A tapered part meets at its apex

A cylinder has two different ends took the drawing office’s construction for a thickened wheel and found both of its halves wrong: the two end circles image as two different ellipses, and the straight sides touch neither of them where its major axis ends. It left one thing standing, because it was never in question for a cylinder. The two sides of a drawn cylinder, carried on far enough, meet at the vanishing point of its axis.

That is true because a cylinder’s two outline rulings are parallel lines in space, and parallel lines share a vanishing point. It stops being true the moment the part is turned rather than extruded. A table leg, a bottle neck, a flowerpot, a paper cup, the shank of a tool: almost nothing made on a lathe or in a mould is a cylinder, because a mould needs a draft angle to release and a lathe cut is usually shaped. Each of those objects is a slice of a cone, and a cone’s rulings are not parallel. They all pass through one point, the apex.

So the drawn sides of a tapered part must meet at the image of that apex, which is a finite point in space and has an image like any other point. The question worth measuring is not whether this is so — it follows from what a straight line is — but how far it takes the meeting point from the place a draughtsman who thinks of the part as a cylinder would send the sides, and how quickly. A taper of a degree or two is invisible on the object. The hope behind the cylinder habit is that it is invisible in the drawing as well.

It is not, and the reason it is not says something about where the evidence for a taper lives on the page.

The sides meet where the apex is drawn #

The part measured here is the cylinder of the earlier essay with one number added. Its axis runs level, 0.62 metres above a floor and 0.8 metres to the side of the principal ray, 3.4 metres in front of an eye 1.55 metres up, turned 35 degrees; its middle section has a radius of 0.45 metres and it is 1.3 metres long. The taper is the angle each side makes with the axis, and a positive taper narrows the part as it goes away from the eye. The middle section stays exactly where it was at every taper, so a sweep changes the two ends and nothing else.

The outline is found the way every quadric has one outline finds it for any solid of this kind. The eye sees the surface turn away along the rulings whose tangent planes pass through the eye. For a cone that condition does not depend on the position along the ruling, so the contour generator is two whole straight rulings, and the drawn sides are their images. Written out, the condition is the cylinder’s own equation with the radius replaced by the radius plus the taper times the eye’s distance along the axis — which at zero taper is the cylinder’s equation. The two routines, the old one for a cylinder and the new one for a frustum, agree there to the last bit of the arithmetic, and that agreement is checked rather than assumed.

The meeting point is then found as the crossing of two drawn lines, each through the two points where its ruling touches the two rims. It is never placed at the image of the apex. That matters, because “the sides meet at the apex” would otherwise be a definition dressed as a result. Computed separately, the apex’s image and the crossing agree to a few trillionths of a pixel at every taper tried on both sides of zero, and to a few billionths close to the one flare, met below, at which the two sides almost fail to cross at all.

At zero taper the dot sits in the ring. At one degree it has moved 54 pixels toward the part; at two degrees, 94. The ring does not move at all, because the axis’s direction has not changed — only the part’s radius along it has. A draughtsman therefore has two candidate points on the board, both computable, and exactly one of them is where the sides go.

The candidate that is right is also the one that has nothing to do with the camera’s orientation in any special way. The apex is a point on the line of the axis, about 26 metres beyond the middle of the part at one degree of taper; it has an image because every point in front of the eye has one. The vanishing point is where the axis’s direction goes, and it would be the right answer only if the apex were infinitely far away, which is what zero taper means.

A quarter of a degree is not nothing #

The natural hope is that a small taper behaves like no taper: that the meeting point stays glued to the vanishing point until the taper is large enough to see on the object, and then moves. Mathematically the question is whether the meeting point’s distance from the vanishing point grows from zero with a flat start, as a square or a higher power of the taper, or linearly.

It grows linearly, and it starts at once.

A quarter of a degree of narrowing puts the meeting point 15 pixels from the vanishing point, half a degree 29, a degree 54. Those are not large on a board, but they are not small against what they are about: at one degree the meeting point has come a seventh of the way from the vanishing point back to the part itself.

The reason the curve has no flat start is visible in where the apex is. At a taper α the apex stands about r/tan α beyond the middle of the part along its axis, and a point at distance t along a line is drawn at a distance from that line’s vanishing point that falls off as one over t. One over r/tan α is tan α over r. So the displacement is proportional to the taper itself from the first hundredth of a degree, and the only thing that sets its size is the part’s radius against its distance and how steeply its axis recedes. There is no threshold below which the approximation is free, only a size below which it stops mattering on a particular sheet.

On the narrowing side the curve bends over, because as the taper grows the apex comes in toward the part, and a nearby point moves less on the page per metre than a distant one. On the flaring side it does the opposite and steepens without limit. That asymmetry is the most interesting thing in the figure, and it gets its own section below.

A part drawn as a cylinder loses its taper #

The size of the error a draughtsman makes by drawing a tapered part’s sides toward the axis’s vanishing point can be put in pixels on the drawing, and then it can be compared with something better than a fixed tolerance: the taper’s own mark on the page.

