A turned part's outline reads its swelling from every side, and a straight taper hides it

A turned part drawn from the side keeps its taper in its sides fitted a drawing of a turned part as one tapered solid — its two rims and its two sides together — and read a one-degree taper to ±0.032° with the part lying across the view, a hundred times better than the rims alone. Every reading in that essay assumed the part was a frustum: a straight taper from one rim to the other, one number of shape beyond the scale.

Turned parts are rarely that simple. A spindle swells in the middle, a baluster has a neck and a bulb, a column has its entasis, and in each the side of a drawing is a curve rather than a line. Every quadric has one outline found the outline of a ball, a dish or a hyperboloid exactly, as the section by one plane; a part of arbitrary profile has no such shortcut. Its outline is the curve along which each of its circles turns away from the eye, and the earlier essay closed by asking whether a drawing reads that profile — how many numbers of it, to what precision a line’s width allows — and whether the side view, which starves the rims, is the one that shows the profile best.

It reads the profile to a few thousandths of the radius. It reads it about equally well from every view, so the side view has no special claim on it. And the reading that assumes a straight taper does not merely miss the swelling. It converts it into taper.

The swelling is a bulge in each side #

The part is the earlier essays’: 0.45 metres in radius at its near rim, 1.3 metres long, 3.4 metres from the eye, narrowing away from the eye by a degree a side. Along it now runs a swelling, a smooth bulge that is nothing at either rim and rises to 15 per cent of the near radius two fifths of the way along — about seven centimetres at its crest, the proportion of a baluster’s bulb. Since the swelling is nothing at the rims, the rims are exactly the plain part’s rims: the two ellipses that the earlier essays read the taper from are unchanged, and everything new lies in the sides.

The side of a turned part is where each of its circles turns away from the eye. For a straight taper that place lies along one straight ruling of the surface and the side is a straight line in the picture. For a swollen part the surface’s slope changes along it, so the place where each circle turns away shifts round the circle as well as out, and the side comes out as a curve with the swelling’s bulge in it. The model the figure fits computes that curve exactly, circle by circle, from nine numbers: where the near rim’s centre is, which way the axis points, the part’s length and taper, and the swelling’s height and the place of its crest. The near radius stays at one, because a drawing has no scale.

The drawing is made the way the earlier essays made theirs: forty-eight points round each rim and twenty-four along each side, every one moved half a pixel across its line at random. Over twenty such drawings the fit reads the swelling as 0.1497 ± 0.0024 of the near radius, its crest at 0.399 ± 0.003 of the length, and the taper as 0.99° ± 0.03. The taper is read as well as the earlier essay read it from a plain part, ±0.032°: allowing the side to curve has cost the taper nothing.

A straight taper fitted to a swollen part #

The figure draws the part’s radius along its length. The true profile and the read one lie on each other; the band of two spreads either way on the swelling’s height is a few thousandths of a radius wide, finer than the line drawing it.

The dashed line is what happens to the same drawings when they are read as the earlier essays read them, as a frustum. The rims are the plain part’s rims, and a rim fit alone would find the right taper, roughly; but the sides are curved, and the straight sides of a frustum can follow them only by leaning. The fit leans them, and in leaning them it tilts the whole solid: the taper comes out as 2.22° ± 0.04 instead of 1°, nearly thirty times its own spread from the truth. With the drawing’s sides straightened into a cone’s, the model then misses every side point by a growing amount from the near rim, and the fit’s residual is about seven times the drawing’s half pixel. Nothing in the frustum reading’s own result says it is wrong except that residual, and a reader who was not looking at it would report a part that tapers twice as fast as it does.

That is the earlier essay’s lesson in reverse. The sides carried the taper because a straight side’s slope against the axis is the taper; a curved side’s slope at any one place is the taper plus the swelling’s own slope there, and a reading that has only one number for both gives it all to the taper.

Every view reads the swelling about as well #

The earlier essays found the rims and the sides failing in opposite views: the rims read the taper best with the part pointing away from the eye and lose it as the part turns across the view, while the sides do the reverse. The expectation was that the profile, living in the sides, would follow the sides and be read best from the side.

