If we then parameterize the line of sight using the circular coordinates $\theta$ and $\phi$ where $\theta$ is the angle with the $z$ axis and $\phi$ is the projected angle with the $x$ axis, then our coordinate system is such that it has the following properties.

-   The $x, y,$ and $z$ axis of our coordinate system correspond to the middle, long, and short axis of the halo respectively.

-   Projections along $\cos(\theta)=1$ are along the short axis of the halo.

-   Projections along $\cos(\theta)=0,\ \phi=0$ are along the middle axis of the halo.

-   Projections along $\cos(\theta)=0,\ \phi=\pi/2$ are along the long axis of the halo.

The nice thing about this particular choice of coordinates is that in the $\cos(\theta)-\phi$ plane, both the long and the middle axis are represented by a single point, whereas the short axis is represented by an entire line. As we shall see, projections along the middle and long axis maximize the lensing cross section of a halo for quad and double lenses respectively, so having that maximum be a single point in the space of lines of sight is a desirable quality of our chosen coordinate system.

Figure \[fig:orient\] shows the ratio $b({{\mathbf{\hat 