The habit is modelled generously. The near rim is drawn correctly, including the correct touch points of the true outline on it, and each side is then drawn from its touch point toward the vanishing point of the axis. The error is the distance from that line to the point where the true side touches the far rim. The taper’s own mark is how far the far rim’s touch points move from where a cylinder with the same near rim would put them — how much of the far end’s drawing is there because of the taper.

At two degrees the habit misses the far rim by 8.47 pixels and the taper moves that rim by 9.42. The ratio is 0.900, and it hardly changes across the sweep. The habit is therefore not a small error laid over a correct drawing of a tapered part. It draws a cylinder. What it throws away is nine tenths of the only evidence of the taper that the drawing contains, and it throws that away at every taper, large or small, because both quantities grow linearly from zero together.

That is a sharper statement than “the habit is inaccurate”, and it changes what the habit should be compared with. A two-degree taper on a part this size moves the far rim by less than ten pixels on a 690-pixel photograph, and a reader might reasonably decide that ten pixels is below what anyone would notice. If so, the habit is harmless — but for the plain reason that the taper was invisible anyway, not because the construction is good. Wherever the taper is large enough to be seen, the habit removes it.

The ratio is not exactly one because the near touch points themselves move as the taper changes: the rulings along which a cone turns away from the eye are not the rulings a cylinder of the same near radius turns away along. The measurement reports the ratio rather than deriving it, and it has been checked at this one part, this one turn and this one camera.

A flare can print its sides parallel #

Now the other side of the swing. A part that widens as it goes away from the eye — a flowerpot lying on its side with its mouth toward the far wall, a trumpet bell seen from the mouthpiece end, a lampshade — has its apex on the near side, out along the axis toward the viewer. For a small flare the apex is a long way out: at one degree it is 17 metres behind the eye. A point behind the eye still has an image in the projective sense, through the principal point on the far side of the frame, as what happens behind the eye makes precise — and that image lies beyond the vanishing point, which is why the meeting point moves past it and away from the part. The sides still converge with depth, only less than a cylinder’s do.

The apex comes nearer as the flare grows, and at one particular flare it lies exactly in the plane through the eye parallel to the picture. A point in that plane has no image. The two drawn sides then have nowhere to meet, and they print exactly parallel — in a correct perspective photograph, of a part that recedes steeply from the viewer.

For this part the flare is 5.92 degrees a side. Short of it, at 80 per cent of that flare, the sides converge with depth; past it, at 125 per cent, they spread with depth. Past it the apex has crossed the eye’s frontal plane and stands in front of the eye, between the viewer and the part — half a metre in front at a seven-degree flare — and the sides meet at its image, which now lies on the near side of the part.

This is worth holding against a claim an earlier essay here has already corrected. The picture whose lines spread takes a Byzantine table whose sides diverge with depth and the standard account of it — that the vanishing point is behind the viewer — and shows the account is wrong for that picture: its meeting point is in front of the eye, and the drawing is a correct image of a leaning plane, as an inverse perspective is a leaning plane develops.

The flared part is a second, independent instance of the same correction, and it is the more striking one because the object is real and the photograph is exact. Its sides spread with depth, and the point they come from is in front of the eye — a real point on the axis, between the viewer and the part — exactly as the icon’s was. The point that really is behind the viewer belongs to the gentle flare, whose sides do not spread at all: they converge, a little less than a cylinder’s. So the standard account has the order backwards twice over. Spreading sides go with a meeting point in front of the eye; a meeting point behind the eye is what a picture with ordinary-looking, converging sides can conceal.

How much flare it takes #

The condition for parallel sides is simple enough to write down, and the formula is more useful than the number. The apex has to lie in the eye’s frontal plane, which happens when

$$\tan\alpha = \frac{r\cos\psi}{d}$$

where r is the part’s radius at its middle, d is the depth of its middle in front of the eye, and ψ is the angle between the part’s axis and the line of sight. At the part drawn here r is 0.45 metres, d is 3.51 metres and cos ψ is 0.808, which gives 5.92 degrees.

Pointing within about eleven degrees of straight away, the part needs a flare of 7.2 degrees before its sides print parallel. Turned to lie nearly across the view, 0.63 degrees does it. The formula says why: cos ψ is the share of the axis that recedes, and a part lying across the view hardly recedes at all, so its sides are nearly parallel on the page to begin with and the smallest flare tips them over.

The distance term says something less obvious. The flare needed falls as one over the distance, so the same object seen from further away prints its sides parallel at a smaller flare. Walked far enough away with the lens lengthened to hold its size — the drawing office’s own limit, which a cylinder has two different ends uses — every flare, however small, draws its sides spreading, and every narrowing draws them converging. That is exactly what a parallel projection of a cone does. The perspective case is the one with a threshold in it; the drawing office’s case is the one without.

Where a draughtsman meets it #

Three practical readings follow, and none of them needs the numbers above to be exact for a particular drawing.

