Perspective is culture dependent, and since the mid-fifteenth century in the west it has been based on a very geometric system. The method was invented, or rather rediscovered since the Greeks and Romans knew how to draw in this way, by Fillipo Brunellesci around 1425. One of the earliest artists to produce a book on how to draw in perspective was Albrecht Dürer. As well as discussing geometric methods, he also illustrated his book with a set of woodcuts showing practical tools for accurate perspective drawing.

Painting a standard perspective picture is best thought of as looking through a window. Dürer’s woodcut shows an artist drawing a woman as if you were looking through such a window. It illustrates some important points about perspective and shows the grid method common to many methods of creating anamorphic images.

The artist has a grid on the window he is looking through. He also has a grid on the paper to use as a reference for what he sees in the window. This is like using graph paper and is a common method artists use when copying paintings too. Notice also that his eye is in front of a vertical pointer. He uses this to ensure he always looks from the same place and so sees the objects he is drawing in the same position on the grid.

### An experiment to try

You can see why he needs to do this with the following experiment. Close one eye and hold up a finger in front of your face about a foot away, and note what the finger points to. Close that eye and open the other one and you can see that the finger points to a different place.

### Projection and section

Light rays travel in straight lines, so the act of looking is to construct a set of lines from the tip of the pointer to each point on an object and where the line intersects the window is the corresponding point on the drawing. Only if you look at the artist’s picture from the same relative position, will you see it the way he saw it. If you look from anywhere else, it will be distorted. So it is important to realise that when viewing a perspective picture created in this way, you can only truly look at it with one eye if you want to see it as the artist did.

Although Dürer’s artist is looking through a flat window, with the window placed so that if he looks from the pointer to the middle of the window, the light ray to his eye is perpendicular to the window. The grid on the window then looks like the grid on the table. If he moved the pointer to one side, then the two grids would no longer correspond. The grid on the window would no longer look regular and he would have to use a similar irregular grid on the table. He could still use the two grids as reference between points on his drawing and points on the objects he is drawing.

Anyone looking at the picture would have to look from this new viewing point at the side in order to see it correctly. If they looked at it centrally it would look distorted. This would then be an anamorphic painting since it could only be viewed without distortion from that point.

### Foreshortening

One of the reasons that grids appear so often in perspective and anamorphic pictures is that they were one of the first objects it was easy to draw in perspective. They were the used in pictures to depict a tiled or paved floor. Look at pictures dating from 1450 to see how many such floors you can find. This also happened when there was a revival of perspective in Dutch art which coincided with the rise of the technology of optics in the 17th Century (look at paintings by Johannes Vermeer and Pieter de Hoogh). This is a typical interior by de Hoogh.

In this painting notice how the distance between the front and back of the tiles in each row gets narrower the farther away from you they are. This is known as foreshortening. Knowing how to construct this spacing correctly is the key to creating a perspective drawing or painting and is described below.

### Vanishing points

The other vital detail you need to understand is that any two lines that are parallel in space meet at a point in the picture called a vanishing point. Furthermore, the vanishing points for sets of lines that are in a plane (or a plane parallel to a particular plane) meet in a line called a vanishing line. If the lines are parallel to the ground, then they meet on a special line called the horizon line.

In most pictures, vertical lines are drawn parallel, because most objects we look at do not show vertical convergence. The exception to this is when you look at tall buildings. The following drawing shows a horizon line (red) with a number of vanishing points. Vanishing points P and Q are for lines which are parallel to the ground plane. These points are thus on the horizon line.

Parallel lines on the roof gable meet at point R. Because these lines are not on a plane parallel to the ground, point R is not on the horizon line. Line RQ is a vanishing line for planes parallel to the ends of the house. Can you find another vanishing point which is not on the horizon line, but which is on PQ?

### Constructing a “one-point” perspective

The first book on perspective was Leon Battista Alberti’s Della Pittura or On Painting which described the results of the architect Brunellesci who is credited with the discovery of perspective around 1425. Before describing how to correctly draw the spacing of a grid in perspective Alberti says that some artists had a logical method which does not work:

“Here some would draw a transverse line parallel to the base line of the quadrangle. The distance which is now between the two lines they would divide into three parts and, moving away a distance equal to two of them, add on another line. They would add to this one another and yet another, always measuring in the same way so that the space divided in thirds which was between the first and second always advances the space a determined amount. Thus continuing, the spaces would always be – as the mathematicians say-superbipardenti to the following spaces. I can say those who would do thus, even though they follow the good way of painting in other things, would err. Because if the first line is placed by chance, even though the others follow logically, one can never know where the point of the visual pyramid lies.”

The word superbipardenti means exceeding by two parts. The method above shows that artists knew by looking that the spacing of a grid gets smaller as the lines of the grid get farther away from you. This is known as foreshortening and you can see this in the de Hoogh painting above. This discovery was a breakthrough in the understanding of the geometry of foreshortening.

One thing artists did understand, before Brunellesci and Alberti came on the scene, was that if the grid was laid parallel to the edge of the picture, then the sides of the grid going away from you were straight lines and met at a point in the distance. This point is called a vanishing point and is described later. Before constructing a correct foreshortened grid, let’s look at the incorrect one these early artists used.

