### Putting into perspective

Here is a draughtboard (draughts board). Once again, we find the receding lines, as with the boxes. We also see find have guess (on ne peut pas dire voir) ?? the horizon line and the vanishing point. We have deliberately chosen not to show the vanishing point and the horizon this time, but you can easily guess their position by virtually prolonging two of the receding lines stretching towards the horizon.

The viewpoint is placed lower in the picture on the left than in the one on the right. Look at the two pictures in turn, imagining yourself standing up and sitting down. As you know, when you stand up the horizon line rises at the same time. You also note that the lower your position, the more foreshortened the squares. In reality they are indeed squares but, as you can see, the height of these squares is greater on the right than on the left. It is thus closer to its true shape. This is a very simple notion which you should keep in mind when you are planning your drawings.

Now let us turn to an extremely simplified scene showing a situation viewed from an aircraft. We usually speak of a “plan view” so as to avoid confusion with an “aerial view” which has different characteristics.

Here we see a floor in a grid pattern, three cube-shaped houses with a four-sided sloping roof and five trees all in a row. Although there’s nothing very realistic about this landscape, it will allow you to practise building up, accurately and precisely, your first scenes in perspective.

This is what the land looked like before the houses were built. The row of well manicured trees is put in perspective. The height of both the foliage and the tree trunks diminishes with distance, while following the straight lines plunging towards the vanishing point. The distance between two trees, in reality of course always the same, gets smaller and smaller as they move towards the horizon. We rely on force of habit, visual logic and reason to remind us that the trees are in fact equidistant. ##

Shortly afterwards three identical houses were built. You will have noticed that the rooftops, the drain pipes and the bottom of the walls parallel to the row of trees also converge towards the vanishing point naturally situated on the horizon line (in green).

Closing one eye, you check that the horizontal lines perpendicular to the visual ray (in yellow) remain horizontal, the vertical lines remain vertical and the other lines are receding.

Before leaving the scene you decide to take a few photos, moving to the right and the left and climbing to the top of a crane which was still on the building site. Your camera is loaded with a special kind of film which can see through walls and show the structure of the objects photographed.

You notice straightaway that your moving about has made a bit of a mess of the lines and forms. The draughtboard squares, which had assumed a trapezium form in perspective, have now become diamond-shaped. It’s worth taking a closer look at this.

You know where you are in the first photo. Everything is as expected.

But in the second photo (which you took after moving to the right) the side of the house nearest to you is deformed.

First of all, you notice that the angles formed by the drain pipe or the base of the gable close to you with the angles of the vertical walls are no longer right angles.

You remember the plan you drew up of the room seen in frontal perspective. The visual ray was parallel to the dresser.

The vertical lines are still vertical but the horizontal lines perpendicular to the visual ray, which would normally remain horizontal and parallel, recede towards a new point situated on the left but still on the horizon line.

You have just entered the realm of angular perspective. Do you know why? This time the plan shows that the visual ray now strikes the gable diagonally and no longer perpendicularly. Since nothing is parallel or perpendicular to the visual ray, the principal point no longer serves as vanishing point and the lines which are horizontal in reality now recede towards two distinct points.

The photograph taken after a movement further to the left is even more characteristic of angular perspective. Here’s the plan:

As you can see, the walls of the houses are presented diagonally, at an angle.

In short, there are two vanishing points, not just one, in angular perspective. Now for the bad news: some viewpoints have **three vanishing points**…

This is exactly what we see in this photograph taken from the top of the crane. The vertical angles of the houses, the tree trunks – in short all the vertical lines which hitherto behaved themselves very well – now converge on a third vanishing point situated well below the photo. This type of perspective is referred to as aerial perspective, the third type of perspective view with which you will regularly be dealing.

So can we count on a maximum of three vanishing points?

To be quite honest – no. When you become more experienced, you’ll see that there is no end to the possible ??number of vanishing points. But don’t worry: with time and practice the intricacies of perspective will soon cease to hold any secrets for you.

You’ll start by drawing mechanically, gradually becoming more intuitive as you go along. You mustn’t think that you need to learn anything by heart; you just need to understand what you see. It’s really a question of sharpening your vision until you get to the point where you incorporate beautiful perspectives without even thinking about it.

