The circle in the square wants a number

The method is in every book on perspective drawing and it is old. To draw a circle lying on the ground:

Inscribe it in a square. Draw the square in perspective. Cross its diagonals to find the centre; run lines through the centre parallel to the sides to find the four points where the circle touches them. Then mark four more points, about seven tenths of the way from the centre to each corner, and draw a smooth curve through the eight.

Seven tenths is a real number and not a fudge. A circle inscribed in a square meets the diagonal at √2⁄2 of the half-diagonal, and √2⁄2 is 0.70710678. The rule states a fact.

The trouble is which line the fraction is laid off along.

Two readings, one wording #

Reading one. Take the square in the room, go seven tenths of the way from its centre toward a corner, and project that point. Exact — measured on the floor, the mark lands on the circle to about two parts in a thousand trillion of a metre, which is arithmetic noise.

Reading two. Take the square in the picture, put a ruler on the drawn diagonal, and go seven tenths of the way from the drawn centre to the drawn corner. On the floor of the picture drawn here, that mark is 54 mm off a circle two and a half metres across.

The wording of the rule does not distinguish them, and only one of them is available. The real diagonal is in the room. What is on the paper is the drawn diagonal, and a ruler laid along it measures a different quantity.

The reason is the oldest fact in this collection. A projection does not preserve ratios along a line. It preserves the cross-ratio of four points, which is a ratio of ratios, and a single ratio of two lengths is exactly one of the four things what a projection destroys says is gone. Seven tenths of the way along a segment in the room is not seven tenths of the way along its picture.

The one case where the rule is right #

Put the square in a plane parallel to the picture plane — a paving pattern on a wall the reader is facing squarely, a clock face, a wheel seen head on — and both readings coincide exactly.

That is not luck. When the square’s plane is parallel to the image plane, every point of it is at the same depth, so the projection is a plain scaling: lengths along a line all shrink by one factor, and a ratio along a line survives. The drawn diagonal is the real diagonal to within a scale, and seven tenths of one is seven tenths of the other.

Which is the case every diagram in every book is drawn in — because a diagram illustrating a construction is drawn flat on the page, and a square drawn flat on the page is a square in a plane parallel to the picture plane.

So the rule is stated for the general case, illustrated in the one case where it holds, and used in the general case. That is the shape this collection keeps finding, and the essays on constructions that are taught and never measured now contain five instances of it. Each rule is exact in one situation. Each book states the situation as though it were the rule.

Where the error goes as the square recedes #

The error is not a fixed penalty. It shrinks with depth, because foreshortening is what causes it and foreshortening goes away.

The decay is close to inverse in the depth, which is what one would expect: the difference between the drawn diagonal and a scaled copy of the real one comes from the far half of the square being smaller than the near half, and that difference falls off as the square’s angular size does.

What it never does is vanish. On a ground plane, with the camera above the ground and looking anywhere at all, the square is always seen obliquely, so the rule is always wrong by something. There is no regime in which it comes right — only a regime in which the error is smaller than a pencil line, and that regime is “the circle is small and a long way away”, which is the regime in which nobody needs a construction.

A second point about that curve, which was the first thing this measurement produced and is worth recording because it was nearly missed: swept across the frame at constant depth the error does not move at all, to four significant figures. The variable that matters is the depth, not the lateral position, because what the rule gets wrong is the foreshortening of the diagonal running into the picture.

A sweep across the frame would therefore have shown a flat line and been read as evidence that the rule was position-independent, which is true and irrelevant. This site recorded the same shape of mistake once before — a test taken along a surface’s own axis is a test taken where it cannot fail — and here it is again wearing different clothes.

The four points that are free #

The tangency points, where the circle touches the four sides, are at the midpoints of the sides. One half.

One half is a fraction with a power of two underneath it, so it is reachable by the diagonals alone — cross the diagonals for the centre, join the centre to the side’s vanishing point, and where that line meets the side is the tangency point. No measurement, no ruler, no number. Measured, those four marks are on the circle to arithmetic noise, on the floor and on the façade alike.

√2⁄2 is not such a fraction. It is not any fraction. It is irrational, and a straightedge working from the square’s own corners and the horizon reaches exactly the rational points of a line and no others. So the four extra points the rule asks for are, in the strictest sense, not constructible from the square.

That is why the rule reaches for a ruler. It has to: the construction it is embedded in cannot produce the number it needs, so the number is imported from the plan and laid off on the paper, and the import is where the error enters.

What to do instead, which needs no number at all #

The eight-point rule is trying to solve a problem that has a better solution, and the better solution has been available in this collection since five marks and the sixth.

