Learning perspective and view points

The nine typical viewpoints

There are nine typical views which will help you many a time. Try to fix these views in your mind by practising as much as you can, first by referring to the model and then gradually on your own.

The 9 typical views of frontal perspective. There are 9 boxes in front of both of these figures. Note that the numbers, from one illustration to the other, are a mirror reflection since the viewpoint represented on the right is that of one or other of the figures facing us on the left. Boxes 4 to 6 straddle the horizon line, so their upper and lower sides will not be visible.

The aspect and number of sides presented by the boxes will of course be the same for the two spectators. Only the floor plan will be seen differently, but this is not what concerns us here. Up to now we have been looking at boxes made of opaque material and with only one, two or three sides visible to us.

But if the boxes were to become transparent, you would be able to see all the other sides and represent them by drawing their edges, even if they are concealed.

You will note that the observations we made in the three preceding pictures remain valid for the sides which were hidden:

• The vertical lines of the boxes remain vertical in the drawing
• The horizontal lines perpendicular to the visual ray remain horizontal
• The horizontal lines parallel to the visual ray all converge on the principal point

If you look at the central box, you see very clearly that the side perpendicular to your visual ray - the front side - is much larger than the rear side. This is an optical phenomenon that one comes across time after time.

This observation seems obvious enough but please remember one very important notion which can be summarised as follows:

The further away an object (or shape), the smaller its size as it appears to us.

Why is this the case?

An illuminated object emits rays in all directions. Each point of the object emits numerous rays, some of which move towards our eyes. Only the rays which reach the crystalline lens are "seen" by our eye. Those that do attain the crystalline lens pass through it and are deflected. The optical properties of the lenses cause all the rays to converge on a single point on the retina. So we can simplify our diagrams by concentrating solely on the ray which passes the focus of the crystalline lens (grey star) and is not deflected. In this way we can determine the place where the image of a point is projected on the retina.

According to this principle, when an object is in front of your eyes, it is "printed" on your retina upside down, like this red triangle.

Each reflected point of an object will follow a trajectory equivalent to the point passing through the focus of the crystalline lens.

The coloured stars dotted around this drawing will follow the same optical principle before appearing on the retina. Thus, as seen by your eye, the red star will be below the blue star and the yellow star above it, whereas in reality it is the other way round.

If several points are situated along the same "path", they will be situated one behind the other as far as your eye is concerned, and you will see only one of them - the point nearest to you.

The points reflected by the objects constitute in their entirety the "picture" projected on your retina. Look at this illustration. The two grey bars marked respectively by a red star and a green star are two objects of the same size situated at two different distances from an observer. They could just as easily represent two vertical poles as a slanting view of two railway sleepers.

By looking at the trajectories of the luminous rays reaching the retina, we understand very well that the object furthest away is projected in a smaller size on the retina than the nearest object.

The apparently diminishing size of the more distant objects is an additional observation which will help you make a convincing representation of reality.

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.

Construction of the pyramid

Construction of basic forms in perspective

Now that you know how to construct a square in perspective, you can also build up all sorts of simple forms which will come in very useful.

Let's start with the pyramid.

Fig. 1 A pyramid is constructed on the base of a square or a rectangle placed on the ground. This quadrilateral must first of all be put into perspective. For the sake of simplicity, I'm going to show you this with a square but it works just as well with a rectangle.

Begin by drawing the diagonals of the quadrilateral in order to define the centre. Fig. 2 Prolong one of the diagonals towards the horizon line and materialise the vanishing point of this diagonal.

Fig. 3 Now draw a vertical at the intersection of the diagonals. This is the centre of the square, seen in perspective. Next, we need to define the height of the apex of the pyramid on this vertical axis. Imagine that this pyramid fits exactly in a cube. In this case, the height of the apex will be equal to the length of a side of a square. But the apex is not in the same plane. It's further away and will therefore be smaller. But how are you to find its height in perspective? You are going to use a side of the square to find the solution. The side nearest to you is free from all perspective deformation so this is the one to choose. After that it's plain sailing in this case since you are in frontal perspective.

