The ellipse the drawing office draws

A hole in the top face of a bracket, drawn isometrically, is an ellipse. Drawing an ellipse with a compass is impossible, so the drawing offices did what drawing offices do and found a construction with a compass that is close enough: four circular arcs, inscribed in the rhombus that the square face becomes, each tangent to two of its sides at their midpoints.

It is called the four-centre ellipse. It is not an ellipse. That much is immediate — four circular arcs have four constant curvatures and an ellipse’s curvature varies continuously — so the only question is how wrong it is, and that question has an exact answer nobody quotes.

The construction, and why it is forced #

Nothing about the recipe is chosen, which is worth establishing before measuring it, because a construction with an arbitrary step in it could be improved and this one cannot.

Each arc has to be tangent to two adjacent sides of the rhombus at their midpoints — that is what makes the curve smooth where the arcs meet and what makes it touch the true ellipse there. A circle tangent to a line at a given point has its centre on the normal to that line at that point. So each arc’s centre is where the two midpoint normals meet, and the whole construction is determined.

In a 60°/120° rhombus — which is what an isometric projection makes of a square — the normal to a side at its midpoint runs to the opposite obtuse vertex. That is why the two large arcs are centred on vertices of the rhombus, which is the step that looks like a lucky simplification and is a property of that specific rhombus.

It is also the step that makes the recipe isometric-only. On a dimetric or trimetric rhombus the midpoint normals do not run to the vertices, the arcs are not tangent, and the construction does not exist. Handing the machinery a dimetric plane is refused rather than answered — the honest response, since the thing being asked for is not defined there.

The measurement #

Take the tangency-forced construction and compare it, point by point, with the ellipse the projector actually draws.

The four arcs reach $2\sqrt2/3 = 0.9428$ of the true semi-major axis. That is 5.72% short, and because the shortfall is at the ends of the major axis it is also the worst departure anywhere on the curve — the largest distance from any point of the four-arc curve to the true ellipse is exactly the major-axis shortfall.

And the minor axis comes out at $\sqrt6 - \sqrt2 = 1.0353$ of the true one, 3.53% long.

So the curve is short one way and long the other, which means it is the wrong shape and not merely the wrong size. A rescaling cannot repair it. Its axis ratio is 0.6339 against the true 0.5774 — an isometric circle drawn this way is visibly rounder than an isometric circle.

Both departures are closed forms rather than measurements: $2\sqrt2/3$ and $\sqrt6-\sqrt2$ are asserted at $10^{-9}$ against the construction carried out on the page, which is the difference between a number that describes this drawing and a number that describes the recipe.

Where the two constants come from #

Both closed forms fall out of the rhombus in a few lines, and the working exposes a third number that says why the curve looks the way it does.

Take a unit circle in a coordinate plane. Isometrically it draws as an ellipse with semi-axes 1 and $1/\sqrt3 = 0.5774$, inscribed in a rhombus whose half-diagonals are $\sqrt2$ and $\sqrt{2/3}$ and whose side midpoints — the tangent points — sit at $(1/\sqrt2,\ 1/\sqrt6)$.

The small arc is centred on the major axis and tangent at two of those midpoints. Solving for the centre puts it at $\sqrt2/3$ from the origin with radius $\sqrt2/3$, so it reaches $2\sqrt2/3 = 0.9428$ — the major-axis shortfall, derived. The large arc is centred at an obtuse vertex, $\sqrt{2/3}$ out along the minor axis, with radius $\sqrt2$, so it reaches $\sqrt2 - \sqrt{2/3} = 0.5975$ against a true 0.5774: a ratio of $\sqrt6 - \sqrt2 = 1.0353$, the minor-axis excess.

The third number is the pair of radii themselves: $\sqrt2/3$ and $\sqrt2$, in a ratio of exactly three. An ellipse of this shape has curvature $a/b^{2} = 3$ at the ends of its major axis and $b/a^{2} = 0.5774$ at the ends of its minor — a ratio of $3\sqrt3 = 5.196$. So the construction supplies a curvature contrast of 3 where the conic wants 5.196, and that is the visible defect: not that the curve is the wrong size but that it is not tight enough where it should be tightest and not flat enough where it should be flattest. An isometric circle drawn this way reads as rounder because it is, by 40 per cent of the curvature contrast.

Two consequences worth having.

