How to become a renaissance artist in 5 minutes or less

During my previous semester at college I took a course on projective geometry, which is basically a “fancier” field of geometry. It’s most important property is that ratios do not change(stay invariant) under projective transformations and that parallel lines meet at a point at infinity.

Parallel lines meet at the infinity point

Before I lose you as a reader, I’ll just refer you to the wikipedia article and move on to the renaissance stuff 😀

Now let’s move on to the tutorial, where we’ll learn how to draw a chess board projectively, just like the artists/painters do it.

Take a look at the following two pictures:

Projectively transformed chess board(left) and normal chess board(right)

As one can clearly see, the picture on the left is basically the same chess board but distorted. All the angles and lengths have changed as these properties do not remain invariant under projective transformations.

Many people like to refer to the left image as “looking at it from a different angle”. And that is in fact correct. If you tilt your screen/phone you’ll also see the “proper” image on the right as tilted or skewed.

So, back to the renaissance stuff !😃

Let’s say we’re given the frame of the distorted image without any black/white squares and our job is to reconstruct the whole chess board projectively. Something like this:

The frame of a projectively distorted chess board

Now here comes the interesting part finally !😆

Remember how all the parallel lines meet at the infinity point in projective geometry. If not, let’s connect them and hopefully that clears things up a bit:

Parallel lines(from the original chess board) meet at points of infinity

Ok, that’s a cool step forward, but is nowhere near enough to redraw the original picture. 🤔
For that we can identify another point, which was projectively transformed. That point is the meet of the diagonals.

Projectively transformed meet of the 2 diagonals

Now comes the powerful part…😊
What happens if we connect the points of infinity with that diagonal point ?

Remember that all the parallel lines meet at an infinity point. The reverse implication also holds true: if 2 lines meet at an infinity point, then they must be parallel. Or better said, must have been parallel in the original euclidean geometry setting.

Points at infinity connected to the diagonal point

Deducing from the aforementioned implication, these lines must have been lines that both go through the diagonal point and are parallel to their respective frame sides. It directly follows that these lines were the ones, splitting the chess board in two equal parts.

Cool, now we have the middle points of all the frame sides.
Now let’s repeat the process for the 4 smaller squares that have originated.

Repeating the process for the 4 smaller squares. Here we focus on 1 of them

Eventually after meticulously connecting all the proper points, we end up with something like this

The fully reconstructed chess board

Oh, we forgot the colors ! 😅

One can just fill them switching between black and white as a normal chess board would look.
We end up with something like this:

Isnt’t it beautiful 😅

Desargues, Descartes, Klein and many others: both mathematicians and painters from the renaissance would use this technique often, if not always, when developing their masterpieces.

Hope you enjoyed ! 😃