PMT-Reflector Geometry

Given that we will be describing the photomultiplier-reflector geometry as a set of spheres, cylinders and tori,it seems like a good idea to do some initial algebra to establish the general solutions of the intersection of a straight line with these shapes and to calculate their normals. We will assume that the line has the form $\underline{r} = \underline{R} + t\underline{u}$, where $\underline{R} = (x_0,y_0,z_0)$ is the particle's current position, $t$ is the distance to the next boundary. We will also assume that the system in question has axial symmetry, is oriented along the z-axis, and that the zero of the coordinate system is at the centre of the geometric shape in question.