I want a way to quickly and efficiently go from un-dispersed to dispersed images reference frame. It is therefore an (x,y) ↔ (dx,dy) transformation but wavelength factors in this, as well as where on the detector we are. It is therefore, in a more general way a transformation from (x,y,lambda) ↔ (dx,dy,lambda), where (x,y) is the location on the detector to use to account for any field dependency.
I can thing of several things I would use this for:
I envision that I will need to compute this several millions on times so I want to keep things as fast as possible. It could be a 2D polynomial solution, which can be inverted either analytically or numerically in the case of higher polynomial orders (as in UVIS).
I want this system to be flexible and extendable to highly distorted spectra so I do not want the calibration to be dependent on the x or y coordinates, or the pathlength along the trace (which is difficult to compute and invert) or lambda directly. The choise of x or y is a bad one for example in cases where the spectra curve dramatically and either of these coordinates become a poor ways to measure where the trace is. Instead I want a system where each of this transformation is given as a function of an intermediate variable t which can then be defined as best fit a particular disperser. Hence the functions will be dx = fx(x,y,t) dy= fy(x,y,t) lambda=fl(x,y,t) and the inverse functions x = fx'(dx,dy,t), y = fy'(dx,dy,t), lambda=fl'(dx,dy,t). The variable t can be defined as t = x in the case of a simple x-direction disperser. It can also be defined as t = lambda, or t = (lambda-l0)/(l1-l0) where l0 an dl1 and the minimum and maximum wavelength in the dispersed spectrum. The latter results in 0<t<1 which could be beneficial to some.