Math and art: two point perspective, measuring points

Test test.

(I'm attempting to embed this on my wix website, so excuse any weirdness as I optimize that end.)

Today, I came across a document describing two-point perspective: read here. For those of you who've never heard of it, two-point perspective is a drawing (composition?) technique where there seems to be two vanishing points at the edges of the image for all non-vertical lines, but where vertical lines are...well, vertical. I see a lot of pictures of building exteriors using this technique because it highlights the depth of the building and allows you to see two sides of it. Interiors tend to use one point perspective, where everything vanishes somewhere towards the middle of the image.

Two point perspective composition

Example of one point perspective.
Photo by Horia Varlan

I first learnt two-point perspective in my eighth grade art class --  super straight forward. Draw the vertical line closest to you, connect the endpoints to the vanishing points, repeat (see the right side of the document). In this simple method, however, the grid distorts in the most foreground area. This is the reason why many drawings using two-point perspective look very 3-dimensional, but not necessarily realistic.

Using measuring points helps define the "compression" of the grid as the lines get farther away (remember how the vertical lines you know are equal width apart actually look closer together from far away?). It gets rid of the distortion, produces far more realistic images, and is accomplished through calculations based upon the degree the building is rotated.

Yup, good old geometry and angles. The details, I haven't had a chance to go into how this method was derived, but it's so simple to apply i'm in awe. A lot of people think drawing is something you just get a sense for, and that "artistic sense" is a foreign concept in engineers. Leonardo da Vinci (one of the earlier noted appliers of perspective, by the way) and this simple extension of two-point perspective disagree.

It's been a good day.