Seeing Things Differently

The topics discussed on this page are a classic example of why artists (and especially curators in art galleries and art historians) need mathematics as much as scientists and engineers.

How to look at a picture without seeing distortion

Where a picture is painted in true perspective, that is as if it were being looked at through a window, it is anamorphic to a greater or lesser extent. Only by looking from the same position as the artist, can all distortion be removed.

It is possible to work out where the artist’s viewpoint would have been, given certain assumptions that can usually be made. If you look in this way then the perspective picture becomes distortion free. Stray from this point and parts of the picture become distorted. Artists have ways of compensating for this in pictures and this is discussed in the section “So how do artists paint spheres” below.

How do you find the artist’s viewpoint?

If you have a painting in perspective, which has some objects whose perspective properties you can use, then you can reconstruct the artist’s viewpoint. It is also possible to do much more, but that is out of the scope of these pages.

The easiest type of painting to work with, is one which has a set of square tiles or other such objects which are on a flat plane such as the ground. This painting of The Annunciation by the Italian Carlo Crivelli which dates from 1486 is a typical example.

The actual painting is 2070 mm high by 1465 mm across. The base of the picture is displayed 1060 mm off the floor. To find the viewing point on the actual painting, we need to work with a reproduction and then scale it up to the size of the actual painting.

The steps to find the artist’s viewing position are:

1. Draw in some lines of the painting which are parallel (in the space of the painting) to a line perpendicular to the viewer. These are the yellow lines.
2. Intersect these lines at the central vanishing point.
3. Draw a horizon line (red) through this line.
4. Draw a line (blue) which is the diagonal of a square and intersect the horizon line in the right diagonal vanishing point.
5. Measure the distance from the central vanishing point to the right diagonal vanishing point.

Since the central vanishing point is directly in front of the artist’s eye, we need to place our eye directly in front of the grid behind the archway. Now we need to find out how far away we need to be. This is the distance from the central vanishing point to the right diagonal vanishing point as measured in step 5. Why this is so is described below.

Using proportion on a measurement on the lines drawn and the actual size of the painting, then the result is that we need to be about 120 cm away from a picture at a height of about 200 cm above the floor the way the picture is hung.

Now an average person’s eyes are about 160 cm above the ground. This means that if you want to get the true or best view of the painting, you need to do so standing on a box 40 cm high (about 16 inches). Moreover, if you try to look at parts of the painting, like the peacock at the top right, it looks distorted.

Why not see if you can find some perspective pictures in a gallery. Buy a postcard and work out the artist’s viewing position. Is it possible to place your eye in the correct position when you go back to the gallery?

How to find out the distance of the viewing point

All lines in the same plane that point in the same direction, end at the same vanishing point in the picture, so we only need to consider one line. This is a view from above with C as the central vanishing point and R the right diagonal vanishing point. That is, R is the point where all diagonals of squares meet in the picture if they have a side parallel to the picture plane, like this:

So we only need to consider the one square shown here where E is the eye point.

Since angle CER is 45 degrees and angle ECR is a right angle, then the triangle ECR is isosceles with ER is the same length as CR. So if we find ER as was done with Crivelli’s Annunciation, then we know the distance of the eye in front of picture (EC).

The mystery of pictures hung in galleries

All this has been known for over five hundred years, but many galleries do not seem to take heed of it when hanging perspective paintings. They have problems of space in many instances, but more often than not the picture becomes anamorphically distorted to an extent that it is not possible to see it correctly. In many cases the pictures do not really come alive.

In some cases the problem may be that curators are not interested in looking. I have seen cylindrical mirror anamorphic pictures displayed flat on the wall, with no chance for you to see how the picture was intended to be viewed, or for you to know what it represents. It is just a very abstract looking picture. There was not even a photograph to show what happens.

Foreshortening and visual illusions

Your brain knows all about foreshortening and uses it to estimate distances and make sense of three dimensional space. Take a close look at the following picture.

If you physically measure the red, blue, and yellow topped cylinders you will find that they are all the same size. However, if you look at it as a perspective picture, especially comparing the cylinders to the foreshortening of the lines, then the blue looks bigger than the red and the yellow bigger than the blue.

Spheres and distortion in paintings

If a painting is created in true perspective, then as you get closer to the edge of the painting distortion increases. You will not see this if you look from the same position as the artist’s eye. But paintings are not normally viewed that way, especially if they are a mural on a large wall. After all, even if someone wanted to get the perfect view, they would have to move towards the painting and could still be looking at it as they did so. Consequently, they would see distortions. If you look at the development of paintings in the century or so after perspective was discovered (say 1430 to 1550) then you will see that artists followed the rules strictly as they learnt about them, but then began to bend them.

