Visual Plane
every picture states the point it is correct from
Constructing a view

Three-point, laid out with a straightedge

This site has recovered a camera from a drawing since its first commit. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.
25 min read 23 figures Projected, not constructedTaught, and never measuredThe round trip

Worth reading first: The measuring point, and the step the method leaves out · One, two and three point are one construction · The point you have to stand at.

There are two directions to work in. A photograph can be handed to a computer, which reads the vanishing points off the drawn edges and returns the camera that made it. Or a room can be handed to a draughtsman, who stands somewhere, picks up a straightedge, and produces the picture.

This site has done the first since its first commit. It has never done the second in three-point.

A box drawn from a known camera, and the camera recovered from the drawingThree vanishing points found from the twelve drawn edges alone give back the focal length to 4e-15 relative.recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across
Fig. 1 The backward direction, which this collection was founded on: three bundles of drawn edges, three vanishing points, an orthocentre, and a focal length recovered to one part in ten to the fifteenth.

The forward direction is not harder in principle and it is harder to write down, because it needs a measuring point for each of the three axes and the third one lives on a line the manuals do not draw.

A box laid out in three-point, with nothing judgedThe marks the construction lands on, with the receding line each is on. The three vanishing points are where the camera puts them; everything else is straightedge work. Along each axis a true length is set out on the **measuring line** — the line of that plane which runs parallel to the picture plane, and so the one line in the picture a ruler is entitled to be used on — and each mark is joined to that axis's measuring point. The crossings are the drawn positions of 4 equal divisions along each of the three runs, and every one of them lands 5.7e-13 px from where the camera projects the same world point. The classical route to the measuring point — swing an arc of radius |E′V| from the vanishing point, with the eye rabatted about that axis's own vanishing line — puts it in the same place to 4.5e-13 px.correct from 16 cm, at 160 mm wide4 divisions per run · worst 5.7e-13 px
Fig. 2 The forward direction. Every mark here was placed by a straightedge from the three vanishing points and one known depth; the box behind it is what the camera projects, and the two agree to three parts in ten million million of a pixel.

What a measuring point is for

Laying out a picture needs two things that pull against each other. Directions come from vanishing points: a receding line is drawn by joining its start to the vanishing point of its direction, and that is exact and easy. Distances along a receding line come from nowhere, because the drawn line is foreshortened by an amount that changes as it goes.

The measuring point, checked against the depths the camera produces6 equal depths of 0.62 m, laid out by the construction, land on the projected positions to 6e-14 px.246VPcorrect from 26 cm, at 160 mm wide34° across
Fig. 3 The construction that supplies them, in the one- and two-point case this site already measured. True lengths are set out along a line where a ruler is entitled to be used, and each mark is carried onto the receding line by a join to a fixed point.

The line where the ruler is entitled to be used is the piece that gets stated without a reason. Manuals call it the ground line and put it at the bottom of the sheet, and the reason it works is that it runs parallel to the picture plane.

A line parallel to the picture plane stays at one depth. So its drawn length is its true length times f/zf/z with the same f/zf/z everywhere along it, and that is the only kind of line in a perspective picture with a uniform scale on it. Every other line is foreshortened by a factor that changes.

The measuring line, the measuring point, and one run of the layoutEvery line the construction draws, for all three runs. The three vanishing points are where the camera puts them; everything else is straightedge work. Along each axis a true length is set out on the **measuring line** — the line of that plane which runs parallel to the picture plane, and so the one line in the picture a ruler is entitled to be used on — and each mark is joined to that axis's measuring point. The crossings are the drawn positions of 4 equal divisions along each of the three runs, and every one of them lands 5.7e-13 px from where the camera projects the same world point. The classical route to the measuring point — swing an arc of radius |E′V| from the vanishing point, with the eye rabatted about that axis's own vanishing line — puts it in the same place to 4.5e-13 px.correct from 16 cm, at 160 mm wide4 divisions per run · worst 5.7e-13 px
Fig. 4 Every line the construction draws, for all three runs. The dashed lines are the three measuring lines, each running parallel to the picture plane inside its own coordinate plane; the faint lines are the joins to the three measuring points.

