Three-point perspective

Tilt the glass up by 22° and the upright edges of the tower meet at a third point 3.71 m above the centre, while the horizon drops 0.61 m below it.

Looking up and looking down

So far the picture plane has always stood upright, as if you were looking straight ahead. Now stand at the foot of a tall tower and look up at it. To keep the whole tower in view you tilt your head back, and the imaginary sheet of glass in front of your eyes tilts with it. Once the glass is no longer upright, the vertical edges of the tower are no longer parallel to it. They cut the glass at an angle, and like any family of parallel edges that cuts the glass, they converge. A box in this view has all three of its families converging: width, depth and height. This is three-point perspective.

Looking up, the upright edges meet at a third vanishing point high above the tower, and the tower seems to taper towards the sky. Looking down from a high window, they meet far below the ground, and buildings seem to shrink towards their bases. In the model, set the Tilt slider to a positive angle to look up and a negative angle to look down; V3 swaps from above the centre to below it.

Where the horizon goes

Something else changes when the glass tilts: the horizon moves away from the centre of the picture. The horizon is still the vanishing line of every horizontal plane, and it is still at eye level, but the centre of the picture is no longer level with the eye; it is wherever you are looking. If you tilt the glass up by an angle τ, the horizon drops to d × tan τ below the centre. With d = 1.5 m and τ = 22°, that is 1.5 × tan 22° = 0.61 m below. The third vanishing point goes the other way, to d ÷ tan τ above the centre, which is 1.5 ÷ tan 22° = 3.71 m. Multiply the two distances and you always get d²: here 0.61 × 3.71 = 2.25 = 1.5². The more you tilt, the closer V3 comes to the picture and the further the horizon falls away from the centre; at τ = 45° both are exactly d from the centre.

You have probably seen this in photographs. Point a phone camera up at a tall building and the sides of the building lean inward in the picture, even though the real walls are vertical. The camera sensor plays the part of the tilted picture plane, so the photograph obeys exactly the same geometry. Hold the camera level and the walls come out parallel again, but then the top of the building may be cut off. A painter has the same choice: keep the picture upright and use two-point perspective, or tilt it and accept the third vanishing point.

Three-point perspective is the natural choice for skyscrapers, towers, deep stairwells and aerial views. It is also the least forgiving, because three points have to be kept consistent at once, and the third one is often far off the page. Many artists draw tall buildings in two-point perspective on purpose, with a level picture, because it looks calmer even though it is less like what the eye does.