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

Laguerre formula — where it appears

The expression of an angle as half the imaginary part of the logarithm of a cross-ratio taken against the two lines to the circular points. It makes an angle a projective quantity rather than a metric one, which is what lets an angle be read out of a photograph at all.

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

horizonv_zthe horizon misses the imaged circle, so the two points are a conjugate pairprotractor on the paper: 80.43° · cross-ratio: 90.000000°correct from 21 cm, at 160 mm wide42° across

An angle is a cross-ratio

A projection destroys angle, which every account of perspective says and a direct measurement confirms. It destroys it only in the sense that the picture no longer carries it for free — because an angle is a cross-ratio taken against two particular points, and once those two points are located in a photograph the world's angles can be read off it directly, with no rectification, no camera, and nothing measured but which lines pass through which points.

foundations · Laguerre horizonthe horizon does not cut the circle — the pair is complexrectified from the two points aloneangles: 2.2e-13°ratios: 5.6e-15length: —circle of radius 1.05 ma dash is a quantity two points cannot buy

The two points a picture hides

The stratification rung says a photograph of a plane is projective, becomes affine when the vanishing line is known, and becomes metric with one more fact. The one more fact has a size and a shape: it is two points, complex conjugates on the vanishing line, and a rectification built from those two and nothing else returns every world angle to a tenth of a picodegree and every ratio of lengths to five parts in a quadrillion — and refuses to name a length.

foundations · Circularpoints

Named alongside it

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

Circular pointsConicDemonstrationline at infinityMetric rectificationProjective stratificationRectificationSimilarityAbsolute conicConic fitCross-ratiodegrees of freedom

All concepts