A cylinder has two different ends

The minor axis is not the axle takes one taught rule about a drawn wheel — that its perspective ellipse has its short axis along the axle — and finds it exact on the principal ray, to 5 × 10⁻¹⁴ degrees, and nowhere else: 5.95 degrees out on an ordinary frame.

A wheel is a circle. Give it a thickness and it becomes a cylinder, and a second taught rule appears: draw the two end ellipses, join the ends of their major axes, and those lines are the sides of the drawing. Both halves of that instruction are wrong.

Two ends, two conics #

The two ends of a cylinder are circles of the same size in parallel planes. It is easy to assume that they therefore image as the same ellipse at two sizes, and the drawing-office construction assumes exactly that: one ellipse template, used twice.

Computed rather than assumed — each circle carried through its own plane-to-picture map as a matrix, so each ellipse is the exact image rather than a curve through sampled points — they are not the same ellipse. For a cylinder standing a little off the principal ray, the two major axes lie 14.4 degrees apart and the two aspect ratios are 0.918 and 0.839.

That is not a small difference. Fourteen degrees is visible at a glance on a drawing, and the aspect ratios differ by a tenth, which is more than a template step.

The outline is two rulings, and where they touch #

The straight sides of a drawn cylinder are not construction lines. They are the images of two lines lying on the cylinder’s own surface — the two rulings at which it turns away from the eye — and they are exactly straight because a ruling is a straight line in space.

Which two rulings they are follows from the eye. The contour generator of a quadric is its intersection with the eye’s own polar plane, and for a cylinder that plane is parallel to the axis, so it cuts the surface in two rulings and not in a curve. Each ruling meets each end circle at one point, and that point is where the outline touches that end’s ellipse.

Those four points are computed here, and none of them is at a major-axis end. On the cylinder drawn, the touch is 28 degrees round the ellipse from the nearest major-axis end, which is 35 pixels away on a 690-pixel frame.

So the construction misses twice over: the two ellipses are different, and the points it joins are not the points the outline touches. Drawn as instructed, the sides of the cylinder are 11.1 pixels away from the true outline.

Distance removes one error and not the other #

There is a case in which the construction is right, and finding it says what the rule is a rule about.

Walk the cylinder away along its own line of sight and lengthen the lens to hold its drawn size — which is the drawing office’s own limit, a parallel projection. The two ends’ major axes converge: 14.4 degrees apart at the drawn distance, 3.5 at four times, 0.87 at sixteen, 0.35 at forty. The difference falls in proportion to the distance, which is what says it is a perspective effect and nothing else.

The offset of the touch does not follow it down. Off the principal ray it falls from 28 degrees to 11.6, to 8.0, and settles at 7.3 degrees — a limit rather than a decay. And the construction’s own error settles at 0.72 pixels rather than at nothing.

On the principal ray it does go to nothing: 0.63 degrees at forty times the distance, on its way to zero. So the rule is exactly right for a cylinder far away and centred, and it is the centring rather than the distance that the drawing office quietly supplies — by drawing one object at a time in the middle of a sheet.

Why the offset survives the limit #

The residual is worth explaining rather than reporting, because it is the part a reader would not predict.

Far enough away, the rays reaching the cylinder are parallel to each other — but they are parallel to the direction to the cylinder, which is not the direction the picture plane is perpendicular to unless the cylinder is on the axis. So the limit of a perspective projection of an off-axis object is an oblique parallel projection, not an orthographic one.

Under an orthographic projection a circle’s image has its minor axis along the projected normal of the circle’s plane, and the tangent lines parallel to that direction touch at the major-axis ends exactly. That is the geometry the template rule is built on. Under an oblique one, the minor axis and the projected normal part company, and the tangency moves off the major axis by the obliquity.

Measured on the cylinder here at forty times the distance: the ellipse’s minor axis points one way and the image of the cylinder’s own axis another, 7.3 degrees apart, and the touch is displaced by the same 7.3 degrees.

