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.