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Perspective, starting from your eye level

One fact does most of the work. Everything else in this chapter falls out of it.

Perspective is usually taught as a list: one-point, two-point, three-point, lines converge. Correct, and also why people can recite it and still draw a wonky box. A rule you cannot derive is a rule you cannot check.

One fact sits underneath all of it. The horizon in a picture is the height of the eye that made the picture — your eye level, not the skyline. Eleven drills measure some part of what follows, most against a real scene built in three dimensions. None is judged by a model or a hand-written answer key, and every number below came out of the drill's own scoring function.

The horizon is your eye level

The line your gaze runs along when it goes neither up nor down is your eye level; in a picture that line is the horizon. It does not move when the ground rises; sit down and it drops with you. Two things follow.

Horizon Read is those two cues and nothing else: five scenes, no horizon drawn.

What Horizon Read measuresThe vertical miss as a fraction of the frame: 100 × clamp(1 − |Δy| / (0.14 × H)). The whole band is 14% of the frame's height — on a 400px scene, 56px out scores nothing and 28px scores 50. No hardware allowance: you drag a line, you do not draw one.

What a vanishing point actually is

A vanishing point is not a place in the world. It is the image of a direction: every line running that way draws to the same point on your page, because as they recede the gaps between them shrink to nothing.

So the point belongs to the direction, not the object. Slide a box across the floor without turning it and its points do not move; rotate it and both slide along the horizon. And a level line, extended forever, gets further away without getting higher — so its image creeps toward your own eye's height. Level directions vanish on the horizon because the horizon is eye level.

One equation runs half this chapter

Two level directions at right angles — the side edges of a box square on the floor — cannot land just anywhere. With P the principal point (where the camera's axis meets the picture) and f the focal length: (vpL − P)·(vpR − P) = −f². Cube From Plane writes it as f² = (u0 − vlx)(vrx − u0) to build its cubes; Vanishing Point Hunt generates its cities from it.

What Vanishing Point Hunt measuresA scene is 0.4 × horizon + 0.3 × left point + 0.3 × right point — the horizon outweighing either point. Full marks inside 1.5% of the frame height for the horizon and 2% of the width for a point; zero at 12% and 15%.

Aiming at a point you cannot see

Finding a point and using one are separate skills, and Vanishing Act alternates them: half its puzzles hide the point and ask you to stroke an edge onward to where it has to be, half give you the point and ask for a new edge pulled into it blind.

What Vanishing Act measuresA hunt stroke earns half its marks on the angle off the true edge-to-point ray, against an 18° base tolerance; half on how near the line passes the point, measured perpendicular to your line rather than along the horizon — 11% of canvas width, 42px floor. On the shallow ray to a distant point, a 1° wobble slides the horizon crossing several times as far as it moves the line. Both multiplied by straightness: wobble off your own best-fit line, free below 1% of the span, costing 40% by 7%.

How to draw a cube — and how to find out you did not

A cube has three sets of parallel edges and three vanishing points, two on the horizon. Start from a square face: Cube From Plane gives you a horizon, one vertical face of a real cube, and three handles to drag until the box closes. A face known to be square, seen through a lens, pins both vanishing points and the focal length through the equation above: one cube grows out of it.

What Cube From Plane measuresEach handle's distance from its true corner over the face diagonal, averaged: 100 × clamp(1 − mean(err) / 0.30). A mean of 30% of the diagonal scores nothing on any device; inside 1.5% is a clean 100.

The critique

Box Check is the closest thing to a teacher this chapter has. You freehand a box, one stroke per edge; each becomes its dominant straight line, and the lines are sorted into three families — not by their angle on the page, since one world direction fans across more than 40° of it, but by which point they aim at. Each gets a number and a line of critique.

What Box Check measuresPer family, the median angle by which its strokes miss their shared point. Inside 0.5° is a flat 100 — as tight as a hand gets — ramping to zero over a span starting at 7.5° for a pen on 150px edges. Underneath it is pixels of slop (150 × tan 7.5° ≈ 19.7px) re-expressed at the edge length you drew, so a small box is not marked harder.

