X axis and Y axis signals to get "projection" on new axis.

Hello folks!

This might be a bit long, so ill include a lot of figures to make it easy.

I have a 2D sensor that records signals simultaneously in the X and Y axis. Heres an example:

The x and y axis are arbitrary: they do not necessarily coincide with the axes of greatest variation (peak-to-peak amplitude) of the signal.

I've found the axis of these signals that DOES give the greatest variation (i've extracted an angle "theta" relative to the X axis that represents this axis of greatest p-p variation).

I did this by finding the greatest distance along the trajectory (X-Y plot - in blue) of these signals (my axis of interest is the red line):

Now, the next and final step would be projecting (and then plotting) the resulting signal along this new axis!

So I've tried a few trigonometry derivations but my equations are shaky at best and frankly I dont trust them. I could use some help with a) some general maths pointers as to what direction to take and/or b) some psuedo code/functions (consider the signals to be 'X', 'Y', and the angle 'theta').

PS: I realize that what I'm doing is extremely similar to Principal Component Analysis (essentially plotting the 1st component), I want to verify that this is indeed the case (so the answer "just PCA it" isn't really what I'm looking for).

Thank you so much to any helpers!

R