Visual Plane
every picture states the point it is correct from
Field

Drawn confidently

The constructions taught in every book on the subject, measured. Some are fine. The point is that nobody knows which until the drawing is asked what solid it depicts.
horizondepicted height ÷ side0.3278corner angles90.000° — forced by the methoddepicted side ratio0.7195lens this drawing implies60° acrossdrawn exactly as the method prescribesthe free step is where the far edges go

The cube that is a box

Measuring what the taught two-point cube construction actually depicts, why the corner angles prove nothing, and which free parameter decides the solid.

 horizonequal stepshalve thetaper byworst error, in metres of depthequal steps to the horizon11.72 mhalve the remaining gap225.13 mtaper by eye347.80 mcorrect from 26 cm, at 160 mm wide34° across

Dividing depth by eye

Three by-eye depth methods measured in metres of error, and why the obvious cross-ratio check fails to catch them.

 drawn: 1.309 : 1subtended at the station point: 1.000000000 : 1the faint circle is the mean radius, for comparisonthe drawn centre sits 3.42 px from the axis's own mark

The sixty-degree cone of vision

The taught cone-of-vision rule, followed through — what a 60° picture's correct viewing distance actually is, and the sphere at the frame's edge measured twice, as an ellipse on the paper and as a circular cone in the eye.

 horizoncentre of the picture0.58° of lean, invisiblethe horizontals are unaffected

The third point put where it looks right

What a three-point drawing depicts when its vertical vanishing point is placed by judgement, why the error is nearly invisible in a book's diagram and large in a picture taken from the pavement, and the one thing that stays exactly right either way.

 0.6× the page48 mm×8.3 at 4000.8× the page64 mm×6.3 at 4001.0× the page80 mm×5.0 at 4001.4× the page112 mm×3.6 at 4002.0× the page160 mm×2.5 at 4003.0× the page240 mm×1.7 at 4004.5× the page360 mm×1.1 at 4007.0× the page560 mm×0.7 at 400distance the picture is correct from, shown 160 mm widea rule about the paperwhich is a rule about the reader

Both vanishing points on the paper

What a two-point layout commits the reader to — the separation of the vanishing points is the focal length, the focal length is the correct viewing distance, and the viewing distance decides how deep the depicted room is.

 centre of visioncorrect from 17 cm, at 160 mm wide50° across

Which way the drawn circle leans

The taught rules for the orientation of an ellipse in perspective, measured against the ellipse the camera actually draws, and the exact statement both of them are approximating.

 correct from 19 cm, at 160 mm wide46° across

The forty-five degree shadow

The forty-five degree shadow rule run exactly as it is taught, with each drawn shadow read back as the light that would have to be behind it, against four real shadows from one sun that read back as one.

 horizoncorrect from 20 cm, at 160 mm wide44° across

Measured down from the waterline

The reflection rule every landscape manual gives, with its two unstated conditions priced separately — the vertical picture plane it needs, and the point on the water it needs the object to touch.

 the pavement implies a horizon 171 px off the topthe horizon the panel drewcorrect from 12 cm at 160 mm wide9.2 px from the truth, 0.17 px from a perspective

The rule that draws another room

Why the constant-ratio spacing rule passes a reader's projective test on any small pavement, what convicts it instead, and the number of braccia at which it finally betrays itself.

 six tangents across 3 of the four arcs1.69% of the width

Six tangents and the point nobody drew

Pascal's theorem dualised into a straightedge test for whether a drawn oval is a conic, and the four-arc isometric ellipse rejected by it without any comparison to the truth.

 the reflected eyecorrect from 16 cm, at 160 mm widetaught rule 1.26 m out

A mirror that is not parallel to the wall

The taught reflection rule against a turned mirror, the two invariants that survive it, and the harmonic test that was vacuous by construction.

 5e-14°2.65°0.59°correct from 16 cm, at 160 mm wideaxle solid, minor axis dashed · 2.65° at the edge

The minor axis is not the axle

The draughtsman's rule that a circle's ellipse leans along its axle, measured across the frame, with a sphere's outline as the rule of the same shape that is a theorem.

 correct from 16 cm, at 160 mm wide35 px from the major axis

A cylinder has two different ends

The two end conics of a drawn cylinder measured against each other, and the tangency of its outline measured against the construction that assumes it is at the major axis.

 -8-6-4-20012345678tile corner along the diagonal, from the near endsigned distance off the chord (px)the constant-ratio rulea correct pavement, crept 4.5%8 braccia, unpaintedone arc, two causes

A steady creep draws the rule's bow

Whether a painter's steady creep can counterfeit the constant-ratio rule's bow in the diagonal of a pavement — measured on correct pavements painted with a compounding creep: the counterfeit's shape against the rule's, the creep it takes at each length of pavement, the arc reading's response to any creep, the transversal gaps' own ratios, and how well the gaps name the rule against its counterfeit under a painter's scatter.

 00.50011.50246812braccia in the pavementthe arc reading's amplitude (1 is the rule's bow)the rule's bowone creep for the handfour pavements of one interior, unpaintedbars: ±1σ at 1 px

A panel's pavements convict the creep one pavement hides

Whether the several receding pavements of one painted interior, read together, separate the constant-ratio rule from a correct construction painted with a creep, where one pavement could not: the arc readings' amplitudes against each pavement's own rule bow, for one steady creep and for creeps tuned to each pavement, the pavements' gap curvatures pooled with their sign, and how both depend on the hand's scatter and on how many pavements the panel holds.

