A circle off the coordinate planes

A drawing office keeps a set of ellipse templates. The isometric one is cut at a ratio of 0.5774 — minor axis to major — and a draughtsman lays it on a drawing whenever a circular hole, boss or shaft end has to appear on a face.

That ratio is exactly right for a circle on one of the three coordinate planes, and it is not a fact about isometric drawing. It is the cosine of one angle.

The two numbers a projected circle has #

A circle lying in a plane, drawn by a parallel projection, images as an ellipse. Two numbers describe it and both have short closed forms.

The major axis is the true diameter, undiminished. A circle has a diameter running along the intersection of its own plane with the picture plane, and that direction is not foreshortened at all, so it comes through at full length. Every ellipse a parallel projection makes out of a circle has its major axis equal to the circle’s own diameter — for a unit circle, exactly 1, in every system and at every tilt.

The minor axis is that diameter times |cos θ|, where θ is the angle between the plane’s normal and the direction of projection. The direction perpendicular to the first one, inside the circle’s plane, is tipped away from the paper by exactly that angle and shortens by its cosine.

So the ellipse is one number: the angle between the face’s normal and the ray. Everything else follows.

Why the template’s number is that number #

An isometric projection looks along the body diagonal. The normal of the xy plane is the z axis, and the angle between the z axis and the body diagonal is 54.7356° — the tetrahedral angle, arccos(1/√3).

Its cosine is 1/√3 = 0.5774, which is the template’s ratio, and the three coordinate planes give the same number by symmetry: all three axes make the same angle with the body diagonal, which is what “isometric” means. So the one template does all three faces of a box, and the drawing office has a tool that covers every case it usually meets.

That is the whole of why the template works, and it is also the whole of its scope.

A face that is not a coordinate plane #

Cut a chamfer, put a hole in it, and the face’s normal is no longer an axis.

The angle between the tilted face’s normal and the body diagonal is 69.63°, and its cosine is 0.3481. The correct ellipse is markedly narrower than the template’s 0.5774 — narrow enough that no amount of careful placement makes the template’s curve fit.

And the ratio is only half of what has changed.

The half that gets drawn wrong #

The major axis of the drawn ellipse is perpendicular to the image of the face’s normal.

That statement is checkable, and it holds at every tilt to arithmetic noise: take the drawn normal as a vector on the page, take the ellipse’s major axis as another, and their dot product is zero to fifteen decimal places whatever the face is doing.

What it is not perpendicular to is anything a draughtsman can see. It is not perpendicular to a drawn edge of the face. It is not along a drawn diagonal. It is not at any fixed angle to the vertical. On a coordinate plane it happens to be perpendicular to the drawn axis that is normal to that plane, which is a drawn line and is why the rule is easy to follow there — and the moment the face tilts, that convenience goes and there is nothing on the paper to align the template against.

This is the third time this site has caught the same shape of error about a drawn circle, and the first two are worth naming because together they make the pattern.

Which way the drawn circle leans took the two rules in circulation for the minor axis of a drawn circle in a perspective picture — that it lies along the disc’s own axis, and that it points at the centre of vision — and found the first exact only on the optical axis, 3.55° out at the edge of a 50° frame, and the second 78.6° out at the same place. Both are printed in the same books.

The ellipse the drawing office draws took the four-centre construction the templates themselves are cut to and measured it against the conic it stands in for: a circular arc has constant curvature and a conic’s varies, so the four-centre curve reaches only 2√2/3 of the true semi-major axis and its minor axis is too long by three and a half per cent.

And this one says that even the exact ellipse, correctly proportioned, is only correct on three planes and is oriented by a line that is not drawn.

How much of a chamfer is a chamfer #

The tilt in the figure is thirty-four degrees, which is a large chamfer. It is fair to ask at what tilt the template stops being usable, and the answer is a matter of what the drawing is for rather than of geometry — so the useful thing is the rate.

Near a coordinate plane the ratio changes slowly: the cosine’s derivative is small there, so a face a few degrees off gives an ellipse a per cent or two narrower than the template, which is inside a pencil line. Past about fifteen degrees the ratio has moved by a tenth of its value and the difference is visible on a hole a centimetre across.

The direction does not behave that way. The image of the normal swings immediately, and the required major-axis direction swings with it — a face tipped five degrees already wants its template turned by a few degrees, and a template turned by a few degrees on a narrow ellipse is a visible error even when the width is right.

So the two halves fail at different rates, and the one that fails first is the one with no rule attached. A draughtsman working by eye will notice the width before the angle and will be wrong about the angle for longer.

The same statement in perspective, where it is worse #

None of this is about parallel projection in particular; the parallel case is simply the one with a closed form.

In a perspective picture a circle still images as a conic — the conic a circle becomes settles which one, and the type is decided by a single incidence — but the neat separation above is gone. The major axis is no longer the true diameter, because there is no direction that escapes foreshortening; the ellipse’s centre is not the image of the circle’s centre, which is what the circle whose centre moves is about; and the axis directions depend on where in the frame the circle sits as well as on how the face is tilted.

That is why the drawing-office rule exists in the parallel world and the perspective world has three competing rules and no agreement. In a parallel projection the answer is one angle. In a perspective picture the answer is a function of position, and a fixed number substituted for a function is the failure mode the whole wrong field is made of.

What the rule actually is #

Stated so that it covers every case:

The image of a circle under a parallel projection is an ellipse whose major axis is the true diameter, lying perpendicular to the image of the plane’s normal, and whose minor axis is the diameter times the cosine of the angle between that normal and the projection direction.