Fig. 4 Raise a vertical line in the angle on the left. Use a compass to copy this measurement on an angle or measure it yourself. This gives you a new point.

Fig. 5 Trace the vanishing line running from this point to the vanishing point of the diagonals. This line bisects the vertical leading to the perspective centre of the square at a point which will be the apex of the pyramid, seen in perspective.

Fig. 6 Now all you have to do to finish off the construction is to join up this apex with the angles of the square.

You notice that if the base square was a circle it would be quite easy to draw a cylinder and of course a cone.

Construction of the cone

Fig. 1 Begin by drawing the square in perspective. This time, if we want the cone to have a circular and not an oval base, the base will need to be a square. [?] Draw the diagonals in order to find the centre of the square, and also put in the lines perpendicular to the two sides in the middle as you will need them in a minute. [J'ai besoin des figures ici ]

Fig.2 Draw the ellipse fitting into the square. Notice that the circle touches the square at four points and that it passes at roughly two thirds of the length of each diagonal.

Fig. 3 Raise a vertical in the middle of the left side of the square.

Fig. 4 Insert the diameter of the circle on the left vertical to obtain the height of the apex which would inscribe the cone in a perfect cube. Move this height towards the centre by inserting it on the diameter perpendicular to the visual ray. Put in a point to mark the apex which is now at the vertical of the perspective centre of the circle. Note the that the perspective centre of the circle is not the geometric centre of the ellipse.

Fig. 5 Now draw the tangents to the ellipse which pass by the apex.

Fig. 6 You can add in as many straight lines as you like, starting from the apex and running to the circle situated at the base of the cone. In this way you will be able to divide up the slices of "cake" in a cone.

Fig .7 If you begin your construction on a square seen in frontal perspective, watch out for aberrations due to a wrong choice of position. The most common mistake is failing to switch to angular perspective once the square has moved too much one side. [?]

Think of it this way: when you look straight in front of you, you can't see an object which is both close and to one side of you. To do so, you have to turn your head and the vanishing point changes.

The resulting deformation is what makes this drawing awkward to the eye.

Fig. 8 This cone is upside down. Here, the precise height of the apex is not given but we may take it that the apex is located on a downward vertical. Notice that there is no prolongation of a diagonal because the square was not exactly at an angle of 45 degrees. From this you may conclude that the two vanishing points are not equidistant from the centre of the cone.

Construction of the cylinder

Construction of an upright cylinder

Fig. 1 (cyl_09) We start off in exactly the same way as with the cone. Draw a square, its diagonals and its bisectors.

Fig. 2 Raise a vertical on the symmetry axis of the figure [?], in other words on the point of the circle at a tangent to the front side of the base square. Put in a point to mark the height of the square which is to frame the circle forming the top of the cylinder.

Fig. 3 Start from this point to build the cube which will encompass the cylinder.

Fig. 4 Don't forget to draw the verticals, here marked in green.

Fig. 5 Place the second ellipse which will be tangential in 4 points and will bisect the diagonals at about two thirds of their length.

Fig. 6 Bring up the first vertical that will complete the right side of the cylinder.

Fig.7 cyl_15 Proceed likewise for the other side.

cyl_17 Look at these two cylinders. One is transparent [?], the other is shaded. On the transparent cylinder, the perspective seems to be wrong whereas the shaded cylinder doesn't give this impression. And yet is strictly traced on the same construction. One has to admit that one doesn't often get the chance to see a cylinder deformed like this. You will already have noticed that this optical phenomenon is created by the insufficient space between the vanishing points.

cyl_18 Here you see that if the vanishing points were to move further apart, drawing towards [?] the apices of this cluster of lines [?], the square would stretch out and the ellipse would be flattened out. Without changing the height in space, the circle of the cylinder base would no longer seem to tip over. [?]

cyl_00 This is what happened in this illustration. The general view shows that the space between the two vanishing points has widened and from the enlargement you can see that everything is better balanced and no longer offends our eye or our sense of logic.

