Visual Plane
every picture states the point it is correct from
Drawn confidently

Which way the drawn circle leans

Two rules are given for the direction a drawn circle's short axis runs in: along the axis of the cylinder, and pointing at the centre of vision. On the optical axis both are exactly right. At the edge of an ordinary frame the first is out by three and a half degrees and the second by seventy-nine. The statement neither of them is comes out of differencing the projection, and it matches the drawn ellipse to a hundredth of a degree.
17 min read 13 figures Taught, and never measuredProjected, not constructedWhat a projection destroys

Worth reading first: The circle whose centre moves · The conic a circle becomes · The circle in the square wants a number.

A round table, a manhole cover, a coin on the floor. Its picture is an ellipse, and drawing that ellipse requires knowing three things: how big it is, where its centre is, and which way round it goes.

The first two have been measured in this collection. The conic a circle becomes settles the shape exactly; the circle whose centre moves settles the fact that the drawn ellipse’s own middle is not the drawn image of the circle’s middle.

The third — the orientation — is the one the manuals give a rule for, and there are two rules in circulation.

Five discs, and which way their ellipses standThe same disc at five places across the floor. Its ellipse's own minor axis is drawn on each; the vertical the rule prescribes is drawn beside it. On the optical axis they agree exactly, and at 4 m across they are 2.73° apart.centre of visioncorrect from 17 cm, at 160 mm wide50° across
Fig. 1 The same disc at five places across the floor, with the ellipse its picture actually is, its own short axis, and the vertical the first taught rule prescribes. On the optical axis the two coincide. At the edge of the frame they do not.

The two rules

Along the axis of the disc. A disc lying on the floor has a vertical axis, and the rule says the ellipse’s short axis lies along it — vertical on the paper, for a level camera. This is the rule the drawing office uses and it is stated in the same words for isometric drawing, where it is exactly true.

Pointing at the centre of vision. The other version says the short axis runs along the line from the ellipse to the middle of the picture, on the grounds that the projection is symmetric about the optical axis.

Both are exactly right on the optical axis, where they are the same statement. Away from it they diverge, and they diverge by very different amounts.

Two rules for the same lean, and the statement neither isThe rule that the minor axis lies along the disc's own axis is out by 3.55° at the edge of this frame. The rule that it points at the centre of vision is out by 79°. The direction of greatest foreshortening is out by 9e-3°, which is what the ellipse's axis is.01230246how far across the floor the disc sits, in metresdeparture from the fitted ellipse's own axis (degrees)points at the centre of visionalong the disc's own axisgreatest foreshorteningone rule nearly right, one badly wrongand no book distinguishes them
Fig. 2 The two rules’ departure from the ellipse’s own axis, across the frame, on a logarithmic scale, with the exact statement plotted beneath them. The disc’s-own-axis rule climbs to a few degrees. The centre-of-vision rule climbs to seventy-nine. The exact one stays at a hundredth.
Two rules for the same lean, and the statement neither isThe rule that the minor axis lies along the disc's own axis is out by 5.60° at the edge of this frame. The rule that it points at the centre of vision is out by 81°. The direction of greatest foreshortening is out by 2e-2°, which is what the ellipse's axis is.02460246how far across the floor the disc sits, in metresdeparture from the fitted ellipse's own axis (degrees)points at the centre of visionalong the disc's own axisgreatest foreshorteningone rule nearly right, one badly wrongand no book distinguishes them
Fig. 3 The same two rules at a nearer depth. The angular spread across the frame is larger, so both rules are worse — and the exact statement is still exact, which is what separates a rule of thumb from a description.

On a 50° frame with the disc at eight metres, the disc’s-own-axis rule is out by 0.77° at one metre off the axis, 2.73° at four metres, and 3.55° at six. The centre-of-vision rule is out by 32.8°, 70.9° and 78.6° at the same three places.

So one of the two is a decent working rule and the other is not merely worse — it is nearly perpendicular to the truth by the edge of the frame. They are given in the same breath in the same books.

Why the second one is so wrong

It is worth understanding, because the reasoning behind it sounds right.

A projection is symmetric about the optical axis, in the sense that rotating the whole scene about the axis rotates the picture. So a circle whose plane is perpendicular to the axis — a coin held facing the camera, off to one side — does image as an ellipse whose short axis points at the centre of the picture. That case is exactly the rule’s, and the rule is right in it.

A circle lying on the floor is not in that case. Its plane contains the optical axis’s own direction of tilt, and the foreshortening it suffers is dominated by depth rather than by off-axis angle. The floor squashes it front-to-back — vertically on the paper — no matter where across the frame it sits.

So the two rules are answers to two different questions and each is exact for its own. The books present them as alternatives for the same question, and one of them has been transplanted from a case where it holds.