Once the measuring line is fixed, the measuring point is forced. Setting out a true length LL along the measuring line and the same true length LL along the receding line puts two marks apart by the world vector L(m^−d^)L(\hat{\mathbf m} - \hat{\mathbf d}) — a fixed direction, whatever LL is. So every join between corresponding marks runs in one direction, and all of them pass through that direction’s vanishing point. That point is the measuring point, and it needed no rabatment and no circle to find.

The measuring line, the measuring point, and one run of the layoutEvery line the construction draws, for all three runs. The three vanishing points are where the camera puts them; everything else is straightedge work. Along each axis a true length is set out on the **measuring line** — the line of that plane which runs parallel to the picture plane, and so the one line in the picture a ruler is entitled to be used on — and each mark is joined to that axis's measuring point. The crossings are the drawn positions of 4 equal divisions along each of the three runs, and every one of them lands 5.7e-13 px from where the camera projects the same world point. The classical route to the measuring point — swing an arc of radius |E′V| from the vanishing point, with the eye rabatted about that axis's own vanishing line — puts it in the same place to 4.5e-13 px.correct from 16 cm, at 160 mm wide4 divisions per run · worst 5.7e-13 px
Fig. 5 A deeper box. The measuring points have not moved: they belong to the directions, not to the object.

The classical route, and it is the same point

A draughtsman with a straightedge cannot compute a vanishing point of a difference of unit vectors. The classical construction gives the measuring point another way, and it is one of the neat things in the subject.

Fold the eye into the picture. For a given vanishing line, drop a perpendicular from the principal point onto it, and swing the eye about that line into the plane of the paper: since the eye stands ff out of the picture plane above the principal point, and the foot is dd away along the paper, the folded eye lands f2+d2\sqrt{f^2 + d^2} from the foot.

Then the measuring point for a vanishing point VV on that line is the point MM on the line with ∣MV∣=∣E′V∣|MV| = |E'V| — an arc swung from VV with the folded eye as radius.

Three vanishing points, three horizons, one orthocentreThe whole picture is the small rectangle; the triangle its three vanishing points form is drawn to the same scale around it. Each **side** of that triangle is the vanishing line of one of the three coordinate planes — the ground's horizon is the side joining the two horizontal vanishing points — and computing each side straight from the camera rather than from two vertices agrees to 4.5e-13. The **orthocentre** is the principal point. So the horizon a perspective drawing is built on is not a privileged line: it is one side of a triangle, and the ground is only special because the ground is where things stand. The eye rabatted about the ground's vanishing line lands 786.3 px from the foot of the perpendicular, which is √(f² + d²) with f = 707.4 and d = 343.4.xupzthe plane of x and upthe groundthe plane of x and zthe orthocentrethe eye, folded flatfocal 707.4 pxrabatment 788.3 px
Fig. 6 The folding, drawn for the ground plane’s own vanishing line. The eye lands 786 pixels from the foot, which is the square root of the focal length squared plus the perpendicular’s own length squared.

The two routes give the same point. Measured across three axes and three box depths, they agree to two parts in ten million million of a pixel — which is what makes the manual’s arc a construction of the same object rather than an approximation that works in the layouts manuals draw.