 00.050the pavements, in the order they were paintedcreep a braccio, fitted from the transversalsthe floora loggia's coffersa stepa table's inlayticks: the creep painteddots: the creep reada hand at one pixelnamed a step → the floor → a loggia's coffers → a table's inlay

A panel's creeps keep the order it was painted in, while the hand is steady

Whether a panel of several pavements, painted correctly by a hand whose creep grows as it works, records the order its pavements were painted in: each pavement's creep fitted from its transversals with its correct construction known, the order read from their ranking, against how fast the hand tires and how steady it is, which pairs of pavements decide it, what more pavements add, and what a creep that depends on a pavement's length does to the reading.

 the axis, from the rimsthe photographdown the axis: the rims' truesizes, near solid, far dashedtaper 1° per side · board at 0.46×read 1.000°

The rims carry a part's taper, as finely as they are drawn

Whether a drawing's two rim ellipses can tell a turned part's taper from its turn: the axis recovered as the one normal that makes both rims circles, the taper read down that axis from the rims alone, why the ratio of the rims' sizes cannot separate the two, and how the reading survives rims drawn by hand with stated scatter, parts turned towards and across the view, and rims drawn with ellipse guides in steps of 5° and 2.5°.

 1° a side, axis 80° from the sight, 0.5 pxtogether ±0.032° · rims ±3.13°

A turned part drawn from the side keeps its taper in its sides

Whether a drawing of a turned part reads its taper better from its rims and its sides together than from its rims alone: the taper read from the rims, from the sides' crossing against the axis the rims supply, and from everything fitted at once as one tapered part, as the part turns from pointing away to lying across the view; how each scales with the care of the drawing; what ruled sides carry against freehand ones; and the smallest taper a side-view drawing shows.

 a 15% swelling at two fifths, 1° of taper, 0.5 pxswelling read 0.150 ± 0.002

A turned part's outline reads its swelling from every side, and a straight taper hides it

Whether a drawing of a turned part reads its profile and not only its taper: a part whose radius swells along it by a stated curve, drawn with its rims and its curved sides, fitted for the swelling's height and place along with the taper; the profile read back against the true one and against the straight taper the earlier reading gives; how the reading changes as the part turns from square to the picture to lying across it; the smallest swelling a drawing shows; where along the part a swelling can be placed; and how each scales with the care of the drawing.

 the axis's vanishing pointthe photographtaper 1° per side · drawn at 0.72×4.2 px missed by the habit

A tapered part meets at its apex

Where the drawn sides of a cone frustum meet, measured against the vanishing point of its axis as the taper runs from a flare to a narrowing, and the flare at which a correct photograph prints the sides parallel.

 station pointmeasuring pointvanishing pointthe picturedrawn at 0.48× · side ratio 1.000 · height 1.000a cube, constructed

A square plan is not a cube

The two-point cube finished with its measuring points and audited mark by mark — the height a square-plan test cannot see, and how far each hand-placed mark may slip before the drawn solid is five per cent off a cube.

 0.050.10.20.5125105101520braccia in the pavement, drawn to one page widthpixelsthe diagonal, by straightedgethe transversals, by fittingboth read on one pavement47× at eight braccia

The rule is exact for a floor that lengthens

Reading the constant-ratio spacing rule as a camera rather than an error — the floor it depicts exactly, the viewing distance each of its tiles names, and the straightedge test that separates it from a square pavement.

 horizondots on the horizon: measuring points and vanishing points · station point 595 px below, not drawnturned 30°, the same lens and near edgea cube to 3e-13

A turned cube punishes the eye, not the construction

Whether the measuring-point construction's tolerance survives a cube turned toward the picture, where one vanishing point runs far off the sheet — and which mark becomes the expensive one instead.

 the constant-ratio rulecorners off the chord by 8.03 pxAlberti's sectioncorners off the chord by 2.47 px

A straightedge convicts the rule on five braccia

Whether a straightedge laid along a painted pavement's diagonal can tell the constant-ratio rule from a correct construction once a painter's hand has scattered every line — measured over hundreds of painted pavements per length: the two pavements side by side, the rule's bow against the hand's wander, the chance the test ranks them rightly against the roughness test that tells two hands apart, the threshold that catches the rule, the scatter at which the test gives out, and which way of laying the straightedge reads best.

 tilted 20°, turned 0°three swung lengths

A slight tilt costs a construction more than a steep one

The marks of a three-point measuring-point construction priced by tilt. By eye a far edge has about 5 px at any tilt; the measuring point has 22 to 38; a horizontal vanishing point falls to 6 px at a 3 degree tilt because of the orthocentre, and the third vanishing point has hundreds to thousands of pixels.

 -7.50-5-2.5002.50012345678tile corner along the diagonal, from the near endsigned distance off the chord (px)the rule's arc, unpaintedthe rule painted: amplitude 0.84a construction painted: -0.178 braccia, 1 px of scatterone shape, one amplitude

The rule's bow is read by its shape

Whether reading the shape of a painted pavement's diagonal, rather than the size of its worst departure, keeps the constant-ratio rule separable from a correct construction where the straightedge failed — measured over hundreds of painted pavements: the rule's arc against a painted rule's and a construction's corners, the chance each reading ranks the pair correctly at every length and scatter, the scatter at which each gives out, and what three corners can say.

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