Construction of a cylinder lying on its side cyl_16.jpg

A cylinder on its side is not so very different from a cylinder standing upright. Things get a bit more difficult when you draw several cylinders resting on each other [?] and placed on the same plane.

Fig. 1 (cylindre_02.jpg) This time you start off in angular perspective. The see-saw [?] we saw a moment ago could occur here since, for reasons of presentation, I have moved the two vanishing points closer together than I should really have done. These should be considered as symbolic diagrams and not as drawings designed to seduce the eye through their aesthetic qualities.

On the base square (vertical this time), draw the ellipse forming the left extremity of the cylinder.

Fig. 2 Draw all the vanishing lines required to construct a glass box encompassing the cylinder.

Fig. 3 Choose a depth by cutting the vanishing lines vertically. Do you see where the cylinder will contact this line?

Fig. 4 Draw the vanishing lines travelling towards the left vanishing point from the right corners of this glass cube.

Fig. 5 Finish the glass cube.

Fig. 6 Draw the diagonals of the furthest square.

Fig .7 Now place the ellipse in its appropriate position.

Fig. 8 Draw the two vanishing lines (in blue) showing the external contours of the cylinder. Notice, too, that the points situated on the diagonals are aligned and lead to the vanishing point.

Fig. 9 Put in as many as you like: they will come in handy for shading and give you an idea of the volume. They will also help you cut up those "slices of cake".

The perspective of prisms

There is one thing of particular interest about prisms: they have at least one inclined plane. This makes it the ideal basic form for constructing sloping streets, the roofs of houses and the volumes of the family of inclined boxes. One example among many would be the lid of a half-open suitcase. Others might include a door ajar, a staircase and banisters, a plane at take-off, the screen of a laptop computer and so on.

And what about this raft? Not bad as an example of an inclined plane!

Fig. 1 … prisme_14.jpg

The prism is inscribed in a regular parallelepiped. Let's see how you get on with constructing a box.

Begin by choosing a vanishing point on the left and another one on the right, on the horizon line. Work out the position of the two blue points forming the edge nearest to you by freely choosing the position of the former and then choosing the height separating it from the latter, making sure nonetheless that it rests on the vertical passing by the former point. Now draw the vertical which determines the depth of the parallelepiped.

Fig. 2 Join the upper corners located at the points marked in blue to the appropriate vanishing point, on the right.

Fig. 3 Continue the glass parallelepiped by drawing the rear side as well (as you see here).

Fig. 4 Now choose the length of the box as you please by tracing a vertical, here marked in red.

Fig. 5 Join the points where this vertical and the receding lines intersect to the vanishing point situated on the left. The box is now almost closed.

Fig. 6 The only thing missing now is the vertical line rising from the corner the furthest away from you. So far, so good. But now you are going to cut this parallelepiped into two and transform it into a prism.

Fig . 7 Just put in the diagonals of the two sides receding on the left. The further diagonal is green and the one nearer to you is black. Do you notice anything special about this about this drawing? Those diagonals you have just drawn seem to be trying to converge towards the top. At any rate they are certainly not parallel.

Fig. 8 In this figure, I have prolonged them for you and you can see that they do in fact converge at a point, here marked in orange.

What's rather more curious is the fact that this point is exactly at the vertical of the vanishing point. This is not a coincidence; it is a law of perspective.

There's no need to formulate this law in detail here. The thing to remember is that parallel diagonals recede towards a single point and that this point is situated on a line perpendicular to the horizon line. Besides if you look at this drawing by turning to your head to the left, you might even believe [= vous pouvez admettre ?] that you have drawn another figure with the dotted line as the horizon line.

