The rims carry a part's taper, as finely as they are drawn

A tapered part meets at its apex followed a turned part’s two drawn sides to where they cross. A cylinder’s sides meet at the vanishing point of its axis; a part narrowing away from the eye by a degree a side meets at the image of its apex, 54 pixels from that point on a 690-pixel photograph, and the distance grows in proportion to the taper. So the sides carry the taper — but only as a gap between their crossing and the axis’s vanishing point, and two lines cannot say where that vanishing point is. A cylinder pointing at the crossing and a tapered part pointing somewhere else draw the same two lines.

The essay ended on the place the vanishing point can come from: the rims. Each rim is a circle in a plane square to the axis, and an ellipse’s shape carries the orientation of the plane its circle lies in. The question was whether the axis can be recovered from the two rim ellipses alone, the taper read from it, and how small a taper survives rims drawn the way rims are drawn — by hand, to a line’s width, or with an ellipse guide whose shapes come in steps. If a one-degree taper on a part this size can be read from hand-drawn rims, the drawing carries its own taper. If it takes five degrees, every modest taper in a drawing is unreadable, and the draughtsman’s habit of drawing every turned part as a cylinder costs nothing any reader could detect.

It can, at the edge of what a line’s width allows; and it depends on which way the part points far more than on the taper.

Two rims agree on one axis #

The part is the earlier essay’s: a turned part 1.3 metres long with a radius of 0.45 metres at its middle, lying level 3.4 metres in front of an eye 1.55 metres up and 0.8 metres to the side of the principal ray, turned 35 degrees from square to the picture. On the left are its two rims as the photograph draws them, as exact ellipses; nothing else on the page is used. The outline’s sides are not used, because each is tangent to both rims — the earlier essay’s practical advice was to draw them that way — so they carry nothing the rims do not.

A picture’s ellipse is the image of many circles, but not of circles in any orientation. Two circles, one picture found that the cone of rays from the eye through an ellipse is cut in a circle by planes of exactly two orientations, each at any distance. One ellipse therefore says which way its circle’s plane faces up to a choice of two. A turned part has two rims in parallel planes, and each rim offers its own pair of orientations; they share one, the true one, and the other two disagree. The axis is the direction that makes both rims circles at once. Found that way, it lands on the axis’s true vanishing point at (825.604, 102.553) pixels, to the third decimal.

The right of the figure turns the view to look down that axis. Each rim becomes a circle, centred on the axis, and the two circles’ centres must both lie on it, which fixes how much further from the eye the far rim’s plane is than the near one’s. With that, each rim’s true size is known in units of the near rim’s distance, and the far rim’s is drawn beside the near rim’s: for the 1° part it is 95 per cent of it, where a cylinder’s would be the same. The difference in radius over the distance between the planes is the taper. It reads 1.000000 degrees.

Nothing in that reading knows the part’s size, its distance or its length. A taper is a ratio — a change of radius over a length — and the rims carry both terms of it up to the one scale a photograph never has. The slider runs the taper from nothing to three degrees, and the reading follows it to the sixth decimal at every step.

A far rim is small for two reasons #

The reading needed the rims’ shapes and not merely their sizes, and the reason is worth seeing on its own. A reader comparing the two rims of a drawn part sees the far one smaller, and a taper would make it smaller; but so would perspective, which draws the further rim smaller by itself, and more so the more directly the part points away. A cylinder whose axis lies 14 degrees from the line of sight draws its far rim at 0.691 of the near one’s long axis; lying 70 degrees from it, at 0.882. A one-degree taper takes each of those down by about five per cent.

So the ratio of the rims is one number with two causes. A cylinder lying 31 degrees from the line of sight draws its rims in the ratio 0.729, and a part tapering by a degree lying 41 degrees from it draws them in 0.719: a hundredth apart, for two parts that differ by a degree of taper. Every ratio on the cylinder’s curve lies on each tapered part’s curve at some other turn. The sizes of the rims are where the taper shows, and they are also where the turn shows, and nothing in the sizes says which is which.