The area is nearly right, which is the trap. The two axis errors pull opposite ways, so the enclosed area is $0.9428 \times 1.0353 = 0.976$ of the true — 2.4 per cent small, against axis errors of 5.7 and 3.5. A reader who checks the construction by eye, or by area, will find it better than it is; the shape error hides in the cancellation.

And the axes can be repaired for nothing. Drawing the same four arcs inside a rhombus stretched by $1/0.9428 = 1.0607$ along its long diagonal and shrunk by $1/1.0353 = 0.9659$ along its short one puts both extremes exactly on the true ellipse. What is left is the curvature error, which is the part no choice of rhombus can fix, because it is a property of using four circular arcs at all. That is the honest split: two thirds of the error is a scaling and can be removed by a draughtsman; the rest is the method.

None of which transfers to a circle that is not in a coordinate plane, where the ellipse has a different shape entirely and no rhombus to be inscribed in — and none of it to a perspective drawing, where the centre of the drawn ellipse is not the image of the circle’s centre and the four tangent points are not midpoints of anything. The four-centre construction is exact about its own tangencies, wrong about its curvature, and confined to the one system whose three axis scales are equal.

Where it touches #

The four tangent points are exact. Each is a side midpoint of the rhombus, each lies on the true ellipse to $10^{-9}$, and the construction is right there and nowhere else.

That is the characteristic signature of a fitted approximation and it is worth naming, because it is the same signature as four correspondences fitting a curved surface: a construction that agrees at exactly as many points as it has freedom, and is wrong at every point it was not given. The four-centre ellipse has four arcs and touches at four points. The rolled-print homography has four correspondences and predicts them perfectly. In both cases the residual at the fitted points is zero and says nothing whatsoever about the rest.

What it costs on a real drawing #

A hole 20 mm across drawn at 1:1 comes out 1.14 mm short in one direction and 0.35 mm long in the other, which is enormous by drawing-office standards and is not what the drawing office was worried about. It never mattered, for a good reason: an isometric of a bracket is a pictorial view. The dimensions are on the orthographic views, the isometric is there so a fitter can see which way round the part goes, and nobody has ever scaled a hole off one.

So the error is harmless in the practice the construction grew up in, and the practice has moved. The four-centre construction now lives on in three places where it is not harmless.

Ellipse templates. A drawing template’s isometric holes are cut to this construction, so a circle traced through one is 5.72% short across its long axis. That was fine when the tracing was pictorial.

Software that reproduces the recipe. A surprising amount of CAD-adjacent code draws isometric circles with four arcs, because the recipe is easy and the alternative needs a parametric conic. There the output may well be scaled off.

Teaching. Every account of the construction calls it an ellipse, so a student learns a curve and does not learn that it is an approximation, which is the failure a ruler on an isometric drawing meets from the other side and the one this site’s taught-and-unmeasured thread is about. The construction is fine. Calling it an ellipse is what is wrong, and the correction costs one sentence and a number.

The curvatures, which say why it goes wrong where it does #

An ellipse with semi-axes $a$ and $b$ has radius of curvature $a^2/b$ at the ends of the minor axis and $b^2/a$ at the ends of the major. For the isometric ellipse with $a = 1$ and $b = 1/\sqrt3$ those are 1.7321 and 0.3333 — a factor of 5.2 between the flattest and the sharpest part of the curve.

The construction’s arcs are 1.4142 and 0.4714, a factor of 3.0. So it is too curved where the ellipse is flattest and too flat where the ellipse is sharpest, and the two errors have opposite signs, which is why the minor axis comes out long and the major short. Four arcs cannot do better: matching curvature at four points would need the radii to be the true ones, and arcs with the true radii through the true tangent points do not close up.

This is not a defect of the particular recipe. It is the general fact that a $C^1$ spline of four circular arcs has eight degrees of freedom and a conic has five, and the four tangency conditions with matched positions and directions use them all up, leaving nothing to spend on curvature.

What to do instead #

Draw the conic. A parametric ellipse is four lines of code and a plotter has been able to draw one since the 1960s, so the construction is a hand-drafting technique surviving into a context that does not need it.

If the four-centre curve must be used — because it is going on a template, or because a compass is genuinely the tool — the correction is a single scale factor in one direction, because the shape error is what it is and cannot be scaled away. Nothing about that is satisfying. The satisfying answer is that the approximation was never the problem: the problem is that it is not labelled as one.

