A turned part drawn from the side keeps its taper in its sides

The rims carry a part’s taper, as finely as they are drawn read a turned part’s taper from its two rim ellipses alone. The axis is the one direction under which both rims are circles; looked at down that axis, the rims are two circles on one line whose sizes differ by the taper times the distance between them. With exact rims the reading is exact. Drawn by hand to half a pixel, it read a one-degree taper to ±0.15° when the part pointed away from the eye, ±0.41° when it was turned to thirty-six degrees from the line of sight, and ±3.1° when it lay across the view, which is to say not at all. The view in which a turned part is usually drawn, from the side to show its profile, was the view in which its rims said least.

A tapered part meets at its apex had found the taper somewhere else: in the part’s two drawn sides, which cross at the image of the apex rather than at the axis’s vanishing point, fifty-four pixels from it for a degree of taper on this part. The sides could not be read alone, because they say where the apex is and only the axis says where it should have been. The essay on the rims closed on the obvious combination. The rims say least where the sides are longest; the sides are shortest where the rims are best. The question was what the two are worth together: the taper read from the rims alone, from the sides’ crossing against the axis the rims supply, and from everything fitted at once as one tapered part, as the part turns.

Together, they read a part lying across the view a hundred times better than its rims alone, and a drawing made from the side carries a taper of a twentieth of a degree.

Three readings of one drawing #

The part is the one the earlier essays used: 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, narrowing away from the eye by a degree a side. The figure turns it eighty degrees from square to the picture, so that its axis lies across the view, eighty degrees from the line of sight. It is drawn the way a hand draws: forty-eight points round each rim and twenty-four along each side, every one placed half a pixel off the true line in a random direction, which on this photograph’s scale is a careful pencil line’s width. The faint lines beneath are the truth; the dots are the drawing.

The rims are the earlier essay’s rims, the same drawings seed for seed, and they read the taper as 1.28° ± 3.13 over eighty drawings — the spread three times the taper and the mean pushed high, the earlier essay’s numbers exactly. Both rims are thin ellipses, and a thin ellipse’s roundness says little about which way its circle faces.

The sides are read the way the essay on the apex suggested. Each drawn side is fitted with a straight line, the two lines are crossed, and the crossing — the image of the apex — is a ray from the eye. The rims, whatever their uncertainty about the axis’s direction, still place an axis line in space, and the apex is where the ray meets it. The near rim’s radius over the apex’s distance from the near rim’s centre is the taper. Read that way the same eighty drawings give 1.02° ± 0.075: forty times tighter than the rims, though the rims supplied the axis.

The third reading treats the drawing as what it is, a picture of one tapered part, and asks which part fits every dot best. The unknowns are seven: where the near rim’s centre is, which way the axis points, how long the part is and how much it tapers, with the near rim’s radius held fixed because a drawing has no scale. Every dot on a rim is compared with that part’s rim ellipse and every dot on a side with that part’s side line, and all one hundred and forty-four distances are made as small as they can be together. It gives 1.00° ± 0.032: a hundred times tighter than the rims, and twice as tight as the sides alone. The slider turns the part through four positions from pointing away to lying across, and the three readings change places.

The rims and the sides fail in opposite views #

The figure turns the part from pointing nearly at the eye, its axis thirteen degrees from the line of sight, to lying across the view at eighty. The rims’ reading grows twentyfold across that turn, from ±0.15° to ±3.13°, for the reasons the earlier essay gave: a part turning across the view draws its rims thinner, so their roundness fixes the axis more loosely, and draws them at more nearly one distance, so the far rim’s smaller size carries less.

The sides’ reading does the opposite, and flattens. Pointing away, the part’s sides are short stubs between two nearly round rims, and their crossing is poorly fixed: ±0.17°, a little worse than the rims. Turned to thirty-six degrees the sides read ±0.075°, five times better than the rims; from there to lying across, ±0.065° to ±0.075°, while the rims fall to forty times worse. A part lying across the view shows its two sides long and nearly undistorted. They converge on the page by 4.85 degrees, of which 2.70 is the perspective a cylinder in the same place would show and 2.15 — very nearly twice the taper — is the taper’s own; and two long lines whose angle is known to a few hundredths of a degree are the clearest thing in the drawing.