Three vanishing points, three horizons, one orthocentreThe whole picture is the small rectangle; the triangle its three vanishing points form is drawn to the same scale around it. Each **side** of that triangle is the vanishing line of one of the three coordinate planes — the ground's horizon is the side joining the two horizontal vanishing points — and computing each side straight from the camera rather than from two vertices agrees to 4.5e-13. The **orthocentre** is the principal point. So the horizon a perspective drawing is built on is not a privileged line: it is one side of a triangle, and the ground is only special because the ground is where things stand. The eye rabatted about the plane of x and up's vanishing line lands 770.7 px from the foot of the perpendicular, which is √(f² + d²) with f = 707.4 and d = 306.0.xupzthe plane of x and upthe groundthe plane of x and zthe orthocentrethe eye, folded flatfocal 707.4 pxrabatment 770.7 px
Fig. 7 The same folding about a different one of the three vanishing lines. The perpendicular is a different length and the folded eye is a different distance out, and the rule is the same rule.

The third axis, which is why this is not two-point

In a two-point drawing the vertical is not foreshortened at all: the picture plane is vertical, verticals are parallel to it, and they are drawn at one uniform scale from top to bottom of the sheet. There is nothing to construct.

The same cube turned 24° — a three-point constructionNothing about the construction changed. The number of vanishing points inside any finite distance is 3, and 1 of them fall on the canvas.horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px
Fig. 8 The one-, two- and three-point cases as one construction with the box turned and the camera tilted. What changes is how many of the three vanishing points are finite.

Tilt the camera and that stops being true. The vertical acquires a vanishing point, verticals converge, and a height set out with a ruler up the side of the sheet is now wrong by an amount that grows with the tilt.

Four figures of the same height, camera level at 1.62 mThe horizon cuts every one of them at 91.0% of its height — the eye height over the figure height — however far away it is.horizon = eye level, 1.62 m91.01%correct from 26 cm, at 160 mm widespread 0
Fig. 9 The condition, which is the same condition the horizon rule needs. A vertical picture plane keeps verticals parallel and puts the horizon at eye level; a tilted one does neither.

So the third axis needs its own measuring line and its own measuring point, and the measuring line is a line in a vertical plane running parallel to the picture plane — which on a tilted camera is not vertical on the paper and is not horizontal either. It is a line at an angle nobody draws, and that is the reason three-point is taught as “two-point plus a third point placed by judgement” rather than as a construction.

A box laid out in three-point, with nothing judgedThe marks the construction lands on, with the receding line each is on. The three vanishing points are where the camera puts them; everything else is straightedge work. Along each axis a true length is set out on the **measuring line** — the line of that plane which runs parallel to the picture plane, and so the one line in the picture a ruler is entitled to be used on — and each mark is joined to that axis's measuring point. The crossings are the drawn positions of 4 equal divisions along each of the three runs, and every one of them lands 5.7e-13 px from where the camera projects the same world point. The classical route to the measuring point — swing an arc of radius |E′V| from the vanishing point, with the eye rabatted about that axis's own vanishing line — puts it in the same place to 4.5e-13 px.correct from 16 cm, at 160 mm wide4 divisions per run · worst 5.7e-13 px
Fig. 10 The construction on all three axes at a shallow box. The third run’s marks are set out along a line that is neither horizontal nor vertical on the paper, and they land where the camera puts them like the other two.

Nothing is judged

The check that matters is that no step consulted a projected point.

The construction is handed the three vanishing points, the principal point, the focal length, the drawn position of one corner and the depth of that corner. A draughtsman has every one of those by deciding where to stand and how big to draw. From there the marks are produced by a ruler along a line and a straightedge to a point, and the comparison with the camera happens afterwards.

It is exact, and it stays exactThe worst distance between a corner the construction places and the corner the camera projects, over a family of boxes from 1.2 m to 5.6 m deep, with 4 divisions set out along each of the three runs. The agreement runs between 12 and 12 decimal places and does not drift with the shape of the box, which is what distinguishes a construction that is exact from one that happens to be close in the layout a manual draws.12.212.312.32345depth of the box (m)decimal places the layout agrees tono drift with the shapestraightedge work, no judgement
Fig. 11 The comparison, over a family of boxes from one metre deep to five and a half. The agreement runs between twelve and thirteen decimal places and does not drift with the shape of the box.

