random point in a constrained two dimensional plane: the
-plane created by mapping the unit sphere to Cartesian coordinates. The same methodology as before
-applies, but now we might have a PDF defined over two dimensions. Suppose we want to integrate this
-function over the surface of the unit sphere:
+Every direction in 3D space can be represented as a point on the unit sphere, where the vector from
+the origin to the point indicates the direction. Standard spherical coordinates are $r$ (the
+distance from the origin to the point), $\theta$ (the polar angle), and $\phi$ (the azimuthal
+angle). Since we're only concerned with directions here, we can just ignore $r$. Thus, to choose a
+random direction, we just need to choose random $\theta \in [0,\pi]$, and
+$\phi \in [0, \frac{\pi}{2}]$.
+
+So now we have a PDF defined over two dimensions: $\theta$ and $\phi$. Suppose we want to integrate
+this function over the surface of the unit sphere:
 
   $$ f(\theta, \phi) = \cos^2(\theta) $$