That is the recurring shape in this collection’s account of taught constructions, and it is worth naming again: every one of these rules is exact somewhere, and the somewhere is not stated.

A circle on the ground, and the two points that get called its centreThe image of the centre and the centre of the image ellipse are 14.6px apart — 4.0% of the ellipse's own width.centre of the ellipseimage of the centrecorrect from 22 cm, at 160 mm wideoffset 14.6 px
Fig. 4 The neighbouring fact about the same drawing. The centre of the drawn ellipse is not the drawn image of the circle’s centre, and the two are separated along the direction of greatest foreshortening — which is the same direction the short axis runs in, and the reason the two errors compound.

The statement both of them are approximating

There is an exact answer, it is short, and it is not either rule.

The image of a very small circle at a point is a very small ellipse, and the ellipse’s axes are the directions in which the map from the ground to the picture stretches most and least at that point. Those are the singular vectors of the map’s derivative — the two directions in the floor which come out perpendicular in the picture, one of them squeezed hardest and the other least.

So: the drawn ellipse’s short axis is the image of the ground direction the picture contracts most. That is a statement about the projection at a point, and it needs no circle at all.

Measured against the ellipse fitted to a projected disc of ordinary size, that direction agrees to under a hundredth of a degree everywhere across the frame — the residue being the disc’s own finite size, since the statement is exact only in the limit.

It is also computed here by differencing the projection rather than by differentiating it on paper, and that is deliberate: the arithmetic that would have to be got right by hand is exactly the arithmetic under test.

A 6.0 m circle on the ground, seen from outside itThe whole conic is drawn, including the part no camera can photograph. Every point of the circle is at least 5.94 m beyond the plane through the eye, so the image is an ellipse and the camera can see all of it. B² − 4AC = -1.45e-1.horizonthe eye stands 8.78 m from the centrecorrect from 26 cm, at 160 mm wideellipse · nearest point +5.94 m
Fig. 5 What is exactly true of the object, for contrast with what is approximately true of the recipe. A circle in a plane images as a conic, computed in one matrix product, valid for every point of the circle including the ones no camera can see.

How much a few degrees matters

Three and a half degrees on an ellipse sounds negligible. It is not, and the reason is the aspect ratio.

An ellipse three times as long as it is wide, rotated by three and a half degrees, has its ends displaced by about a sixth of its own width. On a drawn table top 200 px across and 60 px deep that is ten pixels at each end — a visible skew, and a skew of a particular kind: the ellipse no longer looks symmetric about the vertical, so the table looks as though it is sliding.

The flatter the ellipse, the worse a fixed angular error looks. A disc seen nearly edge-on is very flat, and a couple of degrees of rotation there is the difference between an object lying on a surface and an object hovering.

That is why the drawing office cares. An engineering drawing of a cylinder with its ellipse a degree out reads as a cylinder that has been knocked over slightly, and no amount of care with the outline fixes it.

The four-centre ellipse, and the ellipseThe four arcs are tangent to the rhombus at the four side midpoints and touch the true conic at exactly those four points. Everywhere else they are wrong, worst at the ends of the major axis, where the construction falls 5.72% short — and its minor axis is 3.53% too long, so a hole drawn this way is the wrong shape as well as the wrong size.true ellipse — axis ratio 0.5774four arcs — 5.72% short along the major axisthe four tangent points are exactworst departure 5.72% of the semi-major axis
Fig. 6 The drawing office’s own approximation, kept beside this one. An isometric circle drawn as four circular arcs is a different approximation to a different curve, and it has been measured too — a reminder that the ellipse’s shape has its own literature of shortcuts, independent of its orientation.
Five discs, and which way their ellipses standThe same disc at five places across the floor. Its ellipse's own minor axis is drawn on each; the vertical the rule prescribes is drawn beside it. On the optical axis they agree exactly, and at 4 m across they are 2.74° apart.centre of visioncorrect from 17 cm, at 160 mm wide50° across
Fig. 7 Larger discs at the same places. The lean is a property of the position rather than of the size, so the angles are unchanged and the visible skew is larger — which is the practical reason the rule matters more for big circles.

What the rule gets right, and why it survives

The disc’s-own-axis rule is right at the middle of the picture and drifts by a few degrees at the edge of an ordinary frame. That is a good rule.

It survives because most drawn circles are near the middle. A composition puts its subject centrally; a photograph frames the interesting object; and the discs that end up at the edge of a wide frame are usually incidental — a plate on a table at the side, a wheel half out of shot — where nobody is measuring the ellipse.

It also survives because the error is a rotation rather than a change of shape. The drawn ellipse has the right proportions and is turned slightly; nothing about it looks impossible. A rule whose failure produced a visibly wrong shape would have been caught long ago.

And it survives because the correction is not simple to state. “Along the direction the picture contracts most” is exact, and there is no ruler-and-straightedge construction for it; the honest way to get it is to construct the conic properly out of five conditions, which is what five marks and the sixth does.