Draw the sides tangent to both rims, and let them meet where they meet. The two true sides are tangent to both rim ellipses, because each is the image of a ruling that lies in a plane tangent to the surface along its whole length. A draughtsman who draws each rim correctly and then lays a straightedge against both gets the true outline without ever locating the apex — and gets it for a cylinder, a narrowing part and a flaring one alike. The construction that fails is the one that fixes a point first and draws toward it.

The vanishing point of the axis is still the right tool for the rims, not the sides. The rims are circles in parallel planes square to the axis, so everything about their placement — where their centres sit along the drawn axis, how they foreshorten — is still governed by the axis’s direction. The earlier essays’ warnings about the rims themselves stand: the conic a circle becomes for why the drawn centre is not the ellipse’s centre, and which way the drawn circle leans for its tilt. What changes with a taper is only the sides.

A drawn part whose sides are parallel, or spread, is not thereby drawn wrong. A student’s flowerpot whose sides spread slightly as they go back will be corrected by a teacher who has absorbed the cylinder habit. Whether the correction is right depends on the pot’s flare, its turn and its distance, and the formula above decides it. For a pot flaring by eight degrees a side and lying almost end-on, the student is right.

Two lines cannot tell a taper from a turn #

The meeting point says a great deal about a drawn part, but it cannot say everything, and it is worth being exact about which part of the reading it cannot supply.

Two straight lines on a page meet at one point. That point is consistent with a cylinder whose axis points at it, and with a frustum whose axis points somewhere else and whose apex is drawn there. Given only the two sides, a reader cannot separate how much of their convergence is recession and how much is taper, because both produce a pair of straight lines through a point. It is the same kind of ambiguity the minor axis is not the axle found in a single ellipse: a drawn shape that fits several objects equally.

What breaks the tie is the rims. Their shapes carry the orientation of the planes they lie in, and that orientation fixes the axis’s direction, and so the vanishing point, independently of the sides. A reader who can recover the axis from the rims can then read the taper off the gap between the vanishing point and the meeting point — which, as the swing above shows, grows linearly with the taper and is therefore a usable measurement even at a degree. Whether that recovery is precise enough, from rims drawn at a practical size, to read a one-degree taper is not measured here, and it is the question that remains.

What one part on one floor leaves unmeasured #

One part, one camera, one place. The swing, the habit ratio and the parallel flare are measured on a single frustum at a single position in one 54-degree photograph. The formula for the parallel flare is exact for any part; the 0.900 ratio is not a formula, and nothing here says how it varies with the part’s position in the frame.

The taper is uniform. A turned part with a shaped profile — a baluster, a bottle — is not a cone, and its outline is not two straight lines. Each short stretch of it behaves locally like a cone of its own taper, and the outline is the envelope of their rulings. That is a curve whose shape this measurement does not reach.

The rims are taken as drawn exactly. Every rim here is the exact image of its circle. A draughtsman using a template for the ellipses, which the ellipse the drawing office draws prices, has a second error laid over the one measured here, and whether the two add or partly cancel is not asked.

Visibility is ignored. At a strong flare seen nearly end-on the far rim can be hidden entirely behind the near one, and at a strong narrowing the near rim can hide part of the far. The measurement draws every line whether or not a real part would show it.

The eye is a pinhole. Nothing here is about a lens with distortion, which bends the sides themselves and would move the meeting point for reasons that have nothing to do with the part.

The apex, not the vanishing point #

A tapered part’s two drawn sides are the images of the two rulings along which it turns away from the eye, and those rulings pass through the apex. So the sides meet at the image of the apex, and the vanishing point of the axis is the right answer only at zero taper. The move away from it is linear in the taper from the start: 15 pixels at a quarter of a degree for the part measured here, 54 at one degree. A side drawn toward the vanishing point throws away nine tenths of the taper’s own mark on the far rim, at every taper.

Flaring away from the eye, the apex starts behind the viewer and comes toward the eye’s frontal plane, and when it reaches it — at a flare of r·cos ψ/d, 5.92 degrees for this part — the two sides print parallel in a correct photograph. A larger flare puts the apex in front of the eye and draws sides that spread with depth.

Still open: whether the rims can tell a taper from a turn #

The sides of a drawn part fix one point on the board and nothing else, and the gap between that point and the axis’s vanishing point is proportional to the taper. So a drawing measures a taper exactly as well as it locates the vanishing point of the axis — which the sides cannot supply and the rims can.

The measurement that follows recovers the axis’s direction from the two rim ellipses alone, as the vanishing line of the planes they lie in and the pole of that line with respect to each rim; reads the taper off the gap between the recovered vanishing point and the sides’ crossing; and then scatters the rims by a stated amount — a line’s width, a template step — to ask how small a taper survives the scatter. If a one-degree taper on a part this size can be read from rims drawn by hand, the drawing carries its own taper. If it takes five degrees, then every modest taper in a drawing is unreadable and the draughtsman’s habit, wrong as geometry, costs nothing a reader could ever detect.