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

Tread — where it appears

One step of a stair, and in a turning stair a rectangle rotated from the one below it about a vertical axis. Each tread's front edge has its own vanishing point on the eye-level horizon, and the sequence of those points is a projectivity of that line.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

the horizoncorrect from 16 cm, at 160 mm widecross-ratio 1.366025

The stair that turns has a vanishing point that moves

A spiral stair's treads are one rectangle turned by a constant angle, so every front edge has its own vanishing point and the twelve lie on one horizon to 2.8e-14 pixels. Four consecutive of them read a cross-ratio of 1.353720, and the formula behind it holds no focal length, no principal point and no eye — so a photograph gives the builder's 12° turn back.

construction · Helix the ramp's vanishing linethe ramp's own pointthe horizoncorrect from 16 cm, at 160 mm wide3.43 m out at the sixth tread

A measuring point for a ramp

Stepping true distances along a receding line needs a measuring point, and every printed rule puts it on the horizon. On a 1 in 6.0 ramp the ramp's own point lands every tread to 1.2e-13 pixels and the ground's puts the sixth one-metre tread at 2.57 metres instead of six. A halfway construction separates the two halves of the mistake, and the wrong radius costs 0.083 metres of the 3.43.

construction · Inclined

Named alongside it

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

HorizonVanishing lineVanishing pointCross-ratioDistance pointFixed pointInclined planeMeasuring pointPicture planeProjective invariantProjective lineProjective map

All concepts