Concept

Clip space — where it appears

The coordinates a projection matrix produces before the divide, where the depth survives as a fourth number.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Also named here as homogeneous coordinates — the same set of essays touches all of them, so they are one junction rather than several.

x/z, y/z — the pinholeM·p, then divide by wworst disagreement 4.0e-14 px over 8 verticescorrect from 21 cm, at 160 mm wide42° across · near 0.1 m, far 1000 m

The divide is postponed, not avoided

A renderer does not divide by depth. It multiplies by a four-by-four matrix that carries the depth in a fourth coordinate and divides later, and the postponement is not an optimisation — it is what makes clipping and texture interpolation possible at all. The matrix and this site's pinhole put every point on the same pixel to five parts in a hundred trillion.

pipeline · homogeneous
clipped at the near planedivided without clippingdirection cosine -1.0000 — the far end is drawn 2142 px away20 grid segments, 2.4 m of them behind the eyea reversed line is not a large error, it is a different picture

What happens behind the eye

A point behind the camera has a perfectly plausible image. Dividing by a negative fourth coordinate flips both signs, so the point lands through the principal point on the far side of the frame, and a segment crossing the eye plane is drawn straight, inside the frame, and running in exactly the opposite direction — a direction cosine of −1.0000.

pipeline · clipping
00.2500.5000.7501-10123distance from the eye — log₁₀ metresfraction of the buffer's codes used uphalf the codes by 0.20 ma linear map, for comparisonnear 0.1 m, far 1000 mharmonic mean 0.20 m against arithmetic 500 m

The precision a depth buffer has left

Depth is stored as an affine function of one over the distance, so half of a buffer's codes are spent before the harmonic mean of the near and far planes — twenty centimetres out of a kilometre. The resolution goes as the square of the distance, and the fix that works is not more bits.

pipeline · zbuffer
-0.400-0.200000.2000.4000.6000.8001position across the drawn surfacehow far along the real surface, minus how far along the drawn one(√k−1)/(√k+1) = 0.5195at the page's midpoint, 40.9%depth ratio 10 : 1peak 0.5195 at s = 0.760

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.

pipeline · perspcorrect

Named alongside it

The objects these essays reach for when they reach for this one.

Camera matrixDemonstrationHomogeneous coordinatescentre of projectionpoint at infinityProjective mapResidualConditioningCross ratioDepth bufferDepth divisionFocal length

All concepts