It does not follow them. The swelling’s height is read to ±0.0017 of a radius with the part square to the picture, ±0.0017 turned ten and twenty degrees, ±0.0016 at thirty-five, ±0.0019 at fifty, ±0.0021 at sixty-five and ±0.0024 at eighty. The crest’s place is read to between ±0.0024 and ±0.0047 of the length at every turn. Over the same turns the taper’s spread runs from ±0.14° to ±0.03°, a factor of nearly five, worst when the part is turned ten or twenty degrees and best lying across.

The reason is in what each number asks of the drawing. The taper is a difference between the two ends: the radius at one rim against the radius at the other over the length between them, and how well that is read depends on how the rims and the sides share the work, which the turn decides. The swelling is a bulge in the middle of each side, a departure of the side from straightness, and it is seen square to the line of sight from every view in which the sides are seen at all. Turning the part changes how long the sides are in the picture and how foreshortened, but the bulge stays a bulge across the side, the direction in which the drawn line is read.

So the side view is not where the profile is read best; it is where the taper is. The view in which a turned part is usually drawn costs the swelling a little, ±0.0024 against ±0.0017, and gains the taper a factor of four.

How small a swelling a drawing shows #

The figure shrinks the swelling. With none at all, the fit reads −0.002 ± 0.005 and puts the crest anywhere, 0.58 ± 0.34 of the length, which is what a crest with nothing under it should do. With a swelling of one hundredth of the near radius it reads 0.0098 ± 0.0022, four and a half spreads from nothing; two hundredths, 0.0197 ± 0.0022; five hundredths, 0.0497 ± 0.0023. The spread barely changes with the size, so the smallest swelling a half-pixel drawing shows is about a hundredth of the radius.

On this part a hundredth of the radius is four and a half millimetres, which at 3.4 metres is a little under one pixel of the photograph. No single point of the drawing could show it; the swelling is read from forty-eight side points each half a pixel off, whose departures from a straight line average to a few hundredths of a pixel. The earlier essay found the same arithmetic for the taper, and the rims carry a part’s taper, as finely as they are drawn before it: a line’s width is not the limit of what a drawing measures, because a drawing is many readings of one shape.

The frustum reading is sensitive to the swelling in a costly way. A hundredth of the radius moves its taper from 0.99° to 1.07°, two hundredths to 1.14°, five hundredths to 1.37°: about seven degrees of taper for every unit of swelling, so that a bulge too small to see moves the taper by more than twice its spread. A reader who assumes a straight taper and finds one of 1.07° cannot tell whether the part tapers that much or swells by a hundredth of its radius. The profile reading can.

Where a swelling can be placed #

The crest can be anywhere along the part, and the drawing places it equally well wherever it is: at a fifth of the length it is read as 0.199 ± 0.0025; at two fifths, 0.399 ± 0.003; at three fifths, 0.600 ± 0.0027; at four fifths, 0.800 ± 0.0023. A spread of a quarter of a hundredth of the length is three millimetres on this part. The swelling’s height is read to between ±0.0016 and ±0.0030 at every place, worst near the near rim, where the swelling rises steeply out of a rim that fixes the scale.

The frustum reading’s error depends on the place, and the dependence is the clearest picture of what it is doing. With the crest a fifth of the way along it reads the 1° part as 2.80°; two fifths, 2.22°; three fifths, 1.23°; four fifths, 0.48°. A straight side laid against a bulge leans away from whichever end the bulge is nearer. Since the near radius is fixed, a bulge near the near rim tilts the side down towards the far rim and makes the part look more tapered, and a bulge near the far rim props it up and makes it look less. A swelling at four fifths of the way along would be reported as a taper of half a degree, half the truth, by a reading with no word for swelling.

A tapered part meets at its apex found the sides of a frustum crossing at the image of its apex. A curved side has no apex; the straight line a frustum reading fits to it crosses its partner wherever the bulge has tilted them, and that crossing is where the reading puts an apex that is not there.