Five marks fix the conic, and the sixth is a predictionFive marks on a photographed circle determine one conic — five points and five coefficients, with no fitting left over. The sixth mark was withheld from the fit and the conic passes 4.0e-13 px from it. Nothing about the camera, the circle's size or the plane it lies in was used.horizon12345withheldfive marks fitted, one withheldcorrect from 26 cm, at 160 mm widethe withheld mark is 4e-13 px off the fitted conic
Fig. 8 The construction that has no orientation rule because it does not need one. Five conditions fix a conic exactly, orientation included, and a withheld sixth lands on it. An inscribed circle in a drawn square supplies four points and four tangents, which is more than five.

The isometric case, where the rule is a theorem

The disc’s-own-axis rule is not folklore. In axonometric drawing it is exactly true, and knowing why makes it clear what perspective takes away.

An axonometric projection is parallel: every ray runs in one direction, and the map from the world to the page is linear. A linear map sends a circle to an ellipse whose axes are the singular directions of the map, and those directions are the same everywhere on the page, because a linear map has one derivative. So every circle in a horizontal plane, anywhere in an isometric drawing, produces an ellipse leaning the same way — and for the standard isometric orientation that way is along the drawn image of the vertical.

The rule is therefore a theorem in the drawing office and a rule of thumb in a perspective drawing, and the difference is precisely that perspective is not linear. Its derivative changes from point to point across the picture, and the changing part is what the three and a half degrees measures.

The ellipse the drawing office draws works the axonometric case out in full, including the four-arc approximation that is used to draw it with a compass. Nothing in that essay needs a caveat about where in the picture the circle is, and everything in this one does.

One cube in 5 parallel drawing systemsEvery one preserves midpoints exactly. What separates them is the axis scales, printed beneath each — isometric's are all 0.8165, which is equal and is not 1.elevationx 1.000y 1.000z 0.000cavalierx 1.000y 1.000z 1.000cabinetx 1.000y 1.000z 0.500isometricx 0.816y 0.816z 0.816dimetricx 0.943y 0.943z 0.471axis scales measured from the drawingall 5 preserve midpoints
Fig. 9 Why the two cases differ at all. A parallel projection has one derivative for the whole page; a perspective projection has a different one at every point. Every rule of thumb that is a theorem in the first column becomes a first-order approximation in the last.
The eight-point rule, on the floorSeven tenths along the real diagonal is on the circle to 2e-15 m. Seven tenths along the drawn one misses it by 54 mm on a circle 2.5 m across, and the drawn one is the only diagonal on the paper.the ruler's markthe plan's markcorrect from 19 cm, at 160 mm wide46° across
Fig. 10 The circle’s shape as opposed to its orientation, drawn by the method that gets both wrong in the same drawing. The eight-point rule’s four extra marks are misplaced along the diagonal; the axis rule turns the whole ellipse. The two errors are independent and add.

Reading the lean backwards

The lean is a measurement as well as a drawing problem, and it is a good one.

If the short axis of a drawn ellipse is the image of the ground direction the picture contracts most, then measuring the ellipse’s orientation says something about the projection at that point — and with enough discs across a picture, about the camera. That is the same trade the measuring half of this collection makes: a distortion, measured, is information.

In practice it is a weak measurement and worth being honest about why. The direction of greatest contraction changes slowly across an ordinary frame — a few degrees over the whole picture — so recovering anything from it requires reading the ellipse’s orientation to a fraction of a degree, and the orientation of a fitted ellipse is poorly determined when the ellipse is nearly circular. The discs that are strongly foreshortened, where the orientation is well determined, are the ones far down the picture where the contraction direction is nearly vertical everywhere and there is little variation to read.

So the quantity is available and badly conditioned, which is a specific and useful thing to know. The circles in a picture are much better used for what one conic calibrates the camera uses them for — the circular points of their plane, which are a projective fact rather than a first-order one.

Two points, and everything metric followsThe imaged circular points are where the horizon meets the image of any circle in the plane, and they are a conjugate pair — the first coordinate here is 169.5 − 446.0i. A rectification built from them and nothing else returns the world's angles to 2.6e-13° and its length ratios to 5.9e-15, and no length at all.horizonthe horizon does not cut the circle — the pair is complexrectified from the two points aloneangles: 8.5e-14°ratios: 1.3e-15length: —circle of radius 1.15 ma dash is a quantity two points cannot buy
Fig. 11 What a circle in a picture is actually worth. Its image carries the two points that fix the plane’s metric structure, and those come out of the conic exactly rather than out of its orientation approximately.

What a draughtsman should do

Three cases, and the advice differs in each.

A disc near the middle of the picture. Draw the short axis along the disc’s own axis. The error is under a degree inside the middle third of an ordinary frame, and no construction will beat that by enough to be worth the trouble.