How do we know which perspective to adopt – parallel, angular or aerial? Ce titre ne me plait pas Parallel perspective: a box is presented with a side perpendicular to our visual ray. There is a single vanishing point situated on the horizon.

Angular perspective: there is no side perpendicular to our visual ray, but certain sides are parallel to our horizon plan.

There are two *vanishing points*, both situated on *the horizon*.

Aerial perspective: there is no side perpendicular to our visual ray and no side perpendicular to our horizon plan either.

There are three vanishing points, two of which are situated on the horizon line and the third above or below it.

Here are a few examples of angular perspective views. Try to acquire a feeling for the position of the vanishing points and the deformation of the sides. When you no longer have any problem in drawing a box or a cube, you can accurately depict whatever you want since virtually everything you need to know about perspective is encapsulated in these forms.

Aerial perspective is extremely spectacular but somewhat difficult to use. You will learn from experience that the slightest excess or mistake in choosing the positions of the vanishing points can have disastrous results. Your picture is no longer credible, which is a pity since the whole point of perspective is to present a convincing image to the spectator. So don’t go overboard with your bird’s eye views! This example shows very well that what may be tolerable in a photograph is not necessarily very successful in an illustration.

It is well worth remembering the following precept: The eye is neither a wide-angle nor a telephoto lens. It has its own focal distance which limits the perspective deformations perceived by us. So keep to natural proportions and let nature be your guide.

A cube In your drawings, you will often be required to estimate a direction, approach a form, evaluate the angle of a straight line, imagine the aspect of a concealed part of an object. Indeed, this is the attraction of all artistic endeavour. A rigorous approach has to come from within you; no-one will impose it from the outside.

Here is a frontal view of three boxes. In the foreground, their visible sides are square. If you want to give the impression of cubes, their top sides will have to be square too, this time seen in perspective. The red line determining the depth of the volume must be properly placed. See how its position influences the impression left by each box.

You are no doubt thinking that it’s a bit risky trying to estimate the depth with the naked eye? Is there a more scientific way of creating an exact cube in perspective or do we have to trust the judgement of our eyes?

As you know, a cube is made up of six square sides. The challenge thus consists of knowing how to draw a square in perspective. The choice is very simple: either you trust your eye – and this can be risky – or you apply a precise and exact method.

Here is the exact method. You can either apply it straightaway or bear in mind that such a method exists and refer to it when the need arises. If you want to master this method, you’ll have to b prepared to go into a certain amount of detail. In that case, let’s begin by looking at this illustration. The observer is situated upright on the floor.

His visual cone crosses a vertical plane, framed in yellow and called the plane of the perspective. The intersection of the visual cone and the plane of the perspective is expressed by a blue circle. The vertical and horizontal axes of this circle are represented by a large orange cross. For the observer, the horizontal axis of this circle is merged with the horizon line.

The somewhat darker part of the plan is the picture itself. The green line is the ground line, situated at the intersection of the floor and the plane of the perspective. The red point at the centre of the visual cone is the point where the principal visual ray (here marked in red) meets the picture The two yellow points on the exterior are the distance points, so called because they are situated at the same distance from the central point and the observer’s eye.

The same situation seen from above shows the position of the distance points.

The information given above gives you the vocabulary you need to tackle the next step. Once the square has been put in perspective you will be able to construct exactly the kind of cube you wanted.

Bearing in mind the complex procedure involved, you find it difficult to imagine yourself drawing a chessboard or a floor in a grid pattern.

Actually there is a very simple way of getting round this kind of problem.

How to draw a frontal view of a tiled floor Suppose you want to draw a floor consisting of square slabs. You only need to draw the first square carefully as all the others will flow very easily from it. This time you are going to use your eye to draw the square.

Draw a horizon line and then a second lower line representing the edge of the first row of slabs. Divide this low line into as many segments as you like but make sure they resemble each other. Fix a vanishing point roughly in the middle of the horizon line and draw the receding lines which will express the alignment of the slabs.

Trusting to your eye, draw a first horizontal line, here marked in red. This will determine the depth of the first row of squares seen in perspective. As in the example of the cubes, the choice will be of decisive importance in ensuring that the line does not give the impression that the slab is rectangular in one direction or the other. Once you have chosen the depth, the rest is easy.