The diagonals on a square grid all lie in a straight line:

Tthey can be used as test of whether the artist has used perspective. If the artist uses the 2/3 method Alberti describes then the diagonals form a curved line as shown at the right whereas they should remain straight:

### How to draw a grid in perspective

This method will produce a view of a square grid known as a one-point perspective which is commonly found in perspective pictures, particularly 15th century ones and Pieter de Hoogh paintings including the one above. The grid (usually in the form of a tiled floor) is placed so that one side of the square tile is parallel to the base of the picture.

*Step 1 – * draw a border/frame with the proportions you want.

*Step 2* – draw a horizon line (the red line) parallel to the base of the picture which is level with your eye if you were standing in front of the picture as a window.

*Step 3* – along the bottom of the picture mark off equal distances which will represent the edges of the squares. Extend the base of the picture to get more lines and a bigger grid.

*Step 4 – *mark a point on the horizon, called the central vanishing point, and to it join each of the points you marked. This will give you a picture like the one above.

*Step 5* – choose a point R on the horizon line, which is in proportion how far your eye is away from the picture. Point R is called the right diagonal vanishing point. For why this is so, see “How to look at a picture without seeing distortion”.

*Step 6* – join point R to each of the points you marked on the base of the picture as shown by the blue lines above.

Grid

*Step 7* – where each of the blue lines intersect the lines going to the central vanishing point, draw a line parallel to the horizon line. Note how all of the intersections are on these lines.

*Step 8* – now get rid of the blue lines and the part of the horizon lines outside of the picture.

*Step 9* – now erase the parts of the lines going to the central vanishing point which are beyond the farthest grid line and you have a set squares in perspective going into the distance.

### Investigation

Draw a series of perspective grids which have the same spacing of the points across the bottom of the picture, but the diagonal point R at different distances. What happens to the appearance of the grid?

Place your eye in front of the central vanishing point (C) at a distance CV away from the picture. You should see the same result each time. You can find out more about why this is the case in How to look at a picture without seeing distortion. What is the relationship of the spacing of the lines? This is a topic in itself , described in The spacing of lines in a perspective grid.

### What does all this have to do with anamorphic art?

Anamorphic art is a form of perspective. The above discussion is about drawing on a flat picture plane that is perpendicular to the artist’s line of sight. In anamorphic art, the artist views the picture plane at an angle. This is not quite the same as foreshortening giving rise to visual illusions.

### The spacing of lines in a perspective grid

In drawing a perspective picture, we found out how to space the lines correctly, but what is the mathematical relationship? The early artists who used a ratio of 2/3 were partly correct. They knew each spacing depended on the previous one and there was a ratio involved. The ratio is a special ratio which is called a cross ratio. Cross ratio is a ratio of a ratio. Moreover in a perspective picture it is always a ratio which depends on the vanishing point of a line.

One way to construct the spacing is shown in the diagram above. We want to construct the position of D when we know the position of A and B.

1. Draw any lines CF and CH from C.

2. Draw any line AG from A to intersect CH at H and CF at G.

3. Join BG to give a line and similarly BH.

4. Extend BH to intersect CF at F and intersect CH and BG at J.

5. Join F J and produce it to intersect CA at D which is the required point.

The ratio (AB.CD)/ BD.AC) = -1.

The value is always -1. The values of AB, CD, BD and AC are measured according to their direction, so that if AB is considered positive then CD is negative.

Another instance of the same spacing is the frets on a guitar:

First of all, thank you VERY much for being apparently the only person on the internet teaching this. I can’t believe how hard it is to find decent instructions on how to do this. If you have any other resources for this, please let me know.

Second, you had me up until Step 5: “Choose a point R on the horizon line, which is in proportion how far your eye is away from the picture.” Can you be more specific? How exactly do I measure or gauge how far my eye is from the picture and then how does that translate into where I place this point? I actually did come across another guy who had a formula for how to figure this out, but I’m unable to find him again.

Anyway, how do I figure out where exactly to put that R point? Thanks very much and thanks again for showing how to do this.

Thanks for your kind comments, Tim. I have had other people say that I understand perspective more than most.

When I teach perspective, I normally do it as measured perspective. So here are some extra thoughts.

In perspective you should imagine the picture to be on a glass window and the drawing is what you see from one particular point, the eyepoint..

The distance of the eye from the picture to the central vanishing point is literally that.

In a one point perspective the distance of the point R from the central vanishing point is identical to that distance.

It helps to draw a plan view, ie looking from above. This is where scaling comes in because you often want to draw a smaller picture than the real life version. So if you were looking hrough a window which is 6 feet away, you might use 6 inches on your plan and hence the distance of R on the perspctive drawing.

However, if you want to put your eye in the correct position to see the perspective as it should be, you need to place your eye 6 inches in front of the central vanishing point.

You might find it difficult to focus at 6 inches. The least distance of distinct vision is normally said to be about 10 inches.

What I suggest you do is draw a series of perspective grids with the same spacing along the base. Choose your horizon line and central vanishing point. Then choose R at different positions on each drawing. Place your eye at the measured distance in front to see what the effect is. Note how the spacing of the lines changes.

Look at paintings to find the true distance points.

You might like to join the anamorphic discussion group at http://anamorphosis.grouply.com/.

My other blogs are on paper sculpture and tilings; Look for Sliceforms and Morphing tilings.

If you need more help, please ask.

John S

I agree. This is an excellent introduction. I’ve searched art books in vain for something as clear as this.

Thank you very much for the wonderful article on perspective. I have been searching for one and never found such beautiful explanation. I would also like to go through your blogs on paper sculpture and tilings. Could you please send me the links for the same?