The picture plane: drawing on a window

A 2 m door 6 m from your eye, traced on glass 1.5 m from your eye, comes out exactly 50 cm tall.

Drawing on a window

Stand at a window, close one eye and keep your head perfectly still. Now take a marker and trace the edges of a building across the street directly onto the glass. When you step back, the lines on the glass form a drawing that looks deep: the near corner is large, the far end of the street is small, and the edges of the road lean towards each other. You have made a perspective drawing without knowing a single rule. The rules of perspective are a way of getting the same result on paper, without the window.

The Italian architect and writer Leon Battista Alberti described exactly this set-up in his book On Painting of 1435. Before drawing anything he marked a rectangle on the surface and said it worked for him as an open window through which the scene is seen. Artists still call that imaginary window the picture plane. In the 3D model it is the framed sheet of glass. Behind it is a room with a tiled floor and three boxes; in front of it is a small dark sphere, the eye point. Everything in this course follows from where those three things are: the eye, the glass and the objects.

One eye, one point

Why one eye, and why keep still? Light travels in straight lines. Every point on the building sends light along a straight path into your eye, and the place where that path crosses the glass is where you put your mark. If you move your head, all the paths change and the marks no longer line up with the building. A perspective drawing is correct for one viewing point only. Alberti put it in geometric terms: the lines from the eye to everything it sees form a visual pyramid with its tip at the eye, and a painting is a slice through that pyramid at a chosen distance.

Press Projection lines in the model to see the slice being made. Eight red lines leave the eye and end at the corners of the orange cube. Each one pierces the glass at a red dot. Join the dots in the same order as the corners are joined on the real cube and you get the cube's drawing. That is all a perspective drawing is: for every point you care about, draw the line to the eye and mark where it crosses the picture plane. The faint lines from the eye to the four corners of the glass show the edges of the visual pyramid; anything outside them does not reach the picture.

Why far things are drawn small

The side view above shows why distance shrinks things. The sight lines to the top and bottom of an object of height H, standing a distance D from the eye, form a tall thin triangle. The picture plane, a distance d from the eye, cuts a smaller copy of that triangle. Similar triangles keep the same proportions, so the drawn height h' satisfies h' ÷ d = H ÷ D, which gives h' = H × d ÷ D. A door 2 m tall and 6 m away, drawn on glass 1.5 m from the eye, comes out 2 × 1.5 ÷ 6 = 0.5 m tall. Move the door to 12 m and it is drawn 25 cm tall: double the distance and you halve the size. Leonardo da Vinci wrote the same rule in his notes, measured in braccia, an Italian unit named after the arm and a little over half a metre long: with the glass one braccio from the eye, an object 4 braccia away loses three quarters of its height in the picture, one 8 braccia away loses seven eighths, and one 16 braccia away loses fifteen sixteenths.

The same rule explains why parallel lines seem to meet. The two rails of a straight railway stay 1.435 m apart all the way to the horizon, but in the picture that fixed gap is divided by a distance that keeps growing. The drawn gap gets smaller and smaller and approaches zero. The point where the two rails would meet on the picture is called a vanishing point. The next units show where vanishing points sit and how many a drawing needs.