The profile scales with the care of the drawing #

The last figure changes the care of the drawing. A quarter-pixel drawing reads the swelling to ±0.0012 of the radius and its crest to ±0.0015 of the length; half a pixel, ±0.0024 and ±0.0029; a pixel, ±0.0048 and ±0.0058; two pixels, ±0.0099 and ±0.0117. Each doubling of the scatter doubles the spread, as it did for the taper: the information is in the average of many points, and a line twice as rough is worth a quarter as many.

At two pixels, a quick sketch on this photograph’s scale, one drawing in twenty is so rough that its rims put the axis where none of the part’s sides would be visible, and no reading can start from it; the other nineteen still read the 15 per cent swelling to a fifteenth of itself. The taper at that care is read to ±0.14°. A sketch of a baluster that a draughtsman would call careless still says how far the bulb swells and where.

What a drawing’s outline carries #

Put together, a drawing of a turned part with its rims and its curved sides reads the part’s profile — a swelling’s height and the place of its crest — to a few thousandths of the radius and of the length, at half a pixel, from any view in which the sides are seen. It reads the taper at the same time and as well as the earlier essay read it from a plain part. And a reading that assumes the part is a frustum puts the swelling into its taper, by seven degrees of taper for every unit of swelling, in a direction set by where the swelling lies.

A cylinder has two different ends began this inquiry with a part that has no shape at all beyond its ends, and the minor axis is not the axle with the ellipse that every drawing teacher shows. Each essay since has found more in the outline than the teaching uses: the axis in the rims, the taper in the sides, and now the profile in the sides’ curvature. A turned part drawn by hand is a better record of its shape than the conventions for drawing it suggest, because the hand that draws the side has drawn every circle’s turning-away at once.

A shape read by its shape #

There is a pattern here that this field has met before. The rule’s bow is read by its shape found that a painted pavement’s departure from the true construction is not only a size but a form — one smooth arc, to one side — and that fitting the form with one free amplitude convicts the rule where measuring its worst corner could not, because a hand’s scatter has no such form. The swelling is read the same way. At a hundredth of the radius it moves any one side point by less than a pixel, inside the drawn line’s scatter, but it has a form that the scatter of a hand does not: one bulge, rising from nothing at the near rim, falling to nothing at the far one, the same on both sides of the part. The fit asks only how tall that form is and where its top stands, and every side point contributes a little to both answers.

The rims answer a different kind of question, and the essays on them were reading a different kind of evidence. Which way the drawn circle leans and six tangents and the point nobody drew tested an ellipse’s orientation and its exactness, properties of a closed curve that every point of it shares. A side is an open curve, and what it carries is how its direction changes along its length. That is why the profile appears in the sides alone and why no improvement in drawing the rims would help to read it.

What the reading takes from outside the drawing #

The profile has the form the reading assumes. The swelling here is one smooth bulge, nothing at the rims, with a height and a crest and a fixed relation between its width and its place. A real baluster has a neck and a bulb, a column’s entasis is a curve with no crest, and a fillet is a step. Each would need its own numbers, and a reading with the wrong form would put the difference into the numbers it has, as the frustum reading did.

The sides are drawn as curves, freehand. The earlier essay found ruled sides nearly as good as freehand ones for a straight taper; a ruled side cannot carry a curve at all, and a draughtsman who rules the sides of a baluster has drawn a frustum.

The rims are the part’s ends and are drawn. The near rim fixes the scale; a part drawn with one end hidden loses it, and the swelling’s height is then read relative to a radius the drawing does not show.

Twenty drawings a point. Each spread is fixed to about a sixth of itself, enough to see that the turn moves the taper’s spread fivefold and the swelling’s hardly at all, and not enough to rank the turns between thirty-five and eighty for the swelling.

Still open: how many numbers of a profile a drawing reads #

The swelling here had two numbers, its height and its place, with its width tied to them. A turned part’s profile can be anything smooth, and the question the drawing has not been asked is how many independent numbers of it a side curve carries at half a pixel: a swelling’s width as well as its height, a second swelling beside the first, or a neck and a bulb together.

The measurement that settles what that is worth describes the profile by a growing number of smooth terms along the part, fits each drawing with one, two, four and eight of them, and asks at what count the terms stop being read — when the next term’s spread grows larger than any value a real part would give it — and whether that count depends on the view as little as the swelling’s height did.