A disc near the edge of a wide picture. Construct the conic. Four tangency points and their tangents come out of the enclosing square by joins and meets alone, as the circle in the square wants a number sets out, and a conic through them has the right orientation without anyone deciding what it is.

A cylinder seen along its length. This is the case where the two rules diverge most and where the second one is right. A cylinder’s end faces are perpendicular to its axis, so if the axis points near the camera the end circles are in planes nearly facing the reader — and their short axes do point at the centre of vision. Somebody drawing a row of pipes end-on and applying the floor rule will get every one of them wrong by a large angle.

That third case is, in all likelihood, where the centre-of-vision rule comes from. It is a correct rule for cylinders pointing at the reader, transplanted to discs lying on the ground, where it is nearly perpendicular to the truth.

The field of view is the variable that matters

One more measurement, because it changes when a reader should worry.

The error of the disc’s-own-axis rule grows with how far off the optical axis the disc sits, in angle rather than in metres. So a long lens, which puts the whole frame within a few degrees of the axis, makes the rule nearly exact everywhere; a wide lens makes it wrong at the edges of a picture that a reader is going to look at.

A 25° frame keeps everything within 12.5° of the axis and the rule’s worst error is a fraction of a degree. A 90° frame — which is what a two-point layout with both vanishing points on the sheet produces, as both vanishing points on the paper works out — puts the corners 45° off the axis, and the rule is out by many degrees there.

So the two complaints compound. A wide layout produces both the depth exaggeration that essay measures and the ellipse errors this one does, and both of them are consequences of the same choice about where to put two dots on a horizon.

Seven identical spheres across a 90° frameThe outer sphere images 27% wider than the central one. That is what a correct rectilinear projection does, and it vanishes if the picture is viewed from 8 cm.49 px62 px90° across27% wider at the edge
Fig. 12 The same effect on a sphere, which is the case a reader notices. A sphere off-axis images as an ellipse leaning away from the centre of the picture — the other rule’s case, and the one where it is right — which is part of why the two rules get confused.

The general form

The two rules for the ellipse are a clean example of something worth stating on its own, because it is the mechanism behind most of the taught constructions this collection has measured.

A rule of thumb is a first-order statement, and its case is the point it was expanded about. Both rules here are exact at one place — the optical axis for one, a plane facing the camera for the other — and both degrade away from it. What the manuals lose is not the rule but the point of expansion, which drops out of the statement because at the point of expansion every version agrees and there is nothing to distinguish them.

The repair, wherever it can be afforded, is to keep the exact statement and let the reader take the approximation if they want it. “The short axis runs along the direction of greatest foreshortening, which is vertical for a disc on the floor near the middle of the picture” is one sentence longer than the rule and carries its own domain of validity.

Five discs, and which way their ellipses standThe same disc at five places across the floor. Its ellipse's own minor axis is drawn on each; the vertical the rule prescribes is drawn beside it. On the optical axis they agree exactly, and at 4 m across they are 1.54° apart.centre of visioncorrect from 17 cm, at 160 mm wide50° across
Fig. 13 The same five discs further away. The angular spread across the frame is smaller, so the rule’s error is smaller too — which is the whole content of “it is a good rule for a long lens” drawn rather than asserted.

Where this goes next

centre of the ellipseimage of the centrecorrect from 22 cm, at 160 mm wideoffset 21.1 px The circle whose centre moves The image of a circle is an ellipse, and the image of the circle's centre is not the centre of that ellipse. The gap is 3 or 4% of the ellipse's width in an ordinary view — large enough to see, small enough to be dismissed as a slip by anyone not expecting it, and the reason drawn wheels look wrong. true ellipse — axis ratio 0.5774four arcs — 5.72% short along the major axisthe four tangent points are exactworst departure 5.72% of the semi-major axis The ellipse the drawing office draws Every isometric ellipse template is cut to a construction of four circular arcs, and every account of it calls the result an ellipse. It is not one and cannot be: a circular arc has constant curvature and a conic's varies, so the two can agree at four points and nowhere between. The four-centre curve reaches 2√2/3 of the true semi-major axis — 5.72% short — and its minor axis is 3.53% too long. horizonthe eye stands 8.78 m from the centrecorrect from 26 cm, at 160 mm wideellipse · nearest point +3.96 m The conic a circle becomes A circle photographed is an ellipse, or a parabola, or a hyperbola, and which one is decided by a single incidence: whether the circle reaches the plane through the eye parallel to the picture. Not the lens, not the tilt, not how far away it is. The discriminant of the image agrees with that one test at every point of a sweep, and at the crossing it is zero to 1e-13.

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.

ConicConic fittingDrawing officeEllipseForeshorteningJacobianMinor axisPrincipal pointsingular valuesVanishing point