Shadows

Once you start drawing things with lines in them, the world starts to look different to you. Lines start jumping out at you wherever you look! So I totally am grateful for KT writing me to say, "This is a little random, but I saw this today on Hartman and had to take a picture because the lines made me quite reminiscent of your class."

KT took the non-math-major version of my math-and-art class, so I was especially impressed by her insight.  I asked her what caught her eye, and she said,

I love how the shadow is still so connected to the original bench and not yet distorted by the position of the sun. I had to take a picture of the this because drawing lines to the vanishing point from the individual posts, like we did in class, would have been so satisfying.

Drawing lines to the vanishing points from an actual photograph isn't easy, we'd discovered in KT's class.  First of all, it's not always obvious which points the lines on the paper ought to pass through.  And second, in a real-world photo, the vanishing points are usually very, very far off the edge of the paper.  In our class, when students took photos with their smart phones and then tried to locate the vanishing points, they ended up shrinking the photo massively or else taping on many additional sheets of paper.

The vanishing points for the sun rays are far, far away.

 And yet, if you're good at drawing vanishing points, you can get interesting information from a photograph.  The lines between an object and its shadow (in the photo below, the yellow lines) obviously point in the direction of the light source.  But the shadows of vertical objects (in the photo below, the green lines) point toward the geographic location of the light source.  In other words, comparing the vanishing points of these two sets of lines lets us know something about how high the sun was in the sky, as well as the direction of the sun.

 We touch upon this a bit in our projective geometry class, when we talk about how to draw the shadow of this letter "A".  In both of the two pictures below, the light source is in the same direction. In the first picture, the light source is near (so it's directly above a point on the ground); in the second picture, the light sources is the sun (so it's effectively at a distance of infinity).  Notice how different the shadows are, with everything else -- the location of the light, the location of the "A", and the location of the glass windowpane -- all the same.

The lamppost is sitting on the ground, so the shadow seems to go sideways.

The sun is far, far away, so the shadow seems to come toward us.

In both of the last two pictures, though, we had the luxury of putting the vanishing points exactly where we wanted them -- on or close to the drawing itself. It's a lot harder to get the vanishing points where you want in a photo.  I'm super impressed with KT for her observations, and also for her beautiful photos!