Depth division — where it appears
Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.
Dividing depth by eye
Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.
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.
The near plane can be any plane
Rewrite one row of a projection matrix and the near plane stops being perpendicular to the axis and becomes whatever plane is asked for. Every x and every y is untouched — it is the same projection of the same scene from the same eye — and the depth order is wrecked, which is a clean separation of the two things a projection matrix does.
One plane is nearly free
The near and far planes enter a depth buffer's precision through 1/near − 1/far, and one of those reciprocals is enormous. Pushing the far plane out by a factor of a thousand costs a tenth of a per cent; bringing the near plane in by the same factor costs a factor of a thousand — and an infinite far plane is the limit of the first rather than a separate case.
Along a line of constant depth the page is affine
Stepping a texture by a constant amount per pixel is wrong across a receding floor and exactly right along any line of it that stays at one depth — and on every plane those lines run parallel to its own vanishing line. Turn a 120 px span 1° away from that direction and it is 0.79 px out; roll the camera a hundredth of a degree and a floor drawn to 30 m is out by 0.69 px on its worst scanline.
A tilted span walks a staircase
A span along a banked floor's constant-depth direction is exact, and a renderer visits pixels rather than the span. Snapped to the grid, a 120 px span at a 20° bank costs 0.577 px where the same span along a page row costs 13.26 — twenty-three times better — and it never rises above 1.22 px at any bank. The price is bookkeeping: a band of twenty-four such spans draws 53 of its 1,368 pixels twice.
The map a row of posts is
Walking one bay further down a row is a map of the drawn line to itself, and which map it is settles everything about how the spacings behave. It is parabolic — one fixed point, counted twice, and that point is the vanishing point — which is why the drawn posts crowd toward it and never arrive. Doubling a distance instead gives a hyperbolic map whose multiplier a straightedge can read.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Homogeneous coordinatesProjective invariantTransversalCamera matrixClip spaceCross-ratioForeshorteningHorizonQuantisationVanishing pointdegrees of freedomDemonstration