So the construction is a rule about objects drawn at the middle of the sheet, and the distance it is usually blamed on is not what it depends on.

Two rules, one object, one exception each #

The wheel and the cylinder are the same object with a thickness added, and setting their two rules side by side says what they have in common.

The wheel’s rule says the ellipse’s minor axis is the axle. It is exact on the principal ray and out by 5.95 degrees on an ordinary frame, and the reason is that the ellipse’s axes belong to the conic while the axle belongs to the circle, and only a symmetric view makes them agree.

The cylinder’s rule says the outline touches at the major axis. It is exact on the principal ray and far away, and out by 28 degrees where it is drawn here, and the reason is the same one seen from the other end: the tangency belongs to the conic and the ruling belongs to the cylinder.

Both rules are statements about an orthographic view of a centred object, taught as statements about drawing. Which way the drawn circle leans is a third of the same family, and dividing depth by eye a fourth from a different corner of the subject. The shape they share is worth naming: a rule that is exactly right in the case a draughtsman checks it in, and wrong by a visible amount in the case a picture actually presents.

The two vanishing lines #

The reason the two ends differ has one sentence in it, and having the sentence is better than having the fourteen degrees.

The two end circles lie in parallel planes. Parallel planes share a vanishing line — that is what parallel means, projectively — so the two circles’ images are related by the map that carries one plane to the other, and that map fixes the shared vanishing line pointwise. A map fixing a line pointwise and one further point is a homology, and a homology does not preserve the shape of a conic: it carries a circle’s image to another conic with different axes.

So the two ends being different ellipses is not a small correction to their being the same one. It is the generic behaviour, and their being the same is the special case — which happens when the homology becomes a similarity, and that needs the eye far away and the object centred, which is exactly the pair of conditions the measurement found.

That reading also predicts the direction. The nearer end is drawn larger and rounder and the further end smaller and flatter, which is what the aspect ratios say: 0.918 for the near end and 0.839 for the far one.

What a draughtsman actually does about it #

Three practical statements, and the first is not a criticism.

At the sizes a drawing office works at, the construction is usually good enough. Eleven pixels on a 690-pixel frame is one and a half per cent of the width, and a cylinder drawn near the middle of a sheet at a long viewing distance has a fraction of that. The rule survives because it is nearly right where it is used.

The error grows with the offset rather than with the closeness. A cylinder at the edge of a wide drawing is where it fails — 18.6 degrees between the two ends at the furthest offset drawn here — and that is the opposite of where a draughtsman would expect a perspective error to appear. The ellipse the drawing office draws measures what the four-centre approximation costs; this is a second error of the same family, and it is not the same one.

And the repair is to draw the tangency rather than assume it. The outline’s sides are tangent to both ellipses and the tangency points are where a line parallel to the projected axis touches each — a construction with a straightedge, no harder than the one it replaces, and right at every offset.

The circle behind it #

Every part of this is a statement about the image of a circle, and the conic a circle becomes is the standing account of what that image is: a conic, computed exactly by carrying the circle’s matrix through the plane-to-picture homography rather than by fitting.

Two of that essay’s findings do the work here. The image of a circle’s centre is not the centre of the image ellipse, which is why the two ends’ ellipses are not concentric on the image of the cylinder’s axis. And the ellipse’s axes are properties of the conic rather than of the circle, which is why they move as the circle moves across the frame while the circle itself does not change at all.

What a cylinder adds is a second circle in a parallel plane, and parallel planes do not have parallel images: each has its own vanishing line, and the two vanishing lines meet at the vanishing point of the direction they share. That is the whole reason the two ends differ, and it is a fact about the pair rather than about either circle.

Where a photograph shows it #

The measurement is on a computed picture, and it is worth saying where a reader would meet it outside one.

A tin, a mug, a pipe, a roll of tape near the edge of a wide photograph: the far rim and the near rim of the same object are visibly different ellipses, tilted against each other, and the effect is strongest exactly where a lens is widest. A photograph taken at 54 degrees across — the one drawn here — has 14 degrees of it at a modest offset.

