Two-point perspective

Turn the boxes 30° and the drawing gains a second vanishing point: with the glass 1.5 m from the eye, they sit 2.60 m to the right and 0.87 m to the left of the centre.

Turning the box

Stand on a street corner and look at the building on the corner. You see two of its walls at once, one going off to the left and one to the right, and the nearest vertical edge of the building is closest to you. Neither wall is square to you. In the language of the last unit, the width edges and the depth edges both cut the picture plane at an angle, so both families converge, each to its own vanishing point. The upright edges are still parallel to an upright picture plane, so they stay vertical. Two converging families means two vanishing points: this is two-point perspective.

Both vanishing points lie on the horizon line, because both families are horizontal. Raise the eye in the model and both points rise together. Change the Turn angle and they slide along the horizon in opposite directions: as one moves towards the centre, the other runs away from it.

Where the two points go

Taylor's rule tells you exactly where they are. Suppose the box is turned by an angle θ and the glass is a distance d from the eye. The line through the eye parallel to the width edges also turns by θ, and it meets the glass d ÷ tan θ to the right of the centre; this is written d cot θ. The line parallel to the depth edges meets the glass d × tan θ to the left. With d = 1.5 m and θ = 30°, that gives V1 at 1.5 × cot 30° = 2.60 m to the right and V2 at 1.5 × tan 30° = 0.87 m to the left. The model's readout prints the same numbers. Set the angle to 0° and V2 slides back to the centre while V1 runs off to infinity: you are back in one-point perspective.

There is a neat consequence. The two lines from the eye to V1 and to V2 are parallel to the two sides of the box, and the box has square corners, so the angle between those two lines at the eye is always 90°. You cannot put the two vanishing points anywhere you like. For a given viewing distance d, the gap between them is d cot θ + d tan θ, which works out to 2d ÷ sin 2θ. The smallest gap is 2d, when the box is turned 45°; any other angle pushes them further apart. If you draw the two points close together on a small sheet, you are drawing for an eye held very close to the paper. Seen from a normal distance, boxes near the edges of such a drawing look stretched and their near corners look sharper than 90°.

In practice the vanishing points often fall off the paper. Artists tape extra sheets to either side, or pin a long ruler to the table, rather than move the points inward and distort the drawing. The flat drawing in the model shows the same thing: at small turn angles V1 goes off the sheet entirely.

1. Draw the horizon and mark V1 and V2 far apart on it.
2. Draw the nearest vertical edge of the box.
3. Join its top and bottom to both vanishing points.
4. Place the two side verticals between those lines.
5. Join the tops of the side verticals to the opposite vanishing points; where these lines cross is the back corner of the top face.