Then the test no eye can do. Three sets of lines always meet somewhere, which proves nothing — so the drill finds the orthocenter of the triangle its three shared points make, which for three perpendicular directions seen by one camera is the camera's principal point, and reads the focal length off it as f² = −(a − H)·(b − H). Negative, and no camera anywhere saw your three sets as one box. Positive but under 0.55 of the drawing's size, and that camera is a fisheye nobody sketched through. Either way the round caps at 30.

The verdict you will meet most: "these spread apart as they go back — going away from you they should be closing in." The hand draws the far edge parallel to the near one, or wider to make room, and the picture then describes a box that grows as it recedes.

Foreshortening, and why the far end is rounder

A circle turned away keeps its width and loses its depth: it becomes an ellipse, whose degree — how tall against how wide — says how face-on the circle is. On a cylinder receding from you, the far end is always rounder than the near end: the further a circle sits from the eye, the smaller the angle it subtends, so the more face-on it is seen. Draw a tin can with two matching ellipses and you have not drawn a cylinder. Cylinder Ends gives you a wireframe cylinder whose far end is a flat line to drag out to the right roundness. The truth is not approximated: a circle through a pinhole is an exact conic, solved in closed form, so the ellipse revealed over yours is the answer, not a guess.

What Cylinder Ends measuresYour height-over-width ratio against the true one: 100 × (1 − |yours − true| / zero), zero being 0.18 on a pen, 0.27 on a finger, 0.36 on a mouse. Copy the near end's roundness onto the far one and you are out by 0.20 of ratio at least — 0 on a pen, 26 with a finger, 44 on a trackpad.

Ellipse in Plane asks the harder version: lay an ellipse into a given face of a box. Three errors share a budget of 0.55 — centre weighted 1.4 of the face diagonal, tilt up to 0.8 of a right angle, size and shape capped at 0.38. A centre miss of a tenth of the diagonal costs 0.14, a quarter of the mark: wrong place beats wrong degree. The tilt term fades to nothing as the truth nears a circle.

A cross-contour is a line running around a form — the seam round a bottle. Without a mental model of the form it comes out straight across the shape, reading as a cut. Wrap the Form splits its judgement in two. How far your curve strayed from the true one dies at 11% of the wrap's width — a motor skill, measured against the sheet. How deeply you bowed it dies at 60% of the true curve's own depth, measured against the object rather than the paper, which is why a line drawn dead straight is called too flat on every device.

Even in the world, uneven in the picture

Identical posts at identical spacings give gaps that close up and posts that shrink, faster than they look, because everything in a picture is divided by its own depth — and depth is what grows. Down the Row draws two posts of a receding row and asks for the third.

What Down the Row measuresFoot and top are scored against one zero-point and averaged, so placement counts twice and height once — and that zero-point is 0.6 × the gap the answer sits at, so a row closed to 40px is not judged with the slack of one still 200px apart.

Shadows are not a decision

Once the light is placed, every shadow is already determined. With the sun — far enough away that its rays arrive parallel — it is two facts. The direction is shared by everything in the scene. The length goes with height: a point at height h lands h / tan(altitude) from its base, so a corner twice as tall throws its shadow twice as far. One known shadow is a ruler for every other object; Cast Shadow gives you that ruler as a stick with its shadow drawn.

What Cast Shadow measuresMean distance from your corners to the true ones, over the shadow's diagonal: 100 × clamp(1 − meanErr / 0.28). That ramp is not eased on any device: the difficulty here is the rule, not steadiness. The reveal then asks whether the rays ran the ruler's way (mean cosine below 0.94, about 20° out) and whether the reach was right (under 0.85 short, over 1.18 long).

Short reach is the usual answer. Shadow value belongs to the value chapter.

What none of this can teach you

How to practise this week

Four to six minutes a day.

  1. Days 1–2, the horizon first. Horizon Read, then Vanishing Point Hunt. Watch the direction of the miss, not its size: consistently high or low is a habit.
  2. Days 3–4, aim, then build. Vanishing Act, to learn to commit rather than steer, then Cube From Plane. If your score drops when you slow down, you are steering.
  3. Day 5, take the critique. One box in Box Check, hidden edges included. Read the three family lines before the number. If one says the set spreads apart going back, make the next box's far edges narrower than feels right.
  4. Days 6–7, round it off. Cylinder Ends, Wrap the Form, and Rotate & Place, which scores pose as one number: the rotation angle carrying your box onto the target.

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