The image of a circle is a conic. A conic is determined by five conditions. The four tangency points give four of them — and each tangency point comes with a tangent, the drawn side of the square, which is a second condition apiece.

Four points with their four tangents is eight conditions on a five-parameter object, which is more than enough. In fact four points and two of the tangents already determine the conic, so the construction is over-determined and the surplus is a check: draw the conic through four tangency points touching two sides, and it should touch the other two. If it does not, the square was drawn wrong.

That construction never needs √2⁄2, never needs a ruler on the paper, and produces the exact image rather than an eight-point approximation to it.

It is worth being fair about why the books do not give it. Drawing a conic through five conditions by hand is not a straightedge operation either — it is a curve, and a curve is drawn with a French curve or by eye through points. The eight-point rule is a way of getting more points so that the freehand part is easier. That is a reasonable thing to want. The complaint is not that eight points are worse than five; it is that four of the eight are in the wrong place and nothing in the method says so.

The drawn centre is not the centre either #

There is a second error in the same construction that this collection has already measured, and the two are independent.

The diagonals of the drawn square cross at the image of the square’s centre. The ellipse drawn through the eight points has a centre of its own, and it is a different point: the near half of the circle is magnified more than the far half, so the drawn figure’s own middle sits nearer the viewer than the drawn image of the middle. The circle whose centre moves measures the gap.

A draughtsman who draws the ellipse’s axes through the diagonal crossing is therefore drawing them through a point that is not the ellipse’s centre — a third error, on top of the seven-tenths one, and pointing a different way.

Why the wrong marks are in the right place #

There is something to explain about the picture, which is that the four bad marks do not look bad.

Laid off on the drawn diagonal, the mark on the near side of the square falls outside the true image of the circle and the mark on the far side falls inside it, by roughly the same amount. The two errors point in opposite directions along the same line, and a smooth curve drawn through eight points of which four are alternately outside and inside passes very close to the truth in between.

So the eight-point rule’s output, drawn as a curve rather than as eight dots, is much better than its worst mark. The 54 mm quoted above is the distance from the worst individual mark to the circle; the curve a draughtsman would actually draw through those eight points is nearer than that almost everywhere, because the errors partly cancel.

That is worth conceding, and it is also the reason the rule has survived. A method whose errors alternate in sign produces a result that looks right, and looking right is the only test most constructions ever face. The failure is systematic all the same: the drawn ellipse is consistently too fat toward the viewer and too thin away from them, which is the same distortion the wrong centre introduces and in the same direction, so the two errors add rather than cancelling.

The two errors have the same sign, and both push the drawn figure toward the viewer. A circle drawn by the eight-point rule with its axes through the diagonal crossing is therefore biased twice in one direction — and the bias is exactly the bias that makes a drawn wheel or a drawn arch look, to somebody who knows, as though the draughtsman had guessed.

The size of the thing being drawn #

One more number, because it changes what a reader should do about all this.

The error scales with the circle. Doubling the circle’s radius doubles the miss, at any depth, because everything in the construction is linear in the size of the square. A circle 200 mm across at six metres — a dinner plate on a table — carries an error of about four millimetres, which is below the width of the pencil line drawing it.

A circular pool eight metres across at the same distance carries an error of about 170 mm.

So the practical rule is not “never use the eight-point method”. It is: the method’s error is a fixed fraction of the circle’s own size at a given depth, so it is safe for small circles and unusable for large ones, and the fraction is a few per cent at ordinary drawing distances. Anybody drawing a rose window, a roundabout, a fountain or an amphitheatre is in the second case; anybody drawing a teacup is not.

The drift curve above is drawn in millimetres, for one circle two and a half metres across. Divided through by that circle’s radius it reads as a proportion instead — a few per cent at ordinary drawing distances — and the proportion is the part that transfers to a different circle.

The general lesson, stated once #

The eight-point rule fails in a way that is worth generalising, because the failure is available to any construction that quotes a number.

A construction made of joins and meets carries over from the room to the picture unchanged. A construction that quotes a length or a ratio does not, and the picture is where the quoting happens.

Every instruction of the form “go a fraction of the way from here to there” is in the second category. Every instruction of the form “join these, cross those, take the point” is in the first. The rule mixes them, and the mixture is invisible because both halves are written in the same voice.

The test is mechanical and takes a second: read the instruction, and ask whether carrying it out requires putting a ruler on the paper. If it does, it is a statement about the plan being smuggled into the picture, and it is right only where the picture is a scaled copy of the plan.