Visual Plane
every picture states the point it is correct from
Series

Homogeneous — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · pipeline
  2. 6 vertices with w = 0 · each on its vanishing point to 7e-12 pxcorrect from 17 cm, at 160 mm wideedges extended to where their w = 0 vertex lands

    A point at infinity is an ordinary vertex

    Give a vertex a zero in its fourth slot and it stops being a point and becomes a direction — and the projection matrix draws it anyway, through the same multiply and the same divide, on that direction's vanishing point to seven trillionths of a pixel. Slide the eye ten metres and it does not move. Two of them bound a ground that reaches the horizon, where a ground drawn to ten kilometres stops a fifth of a pixel short.

    part 2 · pipeline
  3. rate vertices at w = 0 · texture exact to 6e-10 checks at half a pixel from the horizoncorrect from 17 cm, at 160 mm wideone-metre checks, no far plane

    A texture reaches the horizon as a rate

    A ground drawn to infinity cannot carry a texture coordinate at its far corners, because a repeating texture has no coordinate there. It can carry a rate — so many checks per metre along the direction — and a second number that is 1 at points and 0 at directions. Interpolated like every other attribute and divided once, that pair is exact half a pixel from the horizon. Give the same corner a value instead and the ground is drawn in reverse perspective.

    part 3 · pipeline
  4. three vertices at w = 0 · directions exact to 3e-14° · angles off by 7.7°correct from 6 cm, at 160 mm wideno far plane, no point in the triangle

    A sky is carried as a direction

    A triangle of sky has all three vertices at infinity, so the weight that lets a texture reach the horizon is zero everywhere and there is nothing to divide by. The attribute that belongs to such a triangle is the direction itself: carried over w like any other, it is every pixel's own ray to 5e-14 degrees. Carry the vertices' azimuth and elevation instead and a 60° triangle is 7.7° out, a 4096-texel sky needs triangles under ten degrees wide, and a triangle across the seam where azimuth wraps is painted with the opposite sky.

    part 4 · pipeline
  5. camera rolled 11.3°, positions snapped to 1/256 px0 lost · 0 doubled

    Ground and sky meet at the horizon without a crack

    A ground of rate triangles and a sky of direction triangles share their vertices along the horizon, and rasterised as a graphics processor does it — positions snapped to a fraction of a pixel, every centre given to one triangle by the top-left rule — they lose no pixel and claim none twice, at any roll and any snapping. What the horizon does have is a sliver: a centre that falls within half a snapping step of it goes to whichever side the snapped edge puts it, and one that falls exactly on it, rolled one way, is given to the ground at a weight of zero. A ground stopped at a far plane leaves the crack the shared vertex never does: f·h/D rows.

    part 5 · pipeline

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