### 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.