The flatness of that curve is the point. A construction that happened to be nearly right in the proportions a manual prints would show a trend; this shows none, because there is nothing in it to trend.

What the composition rule commits the reader toWith the two points one page-width apart the picture is correct from 80 mm — a 90° field of view — and a reader holding it at 400 mm sees a room 5.0 times as deep as the one drawn. Nothing about the drawing changes; the number is the layout's.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
Fig. 12 What the alternative costs, measured earlier here: the third vanishing point placed by judgement, which survives in a book’s layout because the point is thousands of pixels off the paper, and costs ten degrees of lean looking up at a tower.

What the construction is checked against, and why that matters

There is a way to write this essay that would prove nothing. Draw the box with the construction, draw it again with the construction, observe that the two agree, and call it exact.

What is done instead is that the construction is compared with the projection — the same world points put through this site’s camera, which is a dot product and a divide and shares no code with any of the straightedge work. The two routes have the vanishing points in common and nothing else, and the vanishing points are the input to one of them and an output of the other.

The site's camera, written as the matrix multi-view geometry needsK holds the focal length and the principal point; R's rows are the camera basis — right, down, forward — and t is −R·eye. Projecting all 44 scene points through P = K[R|t] and through the camera itself gives the same picture to 1.8e-13 px. Everything in this field rests on the two being one camera, so it is measured rather than assumed.K — focal length and principal point739.90345.00739.9200.0001.0000R — right, down, forward0.980000.1991-0.0182-0.99580.08970.1982-0.0915-0.9759t = −R·eye00.99586.6497focal 739.85 px · 50.0° acrossP projects 44 points where the camera does, to 1.8e-13 pxcorrect from 17 cm, at 160 mm wide50° across
Fig. 13 The projection route’s own check, one level down: the camera as a matrix against the camera as a dot product and a divide. Every claim in this essay rests on the second route being independent of the first.

That is this site’s standing habit and it is the reason the numbers are quotable. A construction checked against itself is a spelling test; a construction checked against a projection is a claim about geometry.

Four points on a line, before and after a projectionLength and the ratio of lengths do not survive the projection; the cross-ratio does, agreeing to 0e+0 relative.horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across
Fig. 14 And the quantity that makes the comparison meaningful in the first place. The construction and the projection agree because both are projective operations on the same objects, and what a projective operation preserves is what they can agree about.

The one decision left

Something has to fix the size of the drawing, and it is not a free parameter in disguise.

The scale on each measuring line is f/zf/z where zz is the depth of the corner the layout starts from. Choose that depth and the drawing has a size; the same room drawn from the same station at twice the depth is the same picture at half the scale. So the “decision” is the choice of how large to draw, which is what a decision about paper is.

Nothing else is chosen. In particular the viewing distance is already fixed before any of this — it is the focal length scaled to the width the drawing will be shown at — and the construction inherits it rather than choosing it. That is the difference between this and Alberti’s pavement, whose free parameter is the viewing distance and which never names it.

Alberti's construction, with the section that fixes the depthsLeft: the panel, six braccia across, its transversals found where the section's rays cross the picture plane. Right: the section, with the eye at its true distance. The transversals agree with a pinhole camera of the same focal length to 6e-14 px.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
Fig. 15 The construction that does leave the viewing distance free. Two draughtsmen following the recipe exactly can produce two pictures correct from two different places, and the recipe does not say which.
The distance point at 620 px — a picture correct from 14 cmThe orthogonals go to the centric point and the diagonal goes to the distance point; the transversals are where they cross. The distance point's offset is the viewing distance, so moving it moves the reader, and the drawing gives no sign that anything has changed.centric pointdistance point, 202 px off the sheet →620 pxcorrect from 14 cm at 160 mm wide33° across
Fig. 16 And what fixes it once the distance point is on the page. The distance from the centric point to the distance point is the viewing distance, drawn.

