Intersection of a line with a sphere

Assuming that the sphere is centred at the origin, the equation of the sphere is $\underline{r}^2 = a^2$, where $a$ is the radius of the sphere. Intersecting this with a straight line will give a quadratic in $t$ whose solutions are:

$$t = -(\underline{u}.\underline{R})\pm \sqrt{(\underline{u}.\underline{R})^2 - (\underline{R}^2 - a^2)}$$ (13.1)

The (unnormalised) normal to a sphere, centred on the origin is simply the vector $\underline{R}$.