Why the arcs cannot be improved #

Four arcs is not a shortage of arcs. It is a shortage of the right kind of freedom, and the count says so.

A chain of four circular arcs meeting smoothly has, before any constraints, eight parameters of shape — two per arc, once each arc’s centre and radius are reduced by the requirement that consecutive arcs share a tangent at their join. The constraints the construction imposes are that each arc pass through two given points with a given tangent direction at each. That is four positions and four directions, which is eight conditions.

Eight conditions on eight parameters leaves nothing. So there is no freedom left to spend on matching the curvature at the four tangent points, and matching curvature is exactly what would be needed to make the departure small between them.

Adding arcs would help — an eight-arc chain has spare parameters and could match curvature as well as position and direction — and nobody does it, because eight centres is no longer a construction anybody would carry out with a compass. The four-centre recipe sits at the point where the arithmetic of the construction and the patience of the draughtsman meet, and that point is not where the error becomes small.

What the true curve costs to draw #

The four-centre construction exists because an ellipse could not be drawn with the tools available, and it is worth saying what the alternatives were, because the recipe’s survival is a fact about instruments rather than about geometry.

A compass draws circles and nothing else. A trammel — two pins in slotted guides, a pencil at the end of a rod — draws a true ellipse and was available for centuries, and it is slow, needs setting up for each ellipse and cannot be used freehand. A string and two pins draws a true ellipse and needs the foci, which have to be constructed first. And an ellipse template draws a true ellipse of whatever sizes have been cut into it, which is a limited set.

Against those, four arcs from a compass is fast, needs no setup, works at any size, and is accurate enough for a picture. That is a good engineering decision and the drawing offices made it correctly.

What has changed is that a computer draws a parametric ellipse in four lines and at any size, so the constraint the decision was made under is gone. What has not changed is the recipe’s presence in textbooks, on templates, and in code that reproduces the textbook. A construction outlives its constraint by inertia, and the inertia is strongest where the construction has a name and the alternative does not.

That is the case for naming the error. An approximation adopted because a compass was the tool is a sentence that expires with the compass; the four-centre ellipse is a name that does not.

What the number is, in the places it is used #

Three settings, and the same 5.72% means something different in each.

On a drawing template, it is a manufacturing tolerance on a tool. A traced circle is short across its long axis by 5.72% of that axis, systematically, for every hole traced through that template — so a drawing full of traced holes is internally consistent and consistently wrong, which is the least harmful case.

In software, it is a silent substitution. Code that draws an isometric circle with four arcs produces a curve that reads correctly at a glance and measures wrongly, and the difference from a true ellipse is well above any rendering tolerance. A CAD-adjacent tool that exports such a curve as geometry has exported the wrong geometry.

And in teaching, it is a missing qualifier. The construction is described as the isometric ellipse and the word “approximation” is often absent. A student who learns it as the ellipse has learned a curve and a false identification, and the second is the part that travels.

The pattern, one field over #

The wrong field on this site exists for exactly this shape of finding, and the four-centre construction would sit in it comfortably if it were not so specifically about isometric drawing.

Its three essays measure a taught rule against what a camera does. The two-point cube construction leaves the depth undetermined and produces a box 1.4 times shallower than it is wide. The three by-eye methods for dividing depth misplace a post by three and a half metres. The sixty-degree cone of vision turns out to be a statement about the reader rather than about the picture, and the marginal stretch it is nominally about is exactly zero from the station point.

In all three cases and in this one, the rule is not stupid and the people who wrote it were not careless. Each is right about the case it grew up in, each was adopted because it was practical with the tools available, and each has outlived the constraint that made it necessary. What goes wrong is the qualifier falling off — and the qualifier is always a number.

The general point, which is about drawing offices rather than about ellipses #

A construction adopted for its practicality carries an error whose size nobody quotes, is taught for a century without the qualifier, and outlives the constraint that made it necessary. That is the same shape as the sixty-degree cone of vision, where the rule turns out to be a statement about the reader rather than about the picture; and as the three by-eye methods for dividing depth, of which the best misplaces a post by three and a half metres.

The pattern in all three is the same, and it is not that the old practitioners were careless. They were solving a problem with the tools they had, and they were right about what mattered. What goes wrong is later, when the qualifier falls off and the construction is inherited as the thing itself. The number is the qualifier: 5.72% short, 3.53% long, and exact at four points.