Thinking about scheme to test anode residuals led to ideas about how to generate tracks parallel to the beam line as artificial test trajectories. Realized that current parametrization of trajectories used by the fitter will not converge for these tracks since the z position of to launch plane position cannot be defined since the launch plane is currently constrained to contain the beamline.
Need to go to a parametrization where the launch plane is allowed to tilt in theta as well.
To specify launch plane start with the x-y plane.
First rotate about the x axis by angle beta. Plane is now the x’-y’ plane where x = x’.
Second rotate about the z’ axis by angle gamma. Plane is now the x”-y” plane.
Now specify offsets in x” and y” to get position in the launch plane.
This is no good. Second rotation does not change the azimuthal direction of the normal to the launch plane. Try again.
First rotate about the z-axis by angle alpha. This sets the phi angle of the trajectory to be alpha + 90
Second rotate about the x’ axis by beta. This sets the polar angle of the trajectory to be beta.
Now specify offsets in x” and y” to get position in the launch plane.
This is better. But still leaves phi not well determined for tracks along beam line. For these an theta_x” and theta_y” approach would be better. But that would mean the variations done in the fit would be with respect to the x” and y” axes. The orientation of those axes (alpha and beta) could not be the fit parameters themselves.
Consider infinite momentum, parallel to beamline: fit would wander around in phi space, possibly not converge. In that case, if one eliminates phi as a parameter, then track can only move in single plane, chosen by the coordinate system. Clearly, here an theta_x theta_y parametrization is needed.
In gluex, the zero degree polar angle tracks is not really an issue. It is arising now because of the desire to make test tracks that are parallel to the beamline.
So, in this artificial case what, can be done?
If really needed, the theta_x, theta_y parametrization can be used.
But can we find out what we want to find out without fitting the track at all? For testing residuals, clearly the answer is yes. A fit is not needed, only a test trajectory. And we know what that trajectory is in advance here.
This means that the test harness needs to be in the context of the full reconstruction. Which is necessary in any case actually.