Fig. 9 And now here's the inclined surface of the definitive prism. When we fill it with colour, we can already see the half-roof of a house beginning to take shape. Now's the time to draw the other half of the roof and to construct it correctly.

Fig. 10 To make the figure easier to interpret I have made the tiles transparent and only left in the frame of the half-roof. Put in the diagonals which will allow you to define the centre of the rectangle corresponding to the floor of the "attic" situated under the half-roof.

Fig. 11 Run another vanishing line past the junction of the diagonals (marked here by the blue point). This vanishing line will bisect the square laid on the ground at a green point and divide it (the square) into two equal rectangles.

Fig. 12 (perspective_prisme_26.jpg) If you draw a diagonal from the further of the two rectangles, it will naturally pass by the green point and continue to the orange point which gives the exact position of the angle of the missing half-roof. In point of fact, the diagonal of the half-square lying on the ground transposes the depth of the square on the vanishing line situated on the left. (perspective_prisme_12b)

Let's go back to the principle at work in this illustration. In 1, you see a black rectangle bisected by a horizontal passing by its centre (obtained form the diagonals). When you draw the diagonal of the upper semi-rectangle you find a point, here marked in orange, which exactly transposes the length of the initial rectangle on the lower line.

In 2, you have the same construction in different proportions. Lastly, in 3, you see that the construction, once put into perspective, remains valid for finding equal measurements along a vanishing line.

Fig. 13 (perspective_prisme_27.jpg) You can put in the edge of the missing half-roof.

Fig. 14 Starting from the angle of the roof, draw another vanishing line (here marked in red) to determine the furthest angle of the roof. This is not an angle that you would be able to see but it helps you to notice something else. If you put in the missing side you see that the two sloping sides of the roof seem to join up at a point near the bottom of the drawing.

Fig. 15 Indeed they meet at a point which is perfectly aligned with the two preceding points, on the same vertical. Your roof is correct and all you have to do is complete it.

The perspective machine

You now have a fairly thorough understanding of perspective. But really there's no end to perspective; one can always make things more complicated or more weird! I would now like to build up a "perspective machine" with you - a machine that will extricate you from all sorts of tricky situations. If you follow the different phases that go into the construction of this machine with a little bit of attention, you will be able to create a brand new one whenever the occasion arises.

It's extraordinary how many artists abandon a particular subject simply because they don't feel up to solving a problem of perspective. But there really is no reason why you should allow yourself to be intimidated in this way. Please believe me when I say that it's very easy to guard against the sort of difficulties we are talking about. Besides, I'm sure you have noticed how a drawing with a well-rendered perspective effect attracts the observer without his realising how this effect is obtained. The fact is that perspective gives such an authentic touch to your drawing that you have already won the observer's heart and mind!

So rid yourself of those preconceived ideas and take it from me, once and for all, that perspective is simple! You are going to begin by drawing a dice in perspective. Then, following the same principle, you are going to see how we draw an entire building without putting a foot wrong as far as perspective is concerned. Imagine that, at some time in the future, you need to draw a room with furniture, a door, a fireplace, a chandelier and various objects scattered about the room. Thanks to the perspective machine, you will be able to handle all this without difficulty.

From the simplest to the most complex challenge

Fig. 1 ((fig ajout 2)) Look at these three sides of the same dice. The green side is seen from above, the blue one from the front and the yellow one from the side. You know that, in reality, that these sides touch each other by one of their edges and from an angle. You know, too, of course that a dice shows several sides at once but that, in perspective, one side at the most can be square, the others undergoing perspective deformations. All this is quite straightforward but I insist on the point because I want to draw one thing in particular to your attention.

On this drawing you see several views of the same object. To represent the real object you will have to make an abstraction: you will have to accept that you can see the object from three sides at once. This in fact is what you see on an architect's plan or on technical drawings intended for use in manufacturing.

You will have to make the same sort of concession, the same "suspension of disbelief" with the perspective machine, but you will very soon get used to the idea.

