Measuring in depth

Posts 1.5 m tall standing 4, 6 and 8 m from the eye are drawn 37.5, 25 and 18.75 cm tall on glass 1 m away: equal steps in the room, shrinking steps on the paper.

Equal spaces are not equal on paper

Vanishing points tell you which way lines run, but not how far along them to stop. That is the hard part of perspective drawing: placing the second post of a fence, the third window of a building, the fifth row of floor tiles. Equal spaces in the room do not stay equal in the drawing. The drawn-height rule from unit one shows by how much. A post of height H at distance D is drawn H × d ÷ D tall. With the glass 1 m from the eye, posts 1.5 m tall at 4, 6 and 8 m come out 37.5 cm, 25 cm and 18.75 cm tall. The posts are 2 m apart in the room, but on the paper the drop from the first to the second is 12.5 cm and from the second to the third only 6.25 cm.

You do not have to calculate every drawing. Artists use a few constructions that put the spaces in the right place with nothing but a ruler. They all rest on one fact: straight lines stay straight in perspective, and lines that cross in the room still cross in the picture. Diagonals are the most useful tool, because the diagonals of a rectangle cross at its exact centre, in the room and in the drawing.

Finding the middle

Suppose you have drawn a wall in perspective and want a door in the exact middle. Measuring with a ruler on the paper gives the wrong answer, because the far half is drawn shorter than the near half. Instead, draw the two diagonals of the wall, corner to corner. Where they cross is the true centre of the wall. A vertical line through that crossing divides the wall into two halves that are equal in the room, even though they look unequal on the paper. Repeat the trick on each half to find quarters, then eighths.

Repeating a space

The same idea works in reverse to repeat a space. Draw the first two posts of a fence. Join the half-height points of the posts with a line to the vanishing point. Now draw a line from the top of the first post through the middle of the second post, and continue it until it hits the line along the bottoms of the posts. That is where the third post stands. In the room, the two posts and the new one form two equal rectangles whose diagonal crosses the middle line exactly halfway, so the step from the second post to the third is the same as from the first to the second. Repeat as often as you need.

Alberti's tiled floor is the most powerful version of this idea. Once a grid of square tiles is drawn correctly, checked with the distance point from unit three, it works as a ruler for the whole picture: every tile is 1 m by 1 m, so you can read off the size and position of anything standing on it. Painters after Alberti drew so many such floors that pictures of this type are called pavimento pictures, from the Italian for floor. The grid also handles curves. Alberti showed how to draw a circle on the floor: mark where the circle crosses the grid lines on a plan, find the same grid crossings in the perspective floor, and join them. The circle comes out as an ellipse.