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

Harmonic range — where it appears

Four collinear points whose cross-ratio is minus one, which is the smallest projectively meaningful configuration on a line. It is constructed with a straightedge alone, from four points or dually from four lines, and the conjugate of a midpoint is the point at infinity.

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

horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across

What a projection destroys

A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.

foundations · Crossratio ABCDjoin and meet only — no length, no angle(A B; C D) = -1.000000000

The quadrilateral that finds the middle

The harmonic conjugate is usually built from four points. Built instead from four lines — the dual construction, the same number of straightedge steps — it lands on the same fourth point to 1.2 × 10⁻¹⁵ of the range's own length, gives a cross-ratio of exactly −1, and refuses the midpoint, whose conjugate is at infinity.

foundations · Harmonic

Named alongside it

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

Cross-ratiopoint at infinityProjective invariantCollineationComplete quadrangleConicDualityHarmonic conjugateHorizonInvariantPencilProjective duality

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