Fig. 2 Take a look at the green side for a moment, the one seen from above. It is green like a square field seen from an aeroplane. The very colour is enough to remind you that you are looking at the top of the dice. Here's the first piece of the machine: the green line running up against the corner of the dice. This is the line representing the picture of which you only see the section. This picture is a glass plane interposed between your eye and the object you are putting into perspective. Its position and its dimensions define the positioning of your drawing.

The thin dotted line is perpendicular to this picture and represents your visual ray. The ray is always perpendicular to the picture, from which you may already conclude that you have chosen an angular view on this dice, with your regard directed exactly on the vertical edge facing you. But you still don't know whether you are going to view the dice from above or below. This is something you will define later on, along with the distance from which you look at it.

Fig. 3 To make the most of your perspective machine you are going to rotate it so that your picture is well and truly parallel to the edge of your sheet of paper.

Fig. 4 Now draw the two lines which prolong the sides of the dice touching the plane of the picture. Put in a point to show your distance from the subject. This point (here marked by a blue star) is necessarily on the visual ray, since it is its origin. This is the point of view.

Finally, draw a parallel at the first of these two diagonal lines. This line, here in red, will pass by the point of view.

Fig. 5 Trace the second line starting from the point of view and parallel to the other side of the dice.

Fig. 6 The position of the red points shows the place where the vanishing points will be situated on the picture plane. These red points are not the vanishing points but the place where they are projected onto the picture plane. They are in fact much further behind, at the horizon.

Fig. 7 The idea is now to turn things round a little, the purpose being to rearrange the picture plane so that it is in the same plane as your drawing board. As soon as you do this, you will have two different views co-existing on your sheet of paper: the outline view and the elevation view which you are going to lay flat.

To give yourself room to do this, you are going to begin by advancing the picture plane towards you as far as the red line, using the arrows as a guide. The junctions of the vanishing lines and the picture plane are marked by crosses which will also be moved in the same way.

Fig 8 This movement takes with it the two points you marked. And the same applies to the blue point marking the vertical edge of the dice resting against the picture plane. As for the distance to adopt for this translation of the picture plane, just make sure your choice fits in with your particular project. You will find out in a minute why this distance is a matter of choice and has no impact on the final result.

Fig. 9 Just imagine you have done the following things: first, you have picked up in both hands a glass pane placed vertically on the table on which you were looking at your dice; next, you have drawn it near to you and then laid it flat on the table by swivelling it forwards and keeping your thumbs on the top. Before carrying out any of the above actions, you have taken care to put in a few points and lines which will come in useful later on.

The glass is now flat on the table but what you had previously drawn when it was vertical remains marked. However, notice the blue line (the horizon line) and of course the two crosses showing the vanishing points on each side.

If you moved this line up or down it wouldn't change anything. But it's best not to shift it sideways as this would distort things and above all make life difficult for yourself to no useful purpose But that's not all!.

The red frame shows the sides of this glass, that is to say the picture plane once it is laid flat on your drawing board.

Once again, I could have made it higher or wider without this making any difference to the whole, and you of course could well decide to make a different choice.

The only things which are really important are the red line at the bottom and the blue horizon line. Here's why : It is at this juncture that you will at last decide on the relative positions of two or three things - and this time your decision will have repercussions on the end result.

Fig. 9 (continued) The bottom line of the red frame is the bottom line of the picture plane, and is referred to as the earth line. The name is well chosen since this is the line on which the dice rests when it is seen face on.

The blue horizon line, for its part, may change position in relation to the earth line and, depending on whether you raise or lower this horizon line you will increase or diminish the height from which it is seen. We will go into this point in a little greater detail further on.

Your perspective machine is now ready for use!

You can already draw the dark blue vertical on this earth line. This vertical represents the edge of the dice at the place where it met the glass. For the moment you have defined the position but not yet the real height. We will come to this in a moment.