Where the drawing goes wrong if a step is skipped

The construction has three runs and each needs its own measuring point. Skipping one is not a small economy, and the size of the mistake is worth having.

The usual skip is the third. A draughtsman lays out the two horizontal runs properly and then measures the height with a ruler up the sheet, as though the picture plane were vertical. On a camera tilted enough for the third vanishing point to matter, the heights come out uniformly wrong — every one of them stretched by the same factor at the front of the box and by a different factor at the back, so the box is drawn as a box of the wrong proportions and the corners still meet.

That is the failure mode this site’s wrong field keeps finding: a drawing that is internally consistent, that satisfies every visual check a reader can perform, and that depicts something nobody chose.

The taught two-point cube, with the two far edges placed 8 points apartThe corner angles are 90° because the method forces them. The side ratio is 0.719, so this picture depicts a box whose depth is 1.39× shallower than its width.horizoncorner 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
Fig. 17 The two-point version of the same complaint. The construction has a step it supplies no rule for, the drawing looks like a cube either way, and what it depicts is a box 1.4 times shallower than it is wide.
What the by-eye step actually decidesSymmetric placement gives a cube for free. 8 points of asymmetry — invisible in the drawing — gives a box of side ratio 0.72, and ±16 points spans 0.52 to 1.94.0.50011.502-10010difference between the two by-eye placements (points)side ratio of the box the drawing depicts (1 = a cube)a cubeeven-handedthe one free choice in the taught methodand it decides the whole solid
Fig. 18 And the size of it, swept. Nothing on the paper announces the difference; the depicted solid is what changes.

What this does not say

It says nothing about whether three-point is worth drawing. It is a great deal of work for a picture a camera produces instantly, and the value of the construction is that it can be checked, which is what this essay does with it.

It says nothing about a draughtsman’s accuracy. Every straightedge here is exact and every intersection is computed; a real drawing has line widths and pencil points, and the error a real one accumulates is set by those and not by the geometry.

And it does not claim the classical route is equivalent in every layout. The arc swung from a vanishing point is exact wherever the vanishing point is on the paper, and a vanishing point thousands of pixels off the sheet makes the arc unswingable — at which point the construction is correct and cannot be performed, which is a different complaint and is the one both vanishing points on the paper is about.

What the composition rule commits the reader toWith the two points one page-width apart the picture is correct from 80 mm — a 90° field of view — and a reader holding it at 400 mm sees a room 5.0 times as deep as the one drawn. Nothing about the drawing changes; the number is the layout's.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
Fig. 19 The layout that keeps the points reachable, and what it commits the reader to. A construction that fits on the sheet is a construction with a wide field of view in it.

What it takes to perform, in practice

The construction is exact and it is not always performable, and the two are different complaints.

Three vanishing points on a sheet of paper is a wide picture. The separation between two vanishing points is the focal length in disguise — both vanishing points on the paper prices exactly that — so a layout compact enough to construct is a layout whose correct viewing distance is a fraction of the page width, and a reader holding it at arm’s length is being shown a room several times deeper than the one drawn.

A wide render read from a screen that subtends much lessA 27-inch monitor at 650 mm subtends 49.3°. A picture rendered at 100° is therefore being read from 2.60 times its own station distance, and the viewing field has already established what that does: depth is stretched by exactly that factor and nothing in the picture changes. The two routes to the number — from two angles, and from a focal length and a display width — agree to 1e-9.051050100150field of view the picture was rendered at — degreeshow many times the depicted depth is stretchedthe screen subtends 49.3°100° → depth ×2.6027-inch monitor at 650 mmsubtends 49.3°
Fig. 20 The consequence, measured. The picture’s own correct distance against where a reader actually stands, and the depth exaggeration that follows from the gap.

An honest layout puts the points off the sheet, and then the construction needs a straightedge longer than the table. Draughtsmen have devices for that — a pin and a length of thread, a trammel, a pair of proportional dividers — and every one of them is a way of drawing a line to a point that is not there.

