Mathematics

The mathematics of Perspective

Ask someone to draw straight railway tracks, as it will look to someone standing between them, and looking in the direction that the tracks are going to. Chances are that he/she will draw two straight lines meeting at the horizon.

What is the correct curve to represent these tracks, as seen by the eye, or as it should appear on a very realistic painting? A bit of thinking reveals they cannot be straight lines ...

Let us first ignore the fact that the earth is not flat. Then, if the correct curves are straight lines, they have to meet somewhere. (We assume they are not parallel. Nobody would draw them parallel, because more distant objects appear smaller on a painting.) The straight lines meet, but this is not realistic: The two railway tracks do not meet!

You may say that I should remember that the earth is not flat. Maybe this is the source of our contradiction. But wouldn't it be astonishing, if the curvature of the earth "compensates"
SO exactly, that what would otherwise be curves now becomes perfect straight lines? No, they cannot be straight lines. We have to find the correct formula.

Along the way we will learn some:

• Trigonometry,
  because there are angles involved, and we would need formulas for how the various sides of a triangle depend on the angles, and also for how the angles depend on the sides of a triangle.
• Calculus,
  because this is what one needs to find the maximum and minimum of graphs of formulas, or to understand how the slope behaves, and so on.
• Programming,
  because for example a cube has eight corners, and to check how it should look from many different angles involves many repetitive calculations. We will also want to graphically represent the results on a computer screen.

To account for the fact that one's eyes have a certain height above the earth, will be another challenge. If we have time, we will also take the curvature of the earth into account. Or we can look at the paintings of scientifically minded artists such as Leonardo. How does the way he represents straight-line objects, such as the edges of buildings, or multiple objects that are far away and lie on a straight line, compare to the results of our investigations? There are several
such angles to pursue, in the mathematics of Perspective.

Participants

• Mr. Gusti van Zyl