The shapes do. How round an ellipse is, and which way its long axis lies, are what say which way its circle’s plane faces, and that is the turn, read independently of the sizes. With the turn fixed, the sizes have only the taper left to explain. That is the whole content of the reading down the axis, and it is why the earlier essay’s question — whether the rims can tell a taper from a turn — has a yes for an answer when the rims are exact.

A line’s width against a degree #

Exact rims are not what a drawing has. A rim drawn by hand wanders from the true ellipse by about the width of the line it is drawn with. The figure draws each rim through 48 points round the true ellipse, each moved across the curve by a stated error, fits an ellipse to them, and reads the taper from the two fitted rims as before, eighty times at each size of error.

The spread of the taper read grows in proportion to the scatter, about eight tenths of a degree for every pixel. Rims drawn to a tenth of a pixel read a one-degree taper as 1.006° ± 0.081; to half a pixel, 1.023° ± 0.406; to a pixel, 1.031° ± 0.814; to two pixels, ±1.63. A taper is told from none when it stands about two spreads clear of zero, so a part this size drawn with rims to half a pixel shows a taper of eight tenths of a degree, and one drawn to a pixel shows a taper of about a degree and a half.

On this photograph’s scale, a pixel is about a third of a millimetre on a page printed at its width, and a careful pencil line is about that wide. So the earlier essay’s question has its answer at its own example: a one-degree taper on a part this size, drawn by hand, sits right at the edge of what the rims can say. Two degrees is plainly readable; half a degree is not. A drawing with its ellipses plotted point by point from a construction, to a tenth of a pixel, carries a taper to a tenth of a degree.

Which way the part points matters more than the taper #

The same rims, drawn to half a pixel, read the taper twenty times better when the part points away from the eye than when it lies across the view. With its axis 14 degrees from the line of sight the spread is ±0.15°; at 36 degrees, the part of the figures above, ±0.41°; at 51 degrees, ±0.98°; lying 80 degrees from it, nearly square to the line of sight, ±3.13°.

Two things go wrong together as the part turns across the view. Its rims become thin ellipses, and a thin ellipse’s roundness says little about which way its plane faces: a small error in its short axis turns the plane a long way. So the axis is read loosely, and the taper is read through the axis. And its two rims stand at nearly the same distance from the eye, so the far rim is drawn at nearly the near rim’s size and the taper’s small difference in size is a small difference between two nearly equal numbers. A part pointing away draws its rims nearly round, which fixes the axis firmly, and at distances that differ, which makes the comparison of their sizes a comparison of unlike things.

The mean drifts as the spread grows, to 1.10° at 51 degrees and 1.28° lying across the view. Once the error is no longer small against the reading, the reading stops being symmetric about the truth. That is a second reason not to trust a taper read from a part lying across the view: it is both loose and leaning.

For a draughtsman this inverts the usual advice. A turned part is drawn most often from the side, its axis across the view, because that shows its profile; that is exactly the view in which its rims say least about its taper. The taper shows best in the view that hides the profile — the part pointing away — and on a drawing made from the side, the taper lives in the sides’ crossing, which the earlier essay found moving 54 pixels for a degree, and not in the rims.

A guide’s step reads as a taper #

The other way rims are drawn is with an ellipse guide: a plastic sheet of ellipses cut at a set of shapes, the shape named by the ellipse angle — the angle whose sine is the short axis over the long, which is the angle at which a circle would have to be seen to draw that ellipse. Guides come in steps of five degrees, and finer sets in steps of two and a half. A draughtsman sets the guide at the true centre with its long axis along the true one and picks the shape nearest the truth. The ellipse the drawing office draws priced the four-arc ellipse that some guides are cut to; this is the step between their shapes.

The figure draws a cylinder that way at five turns and reads the taper it should not have. With five-degree guides, a cylinder whose axis lies 36 degrees from the line of sight reads as tapering by 1.20°; at 22 degrees, by 0.52°; at 14 degrees, by −0.60°, flaring. Lying 51 degrees from the line of sight it reads −2.76°, and at 65 degrees the reading falls apart entirely, −35.3°. Two-and-a-half-degree guides read the 36-degree cylinder as −0.01°, almost exactly right, and the 51-degree one as −1.81°.

