Viewing distance is the lens

There's a slider in the perspective tool that photographers will recognize immediately and draftsmen usually never meet: d, the distance from your eye to the picture plane. It's the only control that changes the character of the drawing rather than its subject — and it's the reason some perfectly correct constructions still look wrong.

The two horizontal VPs are chained

Rotate a box and you'd expect its two vanishing points to move independently. They can't. For the two horizontal families:

$x_1 \cdot x_2 = -d^2$

Their distances from the center of vision multiply to a constant. The minus sign says they always sit on opposite sides. So pull one VP in toward the center and the other is pushed out, hard — it's a see-saw with a fixed product, not two free handles.

Only one thing moves both at once: changing d itself.

Same box, same rotation. The eye moved closer to the window and both vanishing points slid inward — 1.21 → 0.78 and 0.40 → 0.26. The field of view opened from 71° to 96°, and the box started to look stretched and wrong at its corners.

That's the whole story of wide-angle distortion, and it's why the studio advice is put your vanishing points far apart. What that advice really means is: stand further back. VPs crowding into the picture is the symptom; a short viewing distance is the disease.

$\text{FOV} = 2\arctan\!\left(\frac{\text{half-frame}}{d}\right)$

The vertical VP obeys the same kind of rule

Tilt the camera and the horizon drops by −d·tan α while the zenith point appears overhead, chained to it by the same shape of law:

$y_\text{zenith} \cdot y_\text{horizon} = -d^2$

At 22° of tilt with d = 1.4, the horizon sits 0.28 frame widths below center and the zenith lands 1.73 above — and that product works out to −d² exactly.

Which means the dramatic three-point look has a price tag you can read: the more you tilt, the closer the zenith comes, and the more your verticals splay.

Proportion: everything is 1/z

The other half of the projection formula is the one that governs size. For an object of height h at distance z:

$h' = \frac{d\,h}{z}$

Apparent height is inversely proportional to distance. Double the distance, halve the height — exactly, not approximately.

Look at the shape rather than the numbers. The curve is savage up close and nearly flat far away: one step back from z = 4 to z = 5.5 costs the post 27 points of height, while the same 1.5 units from z = 8.5 to z = 10 costs only 7. This is why foreshortening reads as dramatic in the foreground and why distant mountains barely change size as you walk toward them.

It also gives you a measuring trick with no ruler: if a figure in your picture is half the height of an identical figure nearer the front, it is exactly twice as far away. Not roughly. The projection is a division, and division is reversible.

Run it

Open the perspective explorer with the depth row switched on:

1. Read the posts: 100% · 73% · 57% · 47% · 40%. They're equally spaced in the world and collapsing on the page.
2. Drag the viewing distance from 1.40 down to 0.90 and watch both VPs march inward while the FOV climbs to 96°. Stop when the box starts looking like a bad photograph — that's your personal distortion threshold, found empirically.
3. Now try to bring both VPs close without touching d, using yaw alone. You can't. The product is fixed, and that constraint is the lesson.