Three quantities, one angle, no system named. Isometric’s 0.5774, dimetric’s two different numbers for its two unequal faces, cavalier’s front face at a perfect circle, and the chamfer above are all the same sentence evaluated at different angles.

The template version — use 0.5774 and line it up with the axis — is that sentence evaluated at one angle, with the alignment rule replaced by a coincidence that holds at that angle. It is the exact shape of thing the taught-and-unmeasured essays are about: a fixed number substituted for a function, correct at the one place the illustrating diagram was drawn.

One more consequence follows from the major axis being the true diameter, and it is the only free lunch in the essay. Whatever the face, whatever the system, the longest width of the drawn ellipse is the circle’s real diameter at the drawing’s scale. So a reader with a ruler and a drawn hole can read its diameter off the paper exactly, without knowing which face it is on or what the projection is, provided the widest direction is found. That is a rare thing in this field: a metric quantity that survives a projection with no correction and no convention.

The measurement, and how it is taken #

The ratio and the direction above are computed from the projection’s own two rows rather than by fitting an ellipse to samples — the image of the circle is traced by the images of two perpendicular unit vectors in its plane, and the ellipse’s semi-axes are the singular values of the 2 × 2 matrix they form.

That is a closed form, so the site’s rule requires it to be checked against the drawing. It is: the circle is sampled at thirty-six hundred points, drawn, and the extreme radii of the drawn curve about its centre are taken. The two routes agree to a millionth for the coordinate plane and for the tilted face alike.

The check earns its place because the closed form has a trap in it. Singular values come out unordered, and a version that took them in the order the eigen-solver returned would report the major and minor the right way round for some tilts and swapped for others — a defect that shows up as an ellipse drawn at right angles to itself, and that no assertion about ratios would catch because the ratio of a number to its reciprocal is still a number.

The ratio and the ruler are the same anisotropy #

There is a connection to the previous rung that is worth making explicit, because the two essays are about the same number.

A ruler on an isometric drawing measures what happens when a length is taken off the paper in a direction that is not an axis: the answer is anywhere between √(1/2) and √(3/2) of the truth, a spread of √3, with nothing in the picture to say which. That spread is the anisotropy of the plane — the ratio of the largest to the smallest scale factor the projection applies to directions inside it.

The ellipse in this essay is that anisotropy drawn. Its major axis is the direction that is not foreshortened at all and its minor is the direction most foreshortened, so the ratio minor-to-major is one over the anisotropy of the face. On an isometric coordinate plane the ellipse’s ratio is 0.5774 and the anisotropy is √3 = 1.7321, and 1/1.7321 is 0.5774.

So the two results are one result. A circle drawn on a face and a ruler laid on the same face are measuring the same 2 × 2 map, and the ellipse is what that map does to a unit circle — which is what a singular value decomposition is, and is why the semi-axes came out as singular values.

That also settles a question the ruler essay left implicit. The military projection’s horizontal plane has anisotropy exactly 1, so a ruler works on it in every direction — and therefore a circle on the ground in a military drawing is a circle, at true size, at any orientation. One system, one plane, and a compass rather than a template.

Cavalier’s circle, which is a circle #

One case is worth doing explicitly because it is the reason the oblique systems exist.

In a cavalier or cabinet drawing the front face is drawn at true size and true shape. Its normal is along the depth axis and the projection ray is at 45° to the picture plane — but the angle used in the rule is between the normal and the ray, and for the front face the ray is along the normal’s own plane in a way that leaves the face undistorted. The image of a circle on the front face is a circle, at the true diameter, with no orientation to get wrong.

That is precisely what an oblique drawing is bought for: one plane drawn without distortion, so that the shapes on it can be drawn with a compass. The price is the other two planes, whose circles are ellipses at ratios that are neither the template’s nor each other’s — which is why an oblique drawing of a cylindrical part is easy when the cylinder points into the page and awkward when it does not.

Which faces of a real part are covered #

Put the rule against an ordinary machined part and the coverage is worse than the template’s popularity suggests.

A rectangular block has three pairs of faces and all six are coordinate planes, so every hole through it is a template case. Add a chamfer and its face is not. Add a boss on a sloping web, a flange bolted at an angle, a pipe entering a tank obliquely, a countersink whose cone meets a tilted face — none of them is a template case, and each one has its own angle.

The awkward part is that they are not far off. The chamfer in the figure above is tilted thirty-four degrees and its ellipse ratio is 0.348 against the template’s 0.577, and a draughtsman reaching for the nearest template and turning it until it looks right will produce a curve that is too fat and pointing the wrong way, and the drawing will look plausible. Nothing in it will be obviously wrong, and the hole will be a different hole.

That is the ordinary situation with a rule that is stated without its condition. It does not fail loudly on the cases it does not cover; it produces something that passes inspection.

What a reader can take away #

Three things, in decreasing order of how often they matter.

A template is a statement about an angle. It is correct for the planes whose normals make that angle with the ray, and there are three of them in an isometric drawing and none anywhere else in the object.

The orientation is the harder half. Getting the ratio wrong makes an ellipse the wrong width; getting the axis direction wrong makes it the wrong shape in a way that reads as a mistake about the geometry rather than about the drawing. And the correct direction is perpendicular to a line — the drawn normal — that the draughtsman has to construct, because it is not an edge.

And the exact ellipse is not what a template gives anyway. The four-centre curve on the template is four circular arcs, and it agrees with the conic at four points and nowhere between. Two approximations are being stacked: a conic standing in for the true image, and four arcs standing in for the conic. The second is measured in the ellipse the drawing office draws; the first is measured here; and neither is mentioned in the instruction that comes with the template.