A matter of perspective - part II

This is part II of a series of posts I am writing about perspective. As I mentioned in my previous post, the subject captured my imagination for several weeks, and I've decided to share my learning process here. I hope you enjoy it!

Step 3: Doodling

Whenever I get the itch to grok something new, I end up covering endless napkins and filling my little pocket-sized notebooks with doodles. I'm a visual thinker, and so to understand anything complicated, I have to either see a picture of it or, even better, draw one myself. When I finally came up with a picture that seemed like a reasonable model of perspective, it looked something like this:


In the picture, d is the distance between the observer and the object, s is the size of the object, and θ is our measurement of perceived size: the arc that the object fills in our field of vision. Of course, this simplifies the problem quite a lot; it ignores the fact that parts of the circle are closer to the eye than the line we've chosen for s (I'll write about the consequences of this later), and it pretends that all of this is taking place in two-dimensional space. As a starting point, though, I think it's a good doodle.

Our question now is, what happens if we start changing the values of our variables? If our hypothesis is right, then multiplying d and s by the same number should leave θ unchanged (in our Moon/Sun example, the number would be about 400). The problem is, all of the tricks I know with triangles only work with right triangles. Fortunately, you can turn any triangle into two right triangles by cutting it in half. And this is an isosceles triangle, which means both of the resulting right triangles will be identical; they will be mirror images of each other. That means that (a) anything we deduce about one half of the triangle will be the same in the other half and (b) toilets flushed in the two halves will spin in opposite directions. Scientific fact.

Now the triangle's gotten a bit too small, though; let's doodle it again, only a bit bigger.


Technically, θ and s should be θ/2 and s/2, since they're halves of our original triangle, but let's just call them θ and s for now, and we'll remember that they'll need to be doubled later to get the values for the original triangle.

Getting back to our question, we're curious to figure out what happens to θ if we change d and s by multiplying or dividing them both by some number. Let's change d first; we'll divide it by two:


If our hypothesis is right, then to get the object represented by our new triangle to have the same apparent size of our original one, we should have to make it exactly half as large. s with d/2 should be exactly half what it is with d. We could use trigonometry to work this out (that was actually my first approach), but there is a more intuitive way that becomes obvious if you draw the triangles on top of each other:


As you can see the original (blue) triangle is exactly half as wide as the new (orange) triangle when it reaches the new triangle's base. And if you think about it, this is obviously the case; the two triangles have identically wide bases, which means the hypotenuse (the diagonal line) has to travel the same distance from the tip of the triangle to the base in both triangles. Since the old triangle is twice as tall as the new one, the old triangle's hypotenuse will move from right to left at exactly half the speed of the new triangle's hypotenuse.

And so, as represented by the second object I've drawn in the picture above, we can see that an object that is exactly twice as far away from an observer and is exactly twice as large must have the same perceived size. And through logical reasoning, we've proven that that will be true for any set of objects with an equal size/distance ratio.

Technically, we've proven our hypothesis. Yay for us! But I still had lots of questions at this point. What actually happens to θ when we double the size of an object? You'd think it would double, but if you consider the case of a flat object that takes up 100 degrees of your field of view, that would result in the new object needing 200 degrees - it would have to curve around you!

And so, in part III we'll begin exploring some of the interesting wrinkles that turned up when I was working out the math up to this point. I have to warn you, though, that it might take a while to get out, as I could literally become a father at any moment. I hope you're enjoying the series so far, and I'd love to hear from you in the comments if anything I've explained is unclear or if you have any questions.