The reading from everything at once follows whichever source is better and improves on it. Pointing away, where the rims are good and the sides add little, it reads ±0.146° against the rims’ ±0.148°. From thirty-six degrees on it reads one and a half to two and a half times better than the sides alone: ±0.052° at thirty-six degrees, ±0.033° at fifty-one, ±0.029° at sixty-five, ±0.032° lying across. The rims, poor as they are at the taper in that view, still fix where the part is and how it is placed, and the sides no longer have to supply that themselves.

So the earlier essay’s inversion of the draughtsman’s advice inverts back. The rims read the taper best with the part pointing away, the view that hides its profile; the sides read it best from the side, the view that shows it. A drawing has both, and between them there is no view in which a degree of taper is lost. The worst place is the one the rims alone did best in, and even there the reading is a seventh of a degree.

Every reading scales with the care of the drawing #

The readings are all least-squares fits to placed points, and each grows in proportion to the scatter of the drawing. Lying across the view, the rims alone read the taper to ±0.62° when every point is placed to a tenth of a pixel, the precision of ellipses plotted from a construction, and to ±12.4° at two pixels, a quick sketch. Everything at once reads it to ±0.006° at a tenth of a pixel, ±0.032° at half, ±0.064° at a pixel and ±0.13° at two. Turned to thirty-six degrees, ±0.010° to ±0.21°.

Because every curve is a straight line on the figure’s logarithmic axes with the same slope, the ratios between them do not depend on how carefully the drawing was made. A side-view drawing read from everything at once is a hundred times better than its rims alone at any care; to say as much from the rims a draughtsman would have to draw them a hundred times more finely, to a two-hundredth of a pixel, which no hand and no plotted construction reaches. The sides are not a refinement of the rims’ reading. In this view they are the reading, and the rims supply the setting.

A sketch made to two pixels — a line four times the width of a careful pencil’s — still carries the taper of a part seen from the side to an eighth of a degree. The ellipse the drawing office draws priced the shortcuts a drawing office takes with its ellipses in pixels, and those shortcuts land on the rims; the sides are drawn with a straightedge and suffer none of them.

A ruled side is nearly as good as a freehand one #

A draughtsman does not draw a turned part’s side freehand. The side is ruled: a straightedge laid from the point where it touches the near rim to where it touches the far one. The line is straight between its two ends, so its whole error is in where those two ends were placed. That is two readings rather than twenty-four, and on a count of readings a ruled side should be worth a fraction of a freehand one.

It is worth nearly as much. Turned to thirty-six degrees, everything at once reads the taper to ±0.057° with ruled sides against ±0.052° with freehand ones; at sixty-five degrees, ±0.032° against ±0.029°; lying across, ±0.038° against ±0.032°. The direction of a line is fixed by how far apart its readings are as much as by how many there are, and a ruled side’s two readings sit at its ends, as far apart as the side allows. A freehand side spreads its readings along its length, most of them near the middle where they do least for the direction. And the rims carry part of the burden too: the ends of a ruled side are the points where it touches the rims, which the rims’ own dots already place.

For a reader of drawings, this means the conventional way of drawing a turned part — rims with ellipse guides or by eye, sides ruled tangent to them — has its taper in exactly the part drawn with the most precise instrument. The rims are the hard part to draw and the soft part to read; the sides are the easy part to draw and the firm part to read. The earlier essay’s finding that guide-drawn rims read a taper of the guide’s making does not touch the sides at all, though a fit that leans on badly drawn rims for the part’s placing would inherit some of their error, which was not measured.

The smallest taper a side view shows #

The figure answers the earlier essay’s last question directly. A part lying across the view, drawn to half a pixel, at tapers from none to two degrees. From the rims alone the readings are scattered over six degrees at every taper and their means lean high: a cylinder reads 0.23° ± 3.07, a two-degree part 2.34° ± 3.20. Nothing in that column distinguishes a cylinder from a two-degree part, and the earlier essay’s conclusion for a side view stands for the rims.

