As another way of looking at how we generalize, let's look at our principle planes. The principle planes were one of the more important concepts we had for you going from thin lenses up to thick. The discussion on the sine rule and the tangent rule has to do with angles near the axis but in the paraxial approximation, we also made lenses infinitely thin and then later on thicken them again. Well, it doesn't take too much thinking to figure out the principle planes. Unfortunately, in the non-paraxial world can't be planes. And just imagine, I've got a real optic studio diagram here but just imagine that we have a high numerical aperture focus of lenses rays coming from an object at infinity. So, we're taking a collimated laser beam let say and we're focusing it down with a microscope objective or something like that at relatively high numerical aperture. Independent of what this optic is, if we now project those rays from a perfect focus back to intersect with the rays from the object, that's how we'd find the principle plane. It's going to be a sphere, right? It has to be a sphere you can just do the geometry in your head, you can't have it be a plane anymore. So, indeed, in fact if you have a well corrective surface you do get a sphere but in general, the principle plane concept generalizes to something we now call the equivalent refracting surface. It's the same construct that we used before where we project the lines from the object and from the image back and we intersect them and we see near the origin it's a plane but as we go non-paraxial, it bends. That surface is of some conceptual usefulness when you're doing lens design but not nearly so much as the idea of the principle plane when we're doing first order design. But again, it gives you some conceptual tool for what do we do how do we generalize from the paraxial to the non-paraxial? If this was a course in lens design, we'd go a lot deeper into how to think about these things and start doing high numerical aperture lens design. But this is actually a course in system design so we're not going to do that and we're instead going to look at the aberrations themselves and how to recognize them and correct for them in systems.