Next draw the diagonal of any slab in the first row and prolong the line so that it intersects the other receding lines. Each intersection will determine the depth of the following row. You will then be able to draw the lines (here in blue and then green) giving each succeeding row the depth which you chose with the naked eye for the first row. In perspective, the lines determining the depth of each row become narrower as they move further away, but this way you can be sure that your drawing will be strictly accurate because, as you know, the dimensions always seem to get smaller with distance. Actually, there is a name for this phenomenon: linear gradation.

Once you reach the end of the red diagonal, continue by drawing a second diagonal (here in black). Make sure your pencil is sharp as your drawing will lack accuracy if the line is too thick.

If you rotate the picture by 90%, you see straightaway that the same technique can be used to set up a vertical construction. Forget the vertical horizon line and put in a new horizontal line at the level of the vanishing point. You can use each vertical line, for example, to situate windows at equal distances on a façade. The receding lines will help ensure that the height of the windows subject to linear degradation [?] do in fact reflect their “real” constant height and alignment.

Don’t forget that you are still in frontal view here. The facade with the windows is parallel to the visual ray.

The diagonals also recede. Let’s go back to the original paving. You arbitrarily chose the height of a first horizontal in order to define the depth of the first row. You then drew a diagonal so as to establish the apparent depth of the succeeding rows. If you had gone to the trouble of drawing all the diagonals, you would have noticed that they all culminated at the same point – which just so happens to be situated on the horizon.

And if you had traced in the other diagonals of your squares (here in red), you would have observed the same phenomenon, but this time on the other side of the principal point.

The culmination point of the red diagonals is not visible on this drawing; it is prolonged beyond the paper on which you are drawing, but it exists just the same. This happens from time to time and you will see that there are some good ways of getting round this problem. But you didn’t construct your diagonals starting from this point. You created this point by drawing the diagonals of the slabs. We are indeed talking about the vanishing points of the diagonals – whether they are visible or not. This brings us into the world of angular perspective. Here’s why.

Look at this plan view of a grid made up of 4 squares. The diagonals of these 4 squares form another differently-oriented square.

You will take your viewpoint into account in putting this into perspective. You notice that the yellow square is presented by an angle and not by a side. Since you know the vanishing point of the diagonals of the white squares, you also know the vanishing point of the sides of the yellow square. If the yellow square were the base of a cube, it would also be the vanishing point of the sides of this cube.

Here is the yellow square in angular perspective. All we had to do was to place the 4 corners of the square on the grid in perspective. These corners are easy to spot on the plan view. Now it’s child’s play to mount the box resting on the yellow square.

Raise the vertical lines starting from the four corners of the square. This will give you the edges of the box. Choose a height that suits you on the edge in the foreground (indicated by a little black arrow).

Join this point to the vanishing point situated on the horizon line. This will give you a first vertical side in perspective. What’s the next step? Closing the top of the box.

If you were able to access the second vanishing point, you could place the green line receding towards this point, followed by the blue line, and then close the box with the orange line. Since this is unfortunately not the case, we will have to find a way round the problem.

The diagonal of the yellow square is parallel to your visual ray, and therefore recedes towards the principal point. This means that we can draw the diagonal of the lid.

All we need to do is draw a straight line from the point indicated by the black arrow to the principal vanishing point. This transfers the height chosen at the beginning onto the edge situated in the background.

This point is indicated by the red arrow.

Lastly, trace the blue line starting from the vanishing point on the left and intersecting the angle of the lid at the red arrow. Just link up the remaining corners by two lines (marked in orange here) and the top of the box is completed.

If you want to be sure of making a perfect cube, then instead of using your eye to choose a height, you should resort to the same sort of technique as that used when tracing an exact square in the example of the checkerboard floor. The case we have just seen is perhaps a little too good to be true. The square was set at an angle of exactly 45° towards you.

The angle could of course be different and in this case the diagonals are neither perpendicular nor parallel to your visual ray. As a result, the vanishing points are not at an equal distance from the principal point indicated by the red arrow. One is much nearer this point than the other.

Similarly, the diagonal lines, here marked in red, converge towards their vanishing points at very unequal distances from the principal point. The one on the left even goes off the paper. But don’t be impressed by that; the construction is the same as before.

Now it’s just a question of discovering the aerial perspective and then you have virtually everything you need to draw a volume, regardless of the viewpoint chosen.

