When I derive the first order imaging property of a spherical
refractive interface I do it quite differently from the way
Ludwik describes doing it, and I believe most of my colleagues
and textbooks do it my way. I introduce sign conventions at
the outset of the exercise, not after deriving the imaging
condition, and I use the conventions consistently throughout
what follows. I also repeat the derivation for all of the
various combinations which might be encountered to demonstrate
that the formula is general. While I do the variations faster
than I do the first, I do them all, and I don't treat any of
them as being trivial.
The phenomenon of imaging is one in which I find students are
intrinsically interested, and when the y-dependence of the
object distance drops out of the derivation I make it a point
to marvel over it, perhaps even effusively*. This is one
phenomenon for which student interest is assured in advance
(fry an ant with a magnifying glass if they need arousing); it
is the easiest of topics to teach for that reason.
Leigh
* It is important to remind the students at that point that an
approximation has been made. In an introductory course for
physical science students I will indicate how spherical
aberration arises when the approximation is removed, but not
any of the other third order aberrations. I have also used a
textbook at some time which derived the form of the aplanatic
refracting interface (Cartesian ovoid?) starting from Fermat's
principle rather than Snell's law. It was an introductory
course, too. Does anyone know what that textbook was?