Point Source Cones

For these studies we consider the situation in which the light is generated from a point source. In general, six variables are required to fully describe the resulting distribution of the photons: The position of the cone apex (3 variables), the direction of the cone axis (2 variables), and the cone angle (1 variable). Note, however, it is always possible to choose a co-ordinate system where the cone apex lies in the X-Z plane, and the cone axis is parallel to the X-axis. For convenience the Monte Carlo cones in this study will be generated in this system. (The fitting algorithms to be investigated do not have this simplification built in.)

In the following discussion the cone parameters are defined as follows:-
The apex positions: Cx, Cy, and Cz
The axis direction: Ath (angle wrt the Z-axis) and Aphi (azimuthal angle)
The cone angle: Cang.

Note that for this study the cones are generated with Cy=0.0, Ath=pi/2, and Aphi=0.0 .

For each cone 1,000 photons are randomly about the cone axis and projected onto the surface of a sphere of radius 220-inches. For each photon we record its point of intersection with the sphere and the relative distance travelled. I.e. if Ri is the distance from the cone apex to the intersecion point of a specific photon and Rmin is the shortest of these distances, then knowledge of Ti = Ri - Rmin is assumed though knowledge of Rmin is not.

Two dimensionsal projects of the "hits" from two different cones are shown below. In each plot the * makes the apex position, the X marks the intersection of the axis with the sphere, the + marks the closest intersection point, the blue points are the photon "hits" and the green points are photon positions when corrected for their relative distance of travel. Note that the pattern of hits do not necessarily form ellipses and are not even necessarily co-planar.

Some Geometry

It is fairly easy to see that the closest and furthest intersection points of the cone and sphere surfaces are co-planar with both the cone apex and the center of the sphere. Using this property and assuming knowledge of the cone angle, one can completely reconstruct the remaining cone parameters from these two points. (Note: The problem still has six parameters:- 2 parameters each for Pmin and Pmax, Tmax, and the cone angle.)

There remains a twofold ambiguity corresponding to the sign of the unit vector j. This ambiguity can usually, though not always, be resolved by requiring the reconstrcuted cone apex be inside the sphere. Ambiguities that cannot be resolved in this way can only occur when the cone angle is greater than ATAN(R/[R-x]) where R is the sphere radius and x is the shortest distance from the sphere to the cone apex. (For a beta=1 particle producing Cerenkov light in a medium with refractive index 1.5 then x must be greater than 0.1*R to avoid this problem.)