Piero della Francesca and Leonardo da Vinci were among the first to point out that distortion occurs at the edge of paintings. The effect is particularly marked where you have two types of objects, spheres and columns. Leonardo used an example like the following to show the effect. This is logically puzzling at first, since objects which are farther away are larger on the picture instead of smaller as foreshortening would have us believe.

This diagram below shows the cross section of an artist’s view from above of a set of circular columns. The artist’s eye is where the vertical lines meet and the horizontal line corresponds to the picture plane. The coloured circles are the cross sections of a set of columns.

Notice how the red column is directly in front of the artist, but the blue one is quite a way off. Compare the representation of the columns in the picture (the grey lines on the picture plane). Notice how much longer the blue line is than the red one; in fact it is over 50% longer despite the fact that the blue column is farther away from the artist.

As a consequence of Leonardo studying this problem, he came up with a rule of thumb for how you should paint a picture. He said that the angle from the perpendicular to the picture plane and the line to the farthest edge of the painting should be no more than 30 degrees.

Investigation: Plot the length of the line on the picture as the column moves away from the artist. This is a good exercise to do in a spreadsheet. There are a number of parameters that can vary:

  • the size of the column
  • the distance of the artist from the picture plane
  • the distance of the row of columns behind the picture plane.

Investigate various possibilities to see if Leonardo’s rule of thumb is sensible.

Drawing spheres in paintings

Spheres suffer from the same type of distortion when painted in perspective. If the line joining the artist’s eye to the centre of a sphere is perpendicular to the picture plane, then the sphere will appear to be a circle. The spheres do not have to be far away from the centre of the picture before their shape becomes obviously not circular.

The picture above shows a set of billiard balls where the artist’s eye is directly above the central one. Notice how the balls at the end of the central horizontal and vertical rows get progressively elongated as ellipses the farther they are from the centre. The ones at the farthest corners are the most distorted.

Like all perspective pictures, this distortion is truly anamorphic. If you move your eye closer to the screen keeping it above the central ball you will see that the other balls become more realistically round.

Since this picture is small, unless you are very short sighted you may not be able to focus on the screen when you are at the best position. There is a higher resolution version of the picture for you to download and print out in the downloads post.

So how do artists paint spheres

Firstly, artists normally do not paint spheres at the edge of paintings in perspective, although there is an example below. They simply paint a circle since most people move about when looking at a painting, unless it is a small one. In the latter case the angle subtended by the picture is small, so the distortion does not arise. The following case of Raphael’s The School of Athens is a case where there could be a problem, but he skilfully overcame it by bending the rules of perspective.

The painting above is in the Vatican in Rome. It is one of a series of murals on the walls of a room called the Stanza della Segnatura (the room for signing important documents). It was painted in 1510-11. It is appropriate here because it is about Plato’s academy in ancient Greece with its importance of mathematics, especially geometry and its relationship with music in the culture of the time.

Many of the people in the picture are supposed to be famous mathematicians. Some of the people who represent them are prominent people of the time, including Leonardo da Vinci who traditionally is supposed to be Plato (in the arch in the centre) and Raphael has even included himself.

The central vanishing point is the face of Plato. The building and floor is drawn in perspective as the vanishing points show. Even the object in the centre is drawn in the same perspective since its sides extended (shown in red) meet on the (blue) horizon line.


Look at the close up of the right hand corner above  Raphael has pictured a person holding a sphere. Note how the spheres are represented as a circle even though it they at the edge of the painting.

So how has Raphael made the rest of the picture seem natural without the anamorphic distortion inherent in such a large picture? Well, he has painted the people in the front and to the edge with their own perspective, that is Raphael has changed his viewing position . So the people holding the spheres in the excerpt from the picture look in correct perspective.

If you go to the Vatican, Plato’s head is way above yours as you walk on the floor, so you can never position your eye in the correct position for the artist’s viewpoint for the main perspective of the building. Yet the picture works, because the people each have their own viewpoint.

Pirenne  in his book on the optics of paintings and photographs, says that the nineteenth century perspectivist La Gournerie took an engraved version of the picture and stuck the correct perspective ellipses over the ones in the painting. They did not look acceptable as spherical objects, whereas Raphael’s circles do.

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