It is also what makes a drawn cylinder look wrong when it is wrong. A reader cannot say what is amiss with a badly-drawn tin, but the two rims disagreeing by the wrong amount is one of the two things that can be amiss; the other is the ellipse the drawing office draws’s four-centre approximation showing its joins.

And it is a recovery in waiting. Two ellipses with a known relation are more evidence than one: the homology carrying the near rim to the far one has the shared vanishing line in it, and a vanishing line is what the conic a circle becomes needs to run its recovery backwards. A photograph of a tin therefore carries more about the camera than a photograph of a coin, and the extra is exactly the difference this essay measures.

What this does not settle #

The cylinder is right circular and its ends are square to its axis. A tapered cylinder — a cone frustum, which is most turned parts — has end circles of different sizes in parallel planes, and its outline’s rulings meet at the cone’s apex rather than staying parallel. The tangency question is the same one and the numbers are not.

Nothing here is about a fitted ellipse. Every conic above is computed by carrying a circle’s matrix through a homography, so it is the exact image. A draughtsman using a template or a four-centre construction has a second error on top of this one, and the ellipse the drawing office draws measures that one; whether the two errors add or partly cancel is not asked here.

And the outline is taken as the four tangency points and two straight lines. A real drawing also has to decide which arcs of the two ellipses are visible and where the visible arc hands over to the straight side, which is exactly at the tangency points — so getting them wrong puts the hidden-line break in the wrong place as well, and nothing here measures how visible that is.

Only one turn of the cylinder is measured across the frame. The offset from the principal ray is swept and the angle the cylinder is turned to is held at thirty-five degrees. That angle decides how foreshortened each end is and therefore how far apart their aspect ratios can get; a cylinder nearly end-on to the camera has two very flat ellipses whose difference is proportionally larger and absolutely smaller, and which of those a draughtsman would notice is not asked here.

The construction compared against is the strict one. A draughtsman who draws the outline by eye, tangent to both ellipses wherever it looks right, gets a better answer than the rule gives, because the eye is fitting a tangency rather than following an instruction. That is a common shape in this field and it is not measured here.

Still open: what a taper does to the tangency #

A cone frustum is the same object with one number added, and the number changes the outline’s character rather than its size.

For a cylinder the two rulings are parallel in space and their images meet at the vanishing point of the axis. For a frustum they meet at the apex, which is a finite point in space, so their images meet at the image of that point — a quite different place, and one that does not depend on the camera at all. The two constructions a draughtsman has therefore disagree about where the sides converge, and only one of them is right.

The measurement takes a frustum of a stated taper, computes the contour generator as the intersection with the eye’s polar plane, and reports where the sides actually meet against both candidate points — the axis’s vanishing point and the image of the apex — as the taper goes from zero to a few degrees. A taper of a degree or two is what a moulded or turned part has, and the question worth answering is whether the point the sides meet at swings from one candidate to the other quickly or slowly, since a draughtsman choosing the wrong one at a small taper is making an error that grows without warning.

The short version #

A cylinder’s two end circles do not image as one ellipse used twice. Off the principal ray their major axes lie 14.4 degrees apart and their aspect ratios are 0.918 and 0.839, and the difference falls in proportion to the distance — 0.35 degrees at forty times as far — so it is a perspective effect entire.

The outline’s straight sides are the images of the two rulings where the cylinder turns away from the eye, and they touch each end’s ellipse 28 degrees round it from the nearest major-axis end, 35 pixels away. The construction that joins the major-axis ends is therefore 11.1 pixels out. That error does not vanish with distance: off the principal ray it settles at 7.3 degrees of tangency offset and 0.72 pixels, because the far limit of an off-axis perspective view is an oblique parallel projection rather than an orthographic one. On the principal ray it does vanish, which is the case the rule was written for.