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

Constructing a view

Horizon, vanishing points, measuring points. The classical construction drawn alongside the projection it is supposed to produce, so it can be checked rather than trusted.
horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px

One, two and three point are one construction

Why one-, two- and three-point perspective are one construction at different orientations, and what the count is counting.

 horizonequal stepshalve thetaper byworst error, in metres of depthequal steps to the horizon14.72 mhalve the remaining gap228.13 mtaper by eye350.80 mcorrect from 26 cm, at 160 mm wide34° across

The measuring point, and the step the method leaves out

The measuring-point construction for laying out equal depths in perspective, checked against the depths a camera actually produces, and what it fixes.

 horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px

The horizon is at eye level — if the picture plane is vertical

Why the horizon sits at eye level, why it cuts equal-height figures at the same fraction, and the condition the rule is never stated with.

 the panelhorizon — the centric point's heightthe section — the eye, the panel, the ground470 px — the viewing distancethree routes agree to 6e-14 pxsection, distance point, and a pinhole camera

Alberti draws a pavement, and chooses where the reader stands

Alberti's construction reconstructed with drawn intersections and checked against a camera, with the unnamed free parameter identified as the viewing distance.

 centric pointdistance point, 12 px off the sheet →430 pxcorrect from 10 cm at 160 mm wide47° across

The distance point is the viewing distance, drawn

The distance-point construction shown to be the measuring point of the forty-five-degree direction, with its offset identified as the picture's own viewing distance.

 the piazza, in planthe Baptistery, 25.6 m53 m27.2°the cathedral doorthe panel, and the eye it needsthe Baptistery fills 100%the eye is 60 cm back — off this sheet290 mm panel · 27.2° of Baptisterycorrect from 60 cm · 27° across

Brunelleschi drilled a hole in his panel

The viewing distance of Brunelleschi's lost panel computed from its stated size and the Baptistery's angular width, with the one unknown quantity left as the figure's slider.

 correct from 20 cm, at 160 mm wideverticals converge 3.59° · horizon 213 px off centre

The plane is a choice

The plane-change theorem — one eye and two picture planes give a homography for any scene — with the moved-eye counter-case that fixes what a homography of a picture can and cannot be.

 corrected from the picture — 0e+0° between the verticalscorrect from 20 cm, at 160 mm wideagrees with a level camera at the same eye to 3e-13 px

Straightening does not move the eye

Keystone correction built from the picture alone, checked against the level camera it claims to reproduce, with what it costs and the one thing it cannot change.

 horizon = eye level, 1.60 m89.89%correct from 26 cm, at 160 mm widespread 0

The horizon, and the fraction

What survives when the picture plane leans and what does not, separating the horizon's incidence with an upright from the fraction of its drawn height, and the cross-ratio that repairs the second.

 nothing on the ramp images above its vanishing linehorizonuphilllevelcorrect from 10 cm, at 160 mm wideslope recovered 22.0000° against 22° built

The ramp has its own horizon

The vanishing line of an inclined plane, how a ramp's gradient is read off a photograph from two vanishing points, and where the measurement refuses.

 horizoncorrect from 21 cm, at 160 mm wide8 bays · worst departure 8e-13 px

The bay repeated by a straightedge

The diagonal method for repeating a bay in perspective, checked against the projection it claims to produce, and why iterating it accumulates no error.

 horizon175 cm, knownconstructedcorrect from 19 cm, at 160 mm wide46° across

Carrying a height across the room

The straightedge construction that transfers a height from one place on the ground to another, checked against the head the camera projects, and measured for the one thing that makes it usable or useless — where its vanishing point lands.

 the camera's horizoncorrect from 19 cm, at 160 mm wide46° across

A picture with nothing straight in it

Finding the horizon in a photograph with no parallel edges in it, from repeated objects of equal height, with the two arrangements that give nothing and the price of the repeated objects not being quite equal.

 1correct from 19 cm, at 160 mm wide46° across

Seven is not a power of two

What a straightedge on a picture with a horizon can and cannot divide a receding line into, why repeated bisection reaches only the dyadic fractions, and the stepped construction that reaches all of them.

 the ruler's markthe plan's markcorrect from 19 cm, at 160 mm wide46° across

The circle in the square wants a number

The eight-point method for drawing a circle in perspective, run both ways its instruction can be read, with the error of each measured on the floor rather than on the page — and the four points it needs that a straightedge can actually reach.

 horizoncorrect from 20 cm, at 160 mm wide44° across

The line every nosing is on

The three families of parallels a flight of stairs puts in a picture, the fourth direction that nothing in the stair points along, and what a flight drawn with its steps evenly spaced down the page actually depicts.

 correct from 16 cm, at 160 mm wide4 divisions per run · worst 5.7e-13 px

Three-point, laid out with a straightedge

The three-point construction run forward from a station point and a set of dimensions, with measuring points on all three axes, checked against the projection it claims to produce.

 xupzthe plane of x and upthe groundthe plane of x and zthe orthocentrefocal 707.4 pxthree sides, three planes

A drawing has three horizons

The sides of the vanishing triangle are the three coordinate planes' vanishing lines, computed from the camera and compared with the joins of the vertices, with the rabatted eye on each.

 vanishing point 1vanishing point 2where the camera wasassumed centre 50% alongfocal 1129.9 px · 0.283 : 1

The arc every eye stands on

One drawn rectangle leaves the station point free along a semicircle over its two vanishing points; two rectangles on one floor meet the arcs at the eye.