Fig ajout4 All you have to do is put in the real height of the dice seen face on. This will be exactly right since, as far as the picture plane is concerned, heights do not get any smaller as the distance increases.

Fig ajout 5 Now it's time to start up the machine! Start by drawing the four oblique lines joining the height of the dice to the two vanishing points. You begin to sense the cube taking shape in perspective. This is the time to see what happens if one increases the distance separating the earth line and the horizon line.

Fig ajout 6 Here, the height of the red frame has been extended with the effect of moving the two lines further away. Do you see how this influences the perspective ? This time you are situated much higher up.

Before we continue to construct this perspective, I suggest that we look at the perspective machine from another angle.

Fig ajout 1 The table or your drawing board is represented in beige. You can see the sole marks on the floor. These show where the spectator is standing while the blue star represents the point of view. The two blue lines are parallel to the sides of the dice and recede towards the horizon. They bisect the picture plane at the two points marked with a cross. These crosses are the horizon line of the picture plane. The orange star is the point where the principal visual ray crosses the picture plane. The height of the dice has been put on the vertical line.

Fig. 13 From this angle look again at the movement and rotation of the picture plane and lay it flat on the drawing board. Now pull the picture plane towards you.

Fig. 14 Now the picture plane is rotated by 90°. Your identification marks follow this movement.

Fig. 15 Now lower the plane so that it lies on the drawing board.

Fig. 16 Here we see the same set of operations, this time with the diagram "cleaned up" so that it's easier to read. The green arrows show the three successive movements of the corner of the picture plane. Make a mental note of this movement and you will save yourself oceans of time when dealing with perspective.

Fig ajout 5 (deuxième affichage) At this point go back to the construction. You can sense the two sides of the dice receding towards the horizon. You must now be able to define the actual depth. You know that the depth is equal to the height on the real dice, but since its sides recede in perspective, you will have to attack this question in a different way. Here's how.

Fig. 10 Take the external angles of the dice and join them up to the point of view. Each of these lines, which are here marked in black, bisect the picture plane seen from above at a precise point. Since the red frame is the picture plane laid flat, you will draw the vertical parallel lines from these points. You see that they bisect the receding sides at the place which corresponds to their true length seen in perspective.

Fig. 17 You can now clean up your construction even more. The dotted red line is the top of the picture plane. You can now see that you didn't really need it all. Don't draw it any more from now on and save yourself a lot of unnecessary work.

Fig. 17b Finish the top of the dice, using the vanishing points of each side. You have a perfect construction.

Fig. 18 You can use this perspective machine to introduce the most complex of constructions. Look at this ensemble. It is made up of three buildings which are offset or staggered, as you can see from this outline view. Depending on the building, the roofs have two or four slopes , the last one being flat with a terrace on it. The doors and windows are placed at irregular intervals.

How are you going to do this without running into difficulties? This is what we are going to find out in the next step.

Fig. 19 Go back to the perspective machine method. Insert the building plane in the top, choose the angle from which you want to look at it and put in the point of view. Draw the parallels to the walls of the first building and move forward the vanishing points on the horizon line once it has been laid flat on your drawing board. You still need to put in an earth line.

Fig. 20 Draw the earth line and position the building seen from the front on the right side. Transpose the height of the angle of the first building on the axis of the principal visual ray. Notice that this is the same configuration as in the dice drawing. In actual fact you could just as easily place the building seen in outline not only more to the left or the right but also at an angle further removed from 45°. This would work just as well.

Fig. 21 Draw the walls of the first building in perspective by joining its angle to the vanishing points.

Fig. 22 Working in the usual way, you can now put in the points corresponding to the angles of the lower part of the building with the exact proportions in perspective. Remember that it is at the yellow line - that is, when crossing the picture plane - that each point ceases to approach the point of view and is transferred in a straight line onto the perspective view. The drawing is already getting a bit cluttered and you can tidy things up by removing certain construction lines.

