Taught and unmeasured — where it appears
Named by 14 essays across 7 fields — each of them below, with the objects they name alongside it.
The arcs a curvilinear drawing uses
The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.
A ruler on an isometric drawing
Isometric drawing has one scale — 0.8165 — and every account of it stops there. But that number is about three directions and a drawing has infinitely many, so a length measured off the paper and divided by 0.8165 comes back anywhere between √(1/2) and √(3/2) of the truth: 29.3% short to 22.5% long, with nothing in the picture to say which.
One parameter between two surfaces
Wide architectural views are usually made on a projection with a number attached to it — a family running from the flat plane at one end toward the cylinder at the other, with everybody using the value one. That value has never been given a geometric defence. Measured across the family with the same battery of tests as every other surface, the worst angular error over the field has a minimum, and the minimum is at 1.04.
A wall does not get darker as it goes away
The inverse square law is about a point source. A surface is not a point source, and the picture of a wall is exactly as bright at twenty metres as at two — the patch one pixel covers grows as the square of the distance and the light per unit area falls as the square of the distance, and a picture records the product. Which is why aerial perspective has to be the air.
One surface, two images
A water surface reflects what is above it and refracts what is below it in the same photograph, from the same plane. The reflected half has a centre of projection to 1 × 10⁻¹⁴ m and every theorem about central projection applies to it; the refracted half misses its own best-fitting point by 28.6 mm and none of it does. And the landscape manual's rule for drawing a reflection turns out to be the epipole placed at infinity, which is why it costs nothing at zero tilt and 11.5 px at twenty-two degrees.
The ellipse the drawing office draws
Every isometric ellipse template is cut to a construction of four circular arcs, and every account of it calls the result an ellipse. It is not one and cannot be: a circular arc has constant curvature and a conic's varies, so the two can agree at four points and nowhere between. The four-centre curve reaches 2√2/3 of the true semi-major axis — 5.72% short — and its minor axis is 3.53% too long.
Two mirrors make one turn
Reflect a point in two mirrors meeting at 36° and the images arrive nine at a time, every one of them on a single circle about the line where the mirrors meet, to 4 × 10⁻¹⁶ m. The rule taught for the count — three hundred and sixty over the angle, less one — is right at six of nine angles tried and wrong at the rest, because it is a rule about angles that divide a half turn and it is quoted for angles that divide a whole one.
A cylinder has two different ends
The two end circles of a cylinder image as two different ellipses — 14.4° apart in the direction of their major axes and 0.918 against 0.839 in aspect — and the outline's straight sides touch neither of them where its major axis ends, missing by 28° round the ellipse and 35 pixels. Walking the cylinder away removes the difference between the ends and does not remove the offset of the touch, which settles at 7.3° off the principal ray and at nothing on it.
Two mirrors show fewer images than they make
Two mirrors at 55° generate seventy-one images of a point and an eye between them can reach six. The count the field teaches — three hundred and sixty over the angle, less one — is out by as much as sixty-six against the orbit and never by a whole image against what a viewer standing on the bisector actually sees. It is a correct rule about the eye, quoted as a rule about the mirrors.
What survives being copied
A workshop copying a drawing from a drawing is a random walk — the spread across lineages grows as the square root of the generation, with a fitted exponent of 0.5001. A workshop copying the method is not, and its exponent is 0.012, which is no growth at all, and after forty generations two lineages started from different originals end up 5 × 10⁻¹⁵ apart. Copying the marks loses the picture; copying the recipe loses the original and keeps the recipe.
A circle off the coordinate planes
An ellipse template is cut at one ratio, and the ratio is the cosine of one angle: between a coordinate plane's normal and the direction of projection. A face tilted out of that plane needs a different ratio and — the half that gets drawn wrong even when the ratio is close — a major axis pointing somewhere else, perpendicular to the drawn normal rather than to any edge.
The shadow rules that hold here
Drop the foot, run a line from the top at forty-five degrees, take the intersection: the drawing manual's shadow construction is exact in a parallel drawing, to arithmetic noise, at every point of the picture and with one set square. It is where the rule came from, and carrying it into a perspective picture is what broke it.
A lamp behind the camera
A light behind the photographer has no image — the projection refuses it — and the shadows it casts are in front of them, drawn as ordinary shadows. The construction that recovers a lamp from those shadows works anyway, meeting to a ten-thousandth of a pixel at the point the reversed divide puts it, and the taught reading of where the answer lies gets the case exactly backwards.
A tapered part meets at its apex
The sides of a turned part that narrows by one degree meet 54 pixels from the vanishing point of its axis, at the image of its apex, and a quarter of a degree already moves them 15. Drawn toward the vanishing point instead, a two-degree part loses nine tenths of its own taper. Flare it the other way by 5.9 degrees and a correct photograph prints its sides parallel; flare it further and they spread with depth.
Named alongside it
The objects these essays reach for when they reach for this one.
ForeshorteningAnisotropyDemonstrationDrawing systemConicfield of viewFour-centre ellipseOrthographicReflectionArea scalecentre of projectionConformal