Annalisa Crannell, a professor at Franklin & Marshall, has a great essay at Inside Higher Ed on the math of perspective. Crannell, who thinks her students are generally more scared of drawing than they are of math, uses the "fencepost puzzle" to get her students working through the proportions that create realism:
We mathematicians tend to stare at the paper, hoping an answer jumps off the page at us. Artists pull out their pencils and start doodling, often stumbling upon a solution almost by accident. The artists have learned to overcome their fear of drawing something that is "wrong," and so they become the first ones to draw a solution that is right. Another aspect of this problem I love is that, although there is only one correct answer, there are many different correct ways of arriving at that answer. In this way, the problem captures the essence of mathematical research.
One summer a few years back, when I actually had free time, I bought several books and tried to teach myself perspective. For me, it wasn't fenceposts, but columns - I wanted to draw the Cloisters, with their arched passageways. Not just columns, but arches, in perspective: Yeah, I'm an overachiever.
Anyway, it turns out to be incredibly hard to re-derive perspective for artists from first principles and teach it to yourself. I by no means mastered it - my attention shifted to something else and I gave up. But I regret not pursuing it, because if I paint from a photo reference, it often doesn't feel quite right - the camera distorts perspective and does not faithfully reproduce what a person standing in the space with binocular vision sees. One needs to take references with a grain of salt. And contrary to many people's fears, you don't have to lose the spontaneity of your art when you know the rules - you can apply them casually, like an experienced chef who doesn't measure her ingredients anymore, and you can break them precisely when you wish to, to unsettling effect. Knowledge really is power, in art as in so many other things.
Another post found via the endlessly energetic Jennifer Ouellette, who somehow manages to distill all the good stuff out of the internet. How does she do it?