 00.2500.5000.750120406080assumed centre of the picture, % of the way between the vanishing pointsreconstructed proportion of the rectanglethe rectangle that was there, 0.667 : 115.7× across the arcevery one a true rectangle

The proportion is the assumption

How sensitive a rectangle's recovered proportion is to the assumed principal point, why the assumption is usually nearly right, and what a shifted or cropped frame does to it.

 02.5e+35e+37.5e+31e+4200400600800one corner, slid along the picture (px)focal length the quadrilateral implies (px)admitted to 886 pxrefused past 903 px

The quadrilateral no rectangle casts

The admissibility test for a drawn quadrilateral, the vanishing point running out to infinity and back, and how weak the test is in the layouts anybody draws.

 the distance pointAlberti's sectionthe measuring pointcorrect from 12 cm at 160 mm widethree routes, 0e+0 px apart

Three procedures, one panel

The three routes to a perspective pavement agree to the arithmetic floor, so a correctly executed drawing carries no record of which recipe produced it — and the reader's projective test says so.

 2 transversalsnoneunfalsifiable3 transversalsnoneunfalsifiable4 transversals15 transversals26 transversals38 transversals510 transversals713 transversals1017 transversals14independent statements a pavement makesthree unknowns, one equation per mark

Four marks before anything is said

The degrees of freedom a perspective pavement leaves a reader, why a three-tile floor can be attributed to nothing, and how much hand error a longer pavement actually buys.

 10121416-1000100how far the distance point was misplaced, in pixelsthe distance the finished drawing is correct from, in cmas intendedevery point passes the reader's own test4/3 to 8e-15

The slip that leaves no trace

Why a misplaced distance point produces another exactly correct perspective, what it costs the reader in centimetres of viewing distance, and why no reading of the picture can find it.

 the distance pointone mark on the horizonleaves another exact drawingAlberti's sectionone mark per braccio, each from the panelleaves errors each its ownthe measuring pointdividers walked along the measuring lineleaves errors that accumulatea photographnoneleaves a smooth curvethe constant ratioone ratio, applied throughoutleaves a smooth curveone hand step eachand four kinds of trace

One hand step each

The four hand steps in the classical perspective constructions, the four kinds of trace they leave in a finished panel, and the one statistic that separates them.

 0.4000.6000.80011.201.40braccia in the pavement (log₁₀)the worst departure of a mark, in pixels (log₁₀)steppedmeasured from the zerothe same hand, laid off two ways√n against n^0.21

Stepped, or measured from the zero

Why cumulative and independent marks leave different residuals, the fitted exponents that separate them, and why the recipe's name is not what decides which a drawing shows.

 wrong hzcorrectno shapefrom a zerosteppeda rulethe distance pointAlberti's sectionthe measuring pointa photographthe constant ratio60144157203036060read with the panel's own horizon60 drawings each

What a panel says about its maker

The confusion matrix for reading a perspective pavement, what the panel's own horizon adds to it, and the four claims about a drawing's maker that survive being measured.

 to the vertex, 450 px furthercorrect from 22 cm, at 160 mm wide5 courses · 1e-13 px

Dividing to a point off the board

The inaccessible-vanishing-point construction executed and checked against the pencil it claims to produce, with the cost of guessing the vertex measured.

 horizon12.6180339887498954.618033988749895to the measuring point, 0.70 widths offcorrect from 19 cm, at 160 mm wide3 bays · 6e-14 px

The bays that are not equal

What the diagonal method can and cannot repeat, measured as a reachable set rather than as an accuracy, and the measuring point that has no such limit.

 00the third mark, at infinity0 … 12 marks in the unit intervalcorrect from 19 cm, at 160 mm widerank 1 · 3e-15 from the exact rationals

What a straightedge reaches on a receding line

The harmonic net on a receding line — its density by rank, what each unit fraction costs in quadrangles, and the irrational coordinate it never reaches.

 the picture plane, in planto the observer, 208 px further downcorrect from 12 cm, at 160 mm widestation 520 px

Two rules for one pavement

Vignola's two pavement rules executed from the same data, the free parameter neither names, and the disagreement measured in centimetres of the reader's room.

 the horizoncorrect from 16 cm, at 160 mm widecross-ratio 1.366025

The stair that turns has a vanishing point that moves

The cross-ratio of four consecutive treads' vanishing points on a turning stair, and the turn per tread it gives back out of a photograph.

 the ramp's vanishing linethe ramp's own pointthe horizoncorrect from 16 cm, at 160 mm wide3.43 m out at the sixth tread

A measuring point for a ramp

The measuring-point construction moved onto an inclined plane, with the taught rule's error separated into a wrong radius and a wrong line.

 the heads' linethe horizoncorrect from 16 cm, at 160 mm wide59 px, 1.46 m at the far figure

Figures on a street that slopes

The heads-on-the-horizon rule generalised to a sloping street, with the condition restated about the eye and the taught rule's cost measured in metres.

 horizonthe middle of what is leftthe principal pointcorrect from 22 cm, at 160 mm widecrop 20% · centres 81 px apart

The centre of the picture is not the centre of the paper

What a crop does to a perspective picture's principal point, measured against every construction that assumes the sheet's middle is the picture's centre.

 correct from 14 cm, at 160 mm wide12×12 cells · worst 1.57 px

Copying square by square

How far a linearly filled grid cell departs from the projective map it stands in for, measured on a pavement and controlled against a fronto-parallel wall.

 a plan, looking down: the object on the left, the frame, the hook on the rightthe hookthe frame, edge-on160 cmplan at 0.25 px per mm · a hook 160 cm behind a 56 cm framecorrect from 46 cm at 160 mm wide

The hook is the centre, and the eye is not

What it means for a drawing instrument to have a physical centre of projection, measured on Dürer's string frame and controlled against the gridded veil.

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