I have been playing around with a variety of map projections for the purposes of transforming between various representations.
The thing that I still don't quite understand is how to interpret the (x,y) values of the projection. Take the Mercator projection, for example. I get that (x,y) represent locations on the tangent plane, except that they are still in spherical coordinates: e.g.
x = lambda - lambda_0.
How do these (x,y) values translate to actual locations on the tangent plane (i.e. the map) if they're still in spherical coordinates? In my case, I ultimately need to identify actual pixel locations when I render the new projections, but generally speaking it is not clear to me what these spherical (x,y) values mean with regard to the mapping surface.
Perhaps this additional info may help to explain my confusion. I found the image below on slide 60 of this powerpoint. I think it illustrates the step I am confused about. Here it's called the reference transformation. I've been reading through Snyder's "Map Projections: A Working Manual" and I think I've concluded that the rectangular coordinates output from the forward projections are arbitrary in scale. Where I am struggling is how to go from that arbitrary scale to my pixel grid.
I hope it may be more clear if I explain how I have arrived at my confusion.
When resampling an image according to some function (e.g. "undistorting" an image, which is kind of like an inverse projection of some image distortion), the standard algorithm is:
(1) Apply the inverse projection to the normalized pixel coordinates of the image
(2) Find the min/max bounds of that projection and create a grid in that space
(3) Apply the forward projection to the normalized coordinates of that grid to determine where in the original image to sample
(4) Sample according to the output of step (3) in order to render the new image.
I have been trying to follow these steps for converting between an equirectangular map and various projections. It seems my mistake is that, in the algorithm I laid out above, the projected space is always a pixel coordinate system.
Am I correct to understand that each map projection has its own rectangular coordinate system? That is, Mercator rectangular coordinates mean something different than Gnomonic rectangular coordinates, etc. If so, is it based on the class of projection (planar, cylindrical, conic), or specific to the individual projection? Snyder lays out rectangular coordinates in tables in his manual, but they don't seem to correlate with each other.