The pattern is chance, and it is chance of a precise kind. The two rims of the 36-degree cylinder have true ellipse angles of 66.6° and 57.0°. A five-degree guide rounds the first down to 65° and the second down to 55°, by different amounts, so the two drawn rims face in slightly different directions and the reading takes the difference for a taper. A two-and-a-half-degree guide rounds them to 67.5° and 57.5°, both up and by similar amounts, and the errors nearly cancel. Which happens depends on where the true angles fall between the guide’s steps, which depends on the part’s turn and distance, which the draughtsman did not choose for this purpose.

So a drawing made with guides carries no taper that can be read. Its rims are wrong in their roundness by up to half a step, and roundness is exactly what says which way the planes face; the error it puts into the taper is as large as any taper a turned part has. This answers the earlier essay’s last question in its harsher form. For a drawing made with guides, the draughtsman’s habit of drawing every turned part as a cylinder costs nothing a reader could detect, because the guide has already lost what the habit would lose.

What the rims can carry #

Put together, a drawing’s two rims carry its part’s taper exactly when they are exact, and as well as they are drawn when they are drawn. The axis is the one direction under which both rims are circles; down it, the rims are two circles on one axis whose sizes differ by the taper times the distance between them. The reading needs no size, no distance and no outline. Drawn by hand to half a pixel, it reads a degree of taper on a part pointing away from the eye and loses it on a part lying across the view; drawn with an ellipse guide, it reads the guide.

A cylinder has two different ends found that a cylinder’s two rims are two different ellipses even with no taper at all — 14.4 degrees apart in the direction of their long axes and 0.918 against 0.839 in roundness for the cylinder it measured — and that difference is the turn, drawn twice. It is the difference this reading uses. The drawing-office habit of drawing both ends with one ellipse shape, which that essay found wrong for a cylinder, is wrong here in a further way: two rims drawn alike say the part’s planes face one way and its rims stand at one distance, and a taper read from them is a taper of the habit’s making. The minor axis is not the axle began this sequence with a single wheel, whose one ellipse could not by itself say which way it faced. Two rims can, and that is all a taper needed.

What the reading takes from outside the drawing #

The camera’s focal length and centre are known. The cone of rays through an ellipse is a cone from the eye, and its circular sections depend on where the eye is relative to the page — the focal length and the principal point. A drawing whose viewing distance is not known has a family of possible axes, and with it a family of tapers. The conic a circle becomes found how far a circle’s drawn centre already sits from its ellipse’s centre with the camera known; nothing here measured what an unknown camera costs a taper.

The rims are the part’s rims. A rim hidden in part behind the near end, as a part pointing away hides its far rim’s lower edge, is drawn as an arc and fitted as one. An arc fits an ellipse less firmly than a closed curve, and most strongly in its roundness, which is what the reading needs most.

The scatter is independent from point to point. A hand drawing a rim errs smoothly — a curve too round all along one side, not jittered across it — and a smooth error is an error in the ellipse’s shape rather than noise to be averaged. Half a pixel of smooth error is probably worth more than half a pixel of jitter to the taper, and was not measured.

The guide is set exactly at the true centre and along the true long axis. A draughtsman sets it by eye on the drawn centre lines, and which way the drawn circle leans found a circle seen off the principal ray leaning away from the lines a draughtsman sets it on; those errors would add to the step’s.

Still open: whether the sides and the rims read the taper better together #

The rims read the taper from their shapes and sizes. The sides read it from their crossing, which moves 54 pixels for a degree of taper on this part, but only against a vanishing point they cannot supply. Each source is weak where the other is strong: the rims say least about a part lying across the view, which is when its sides are longest and their crossing best defined; a part pointing away has rims that read well and sides so short their crossing is barely fixed.

The measurement that settles what the two are worth together draws a part’s rims and sides by hand with stated scatter, reads the taper three ways — from the rims alone, from the sides’ crossing against the axis the rims supply, and from all of them fitted at once as one tapered part — and asks how the spread of each changes as the part turns from pointing away to lying across the view, and whether a drawing of a part seen from the side, the view a turned part is usually drawn in, carries a degree of taper after all once its sides are counted.