From everything at once the readings sit on the line where read equals true, with the same spread at every taper: a cylinder reads −0.001° ± 0.032, a quarter-degree part 0.249° ± 0.032, a two-degree part 2.000° ± 0.032. The means are right to a thousandth of a degree. A taper is told from none when it stands about two spreads clear of zero, so a side-view drawing made to a line’s width shows a taper of six hundredths of a degree, and a quarter of a degree stands eight spreads clear.

That is finer than most of the tapers drawings are made to show. A taper of a degree a side on a part this size changes its radius by twenty-three millimetres over its length; six hundredths of a degree is under a millimetre and a half. A side-view drawing of a turned part, then, carries its taper after all — provided it is read as a drawing of one solid, rims and sides together, rather than as a pair of ellipses joined by two lines.

What a drawing’s outline carries #

Put together, a turned part’s taper is in both of its drawn features and its two features fail in opposite views. The rims carry it best when the part points away, by their shapes, which fix the axis, and their sizes, which compare across a depth; they lose it as the part lies across the view. The sides carry it best when the part lies across, by their convergence, which there is a cylinder’s perspective plus very nearly twice the taper; they lose it as the part points away and they shrink to stubs. Fitted together as one solid, the drawing reads the taper to a seventh of a degree at the worst view and three hundredths of a degree at the best, from lines drawn to half a pixel.

A cylinder has two different ends found that even a cylinder’s two rims are different ellipses, and that the difference is the turn drawn twice; the rims’ reading uses that difference. Two circles, one picture found that one ellipse leaves its circle’s plane a choice of two, and two rims settle it. The sides were missing from both, because a cylinder’s sides meet at the axis’s vanishing point and say nothing a cylinder’s rims do not. A tapered part’s sides meet somewhere else, and where they meet, measured against where the rims say they should have met, is the taper read in the view that hides it from the rims. 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, and every step since has added one more drawn feature and found it carrying something the others could not. The outline’s sides are the last of a frustum’s features, and the one the drawing office draws most carefully.

What the reading takes from outside the drawing #

The camera’s focal length and centre are known. Every reading here lifts the drawing back into space through a known eye, and the conic a circle becomes is a reminder that an ellipse’s relation to its circle depends on where the eye is. A drawing whose viewing distance is unknown has a family of possible parts. The sides’ convergence for a part lying across the view hardly depends on the eye, which suggests the fitted taper there would survive an unknown focal length better than the rims’ would, but that was not measured.

The sides are tangent to the rims and drawn as the part’s true outline. A draughtsman who draws a tapered part’s sides toward the axis’s vanishing point, as though it were a cylinder, puts the cylinder’s convergence on the page and the reading will find a cylinder, correctly reading the drawing and wrongly reading the part.

The errors are independent from point to point. A freehand side errs smoothly, and a ruled side’s two ends are placed by one eye in one sitting. A side bowed by half a pixel along its length is a different error from half a pixel of jitter, and since the sides now carry the reading, it matters more than it did for the rims.

The rims are drawn where they belong. A draughtsman places a rim’s ellipse by eye on centre lines, and which way the drawn circle leans found a circle seen off the principal ray leaning away from the lines it is set on. A rim set a little wrong in its lean misplaces the part the sides are fitted to, and through it the taper; how much, with the sides carrying the reading, was not measured.

The whole of both rims is drawn. A part lying across the view shows both rims nearly whole; a part pointing away hides much of its far rim behind the near one, and fits it as an arc. That costs the rims in the view where they read best, and the sides cannot make it up there because they are short.

Still open: whether a part’s profile reads more than its taper #

Everything here assumed the part is a frustum: a straight taper from one rim to the other, two numbers 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 wheel’s hub flares into its spokes, and in each case the side of the drawing is a curve rather than a line — the outline of a surface of revolution, which a drawing shows best from the side. Every quadric has one outline found the outline of a ball, a dish or a hyperboloid exactly, as the section by one plane; a turned part of arbitrary profile has no such shortcut, and its outline is the curve along which each of its circles turns away from the eye.

The measurement that settles what that is worth draws a turned part whose radius changes along its length by a stated curve, with its rims at the two ends and its side outline drawn freehand, and asks whether everything fitted at once reads the profile — how many numbers of it, a swelling’s height and where it sits along the part, to what precision a line’s width allows — and whether, as for the taper, the side view that hides the rims’ information is the one that shows the profile best.