Midpoint — where it appears
Named by 16 essays across 6 fields — each of them below, with the objects they name alongside it.
Parallel projection is not primitive perspective
Isometric and oblique drawing are not what people used before they worked perspective out. They are a different answer to a different question, and the difference is one measurable quantity — a parallel projection preserves the ratio in which a point divides a segment, and a perspective projection destroys it by 7% of the segment's drawn length at a comfortable depth, rising to 13% over the range the slider covers.
A centre and a measure are exclusive
Eight drawing systems, measured on five questions, with the pinhole as a row rather than the header. Exactly one has a centre of projection and it is exactly the one with no true measure — and loosening the measure test by a hair lets it in, which is what says the boundary is real.
The circle whose centre moves
The image of a circle is an ellipse, and the image of the circle's centre is not the centre of that ellipse. The gap is 3 or 4% of the ellipse's width in an ordinary view — large enough to see, small enough to be dismissed as a slip by anyone not expecting it, and the reason drawn wheels look wrong.
A map along, and a picture across
The midpoint of a segment lying along a scroll's length images to the midpoint of its image, exactly. The midpoint of a receding one lands 22% of the way off. One projection, two answers — and the direction that keeps measure keeps something a perspective picture never offers.
What perspective gave up
The field ends by turning its own battery on the system it has been comparing everything against. Four quantities a pinhole destroys that the other systems keep, each measured by the same computation as the systems it is set against, and each the price of the one thing perspective has and they do not.
A texture does not interpolate on the page
Walking across a drawn surface at a constant rate walks across the real one at a rate that changes, and the worst gap is a closed form in the depth ratio alone — 0.52 at ten to one, more than half the whole range. It is exactly the error a person makes dividing depth by eye, made by a machine, and the fix is the fourth coordinate the pipeline kept.
The diagonals find the middle
Three taught methods for spacing a receding row are all wrong, and the best of them misplaces a post by three and a half metres. There is a fourth, it needs no measurement and no vanishing point, and it is exact at every camera and every depth — because 'the diagonals of a rectangle cross at its centre' is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.
What one picture of a plane determines
A photograph of a tiled floor fixes that floor's geometry up to a projectivity, and no further. Supplying the horizon buys back the midpoint — exactly, and nothing else. Supplying the image of one circle buys the right angle and the ratio of two lengths at right angles. Three stages, three prices, and a quantity a stage does not determine has no value rather than a wrong one.
A pond in a scroll is not an ellipse
A pinhole draws a round pond as an exact ellipse whose widest row is 3.58 px off the row of the pond's centre — the drawn-circle error every perspective textbook warns about. A handscroll draws the same pond widest exactly on its centre's row, and draws it as a quartic that no conic fits: the best ellipse misses it by 3.08 px. Each keeps what the other loses, and off to one side the pinhole's pond leans 9.67 px while the scroll's does not lean at all.
A point under water has two depths
The apparent depth of a submerged point is not one number even along one line of sight. The rays it sends to an eye pass through two focal lines, and at 60° from the vertical a point 1.50 m down has an image 0.740 m down and another 0.320 m down. Two eyes side by side read the first, a head moving up and down reads the second, and a pair of eyes tilted between them reads neither — their two rays miss each other by as much as 6.39 mm.
The tenth row has neither
A crossed-slits camera divides by depth in both page directions and its rays miss any common point by 0.354 m. Put on the comparison table it prices 38.8% on length, 69.9% on area and 58.2% on angle against a pinhole's 39.5%, 70.8% and 60.9%, and its midpoint drift is 15.6% — the pinhole's own figure. It keeps a true scale in no direction at all, and it bows a straight run of ground by 1.11 px, which no row with a centre does. Giving up the point buys nothing and costs a third thing besides.
The stick a stereo pair puts back
Two eyes side by side reconstruct a submerged stick exactly as the sagittal image — kinked 14.96° — and two eyes one above the other exactly as the tangential one, kinked 9.59°. Roll the baseline between them and the two rays to a point miss each other by up to 3.13 millimetres, past the 2.85 a pixel covers at that range, and the reconstruction is a third stick that is neither — 551 millimetres of a one-metre stick, with its tip at 0.405 metres against a true 0.866.
The residual does not warn
Fit a straight line to the stick a stereo pair puts back and the fit looks best exactly where the reconstruction is least supported: the residual is 0.535 mm at a level baseline, where the two rays meet perfectly, and 0.038 mm at sixty degrees of roll, inside the band where they miss by more than a pixel covers. Over the same sweep the fitted line is 285 to 584 mm short of the stick's metre — fourteen thousand times its own residual at worst.
What one oblique drawing shows
A parallel projection is a linear map from three dimensions to two, so it has a direction it throws away. Slide any point along that direction and its mark does not move — which makes every parallel drawing the drawing of a three-parameter family of solids, and the depth scale a convention chooses is one direction through the family rather than a boundary of it.
A fit weighted by the miss trusts only the surface
Weight each point of a reconstructed underwater stick by how well its two rays meet, and the fit hands all but a ten-billionth of its trust to the one point where the stick enters the water — and reports a residual of nothing at every rolled baseline. Floored at the reading error, the weighting changes the answer by a few thousandths of a millimetre. And the miss itself, the one honest number, is exactly zero at a level baseline where the stick comes back 285 mm short.
The ball at the edge of the frame
A ball photographed near the edge of a wide picture is not drawn as a circle. It is an ellipse, longer along the radius from the centre of the picture than across it, and the centre of that ellipse is not the image of the centre of the ball. Both are properties of the flat sheet the picture is on, and an exact pinhole produces both.
Named alongside it
The objects these essays reach for when they reach for this one.
Parallel projectionBaselinecentre of projectionCross-ratioDemonstrationDrawing systemPushbroomStereo pairTriangulationAffine mapApparent-depthAstigmatism