Fig. 23 Here is a cleaner view of the picture. You can understand how the walls of the three buildings are built up.

Fig. 24 Now put in the windows by using the view of the side of the building on the left of the drawing.. Check out the heights of the doors, the windows and the spaces between them. The yellow horizontals will help you transpose these heights onto the angle of the first building. In order to work out the heights, send back these points towards the vanishing point, as here with the dotted black lines. These doors and windows are aligned between two verticals (determined in the same way as before).

Fig. 25 Proceed in the same way for the other side. Look for the position of the verticals wherever you need to put in openings. In this case you can only place the windows in the first building. For the other buildings you will need to resort to a little "trick of the trade": snaking.

Fig 26 By using the left and right vanishing points alternately you can wind or "snake" a measurement taken on the angle serving as your reference through to a point which was not itself equipped to allow you to make a direct construction.

Fig. 27 Here is a more detailed view of this technique.

Fig. 18 ((a nouveau)) You will have noticed that the top of the sloping roof was level with the base of the highest windows , and that its gutter was level with the base of the windows on the intermediate floor.

Fig 29 ou 28 +29 a resouder So you won't have any trouble in placing the gutter side, highlighted here between the two green points at the bottom.

Fig 28 +29 a resouder To find the apex of the roof which will allow you to draw its correct slope, begin by tracing the vertical passing by this apex. This is the green vertical running down from the yellow line. You have already done this fro the other vertical markers of the windows. There's no difference here and no doubt you are beginning to get used to locating where a given point of the object seen in outline crosses the picture plane on the line linking it to the point of view. But you still need to find at what height the apex of the gable will be placed on the green vertical.

We noticed a moment ago that the top of the height of the roof was exactly under the windows of the top floor. The arrowed orange line gives you the height on the angle of the building. The pink arrows snake their way along the front up to a green point. Finally, the little green arrow deviates this point to the place you want along a line returning to the vanishing point on the left. You can now draw the first line which expresses the slope of the roof, but you can also draw the roofing from this point.

Fig. 30 And here is the figure with these two lines drawn in. You notice that the roof slope line meets the vertical running up from the left vanishing point at a precise point. As you will remember, there is a name for the point where the vertical meets the edge of the roof: the vanishing point of the inclined plane of the roof.

Fig. 31 So all you have to do is to draw the last side of the roof, starting from this point. Your roof is finished! Why not now try your hand at drawing the pyramidal roof at the top of the building unit nearest to you? How are you going to go about this task? Well, there's no need to draw the vertical along which the top of the pyramid passes; it's the same as the vertical of the angle of the building. This is what I suggest you do. Start by prolonging the building upwards and then by bisecting it at the apex of the pyramid. After that, it's pretty plain sailing.

Fig. 32 This will give you a square seen from below since it arrives above the horizon line. Notice how the additional " floor " has been measured on the front view situated on the right. After that it has been constructed in glass, using the vanishing points.

Fig. 33 Lastly, the diagonals of the square have been put in, giving this green point which is the top of the roof.

Fig 34 Get rid of the construction lines, points, etc. you don't need, and finish off the roof. That leaves us with the last rooftop terrace of the most distant of the three buildings. You could use the snaking technique but you would be left stranded above the middle building. There is a simpler way of attacking the problem. You are going to extend the building temporarily.

Fig. 35 Start by transferring the height of the terrace to the angle of the building as shown by the orange line culminating at the pink point. Draw the vanishing lines culminating at the black point and both passing at the foot of the outer buildings. This creates a perspective link between the two and allows you to continue with your construction..

Fig. 35b At the vertical of the black point draw a line (here marked in green) upwards simulating the most advanced angle of the box encompassing the construction as a whole.

If you start from the vanishing point on the left and pass by the pink point you will bisect the green vertical at the exact height of the terrace, but much too far in front. Transpose this point with a vanishing line towards the vanishing point on the right, and you will find the height of this terrace on the right building.