The vanishing point runs to infinity and the measurement does not careAs the camera comes level the vertical vanishing point leaves the canvas, the page and eventually the plausible — 7.2 × 10⁹ px at a tilt of one part in eight million. The recovered height stays exact to 2e-16 relative the whole way. At exactly level the method has nothing to work with and refuses.0510-6-4-20how far the camera looks down, over eight metres (metres, log scale)where the vertical vanishing point falls (log₁₀ pixels)the vanishing pointthe error in the recovered heightthe error curve is offset by 17 decades to be visiblea flat line at machine precision
Fig. 21 A vanishing point four page-widths outside the frame. The construction is unchanged and the instrument has to be.

None of that affects the geometry. It affects who can perform it, which is a fact about drawing offices rather than about pictures, and this site’s position is that the two should be kept apart.

Which comes first, the station or the drawing

There is a chicken-and-egg reading of the construction worth heading off, because it decides what a draughtsman actually chooses.

Nothing here starts from the vanishing points. It starts from a station: a place to stand, a direction to look, and a picture plane. Those three give the three vanishing points by projection, and the vanishing points are outputs.

A draughtsman working the other way — putting the vanishing points where they look right and calling that a choice of composition — has chosen a station without knowing which one, and this collection has measured what that costs three times over.

Where the reader has to be for a 40° picture to be correctShown 160 mm wide, this picture is a correct projection only from 22 cm away. Drawn to scale.the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide
Fig. 22 The station, which is what is actually being chosen. Everything on the sheet is a consequence of it, and choosing the consequences instead is what leaves a construction with a free parameter nobody names.
What the composition rule commits the reader toWith the two points one page-width apart the picture is correct from 80 mm — a 90° field of view — and a reader holding it at 400 mm sees a room 5.0 times as deep as the one drawn. Nothing about the drawing changes; the number is the layout's.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
Fig. 23 And a measurement of the alternative: the third vanishing point placed by judgement, in a layout where the error is invisible and in one where it costs ten degrees of lean.

The order matters for a practical reason as well. Choosing a station first means the field of view is decided before anything is drawn, so a picture that turns out to need a ninety-degree field is caught at the beginning rather than discovered when the vanishing points will not fit on the table.

The transferable form

The measuring point looks like a trick and it is a consequence.

A construction that carries a true length into a foreshortened part of a picture works because the difference of two unit vectors is a fixed direction, and a fixed direction has one vanishing point. The straightedge is doing what the algebra says, once, rather than approximating it.

That is why iterating it accumulates nothing, why it holds at every depth rather than at one, and why it transfers unchanged to the third axis where no manual takes it. It is the same reading the bay repeated by a straightedge gives its own construction: exactness in a drawing comes from the operation being a projective one, not from the draughtsman being careful.

And the negative half is worth carrying too. Every taught rule this site’s wrong field has measured fails because it fixes a quantity on the paper — an angle, a ratio, a fraction along a drawn diagonal. The measuring point fixes a quantity in the world and lets the paper come out however it comes out, and that is the whole difference between the two.

Where this goes next

246VPcorrect from 26 cm, at 160 mm wide34° across The measuring point, and the step the method leaves out Laying out equal depths correctly needs a second vanishing point that most treatments never introduce — the one belonging to the diagonals. With it the construction lands on the projected divisions to eighty femtopixels. Without it, depth is placed by judgement and the drawing depicts something nobody chose. horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px One, two and three point are one construction The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity. horizoncentre of the picture0.58° of lean, invisiblethe horizontals are unaffected The third point put where it looks right Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.

What links here

Computed from the collection, not written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Camera tiltDemonstrationDrawing systemFocal lengthGround lineMeasuring pointOrthocentrePrincipal pointRabatmentStation pointthree-point perspectiveVanishing point