Aerial perspective is encountered in frontal or angular view and is based on adding a viewpoint, making a total of two in frontal view and three in angular view.

Let’s now look at this square in angular perspective. Imagine that, instead of being a slab, this square in perspective is the roof of a building, perhaps even a skyscraper, which you are approaching in a helicopter. Just to show you how easy it is, this example starts from an angular view with, as it happens, vanishing points which are not equidistant from the principal point. This illustration shows the effect given by this angular perspective. You can sense that there’s something a bit rickety about this building; it even seems to be wider at the base than at the top. This is not the case of course, but your eye deceives you. The corner of the building also seems to make an angle of less than 90% at the bottom, as though distorted by a wide-angle lens. This, as you now know, is due to linear degradation [?]. The square forming the base of the building is far removed from the terrace. It should be smaller whereas in fact it is larger!

To get everything back to normal we need to add a vanishing point.

You will place this vanishing point on the vertical line, here marked in green, which intersects the horizon at the principal point. The vertical lines will recede towards this point. This time we appear to be looking down on the building and it is “deformed” in an acceptable way. Choisir les points de fuite What still gives it a curious look is the choice of the two vanishing points on the horizon. They are too close together. This is why you will sometimes need to work on vanishing points placed outside the confines of the sheet of paper.

One way to solve the problem is to lay your paper on a fairly large surface such as a drawing board. This allows you to mark the vanishing points with a pin, for example. But there will always be times when the vanishing points would have to be positioned several metres away from the principal point and then of course the board would be too small. In such cases you will want to use the method described below.

Suppose you wanted to set up a fairly accurate parallelpipedal volume in angular view – a building, for example. But the scale is of no importance: the volume could just as well be a box of matches. The principle is the same. Draw your horizon line and then a vertical which you can place wherever you like. This vertical line will be the vertical edge nearest to you. If you are drawing a building, for example, it could constitute the corner of the street.

Starting from this vertical line, draw two receding lines, here marked in red, culminating outside your sheet of paper. The two vanishing points of this angular view do not need to be equidistant from the vertical line, and anyway you can’t see them. The angles formed by these receding lines with the vertical will be fairly open since the vanishing point is remote. This means you probably won’t be bothered by that unpleasant distortion that sometimes occurs when two vanishing points are too close together. Now put in a point to mark the height you want on the vertical. You see straightaway that you can’t join this point with the vanishing points because you don’t know its position.

Measure the height between the lower angle and the horizon on the vertical line. Divide it by the number you want and trace in the divisions. Make sure that the divisions are simple. In this case, the height is 50 mm so naturally you would want to take five divisions of 1 cm each. Add similar divisions above the horizon so as to exceed the height of the defined edge.

Using a vertically-placed graduated ruler, look for the place where you can easily draw five other divisions which would be a sub-multiple of the previous divisions. Here you can look for the point at which the height between the horizon and the left vanishing point comes to 25 mm. This corresponds to 5 divisions of 5 mm.

Now trace your 5 mm divisions on either side of the horizon, once again exceeding above and below.

This operation would be impossible on the right since the lines are more than 25 mm long at the point where they are interrupted. So in this case you will seek a vertical nearer the centre which is a multiple of 5, for example 5 times 7 mm, which would mean looking for a vertical giving a distance of 35 mm between the two lines.

Now join up your divisions and you have a grid to guide you.

You won’t need a vanishing point any more. Trusting to your eye, just position your receding lines between two lines on the grid. The denser the grid, the more accurate your lines will be.

If the line you draw is poorly directed (as in this bad example), it will not seem to follow the network of lines making up the grid. This will be immediately apparent to the eye.

It’s now time to sum up what you have learnt about perspective.

You have observed that:

- The image of objects received by the eye is subject to distortion
- By rigorously reproducing these distortions we give depth and perspective to a drawing
- The horizon rises with the observer
- There are three perspectives: frontal, angular and aerial
- The distortion of an object stems from the angle by which your visual ray reaches it
- This angle and the observer’s height define the point of view
- You will use one of the three perspectives depending on the point of view chosen

What else do we need to observe or learn? Actually, quite a lot, but this will be easy thanks to what you already know. In particular, you have learnt how to draw a square and boxes. We deliberately started with that because in fact all geometric forms are contained in the box and can be built up in a cube or a parallelepiped.