Edit - picture for this scenario is linked at the bottom; apologies but first time posting here, much to learn and not just about geometry:)
So, imagine I have two parallel beams. They can be deformed, but when they are, they remain at a fixed vertical offset from each other. At a point on one of the beams is a measurement device which can measure the distance between the two beams at two angles. The blue lines represent a measurement at +/- 60 degrees (relative to the y-axis) and the red lines +/- 30 degrees.
At rest this system yields four measurements: D1 through to D4 and they're shown in the view labelled "Undeformed".
Now imagine that these beams are deformed - since they are fixed to each other such that the vertical offset 'x' is maintained, they move as shown in the view labelled "Deformed". This yields a new set of measurements D1' through to D4' - note these are with reference to a datum (marked as 'O') which has shifted in relation to the original datum point (labelled as 'O' in the "Undeformed" view). Since I have drawn this in Visio I had to go with the shapes available to me, but I am modelling this movement as a sine wave to a first approximation (and typically, I'm concerned with a half wavelength shape like the curve shown here). I have written a simulation of this deformation in python to yield data sets of the measurements D1'-D4' for various degrees of shift or deformation - modelled as various wavelengths and amplitudes of deformation.
Now to the problem - I thought that with the four measurements I have (or in this case since it's symmetrical, only two are relevant ?) I would be able to have a go at calculating the 'shift' which has occured between the datum in the undeformed state and the deformed state. I even put aside that it was a sine wave distortion and modelled it as straight lines between my known data points, but I cannot see how I am able to calculate the shift which occurs in the datum points when the deformation takes place.
I am beginning to think that I will have to use the data from my simulation to produce some sort of model which can take the four measurements and estimate the shift. I am concerned about this and before I embark upon it, I am keen to establish if I have exhausted the geometric approach or just my geometric understanding!
A further wrinkle is that in most cases, the datum for the measurements isn't at the peak of the deformation as I have pictured it, but will be offset from it. Since the 30 and 60 degree measurements are referenced to an axis which is at normal incidence to the beam (where the measurements originate), this will result in the measurement system itself being rotated when not at the peak of the deformation - losing the symmetry. Mind you, with the way this is looking at the moment, the asymmetry is something I may grab onto as it provides some more data at least!
So, my question is - can this be solved geometrically and I have just missed it, or is it more complex than this?
Many thanks for considering this problem....
