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

Metric rectification — where it appears

Bringing a photographed plane back to a similarity of the real one, which needs the images of its two circular points and buys angle and ratio but not length. It is the last step of the stratification, and the two circular points are exactly the extra fact the affine reading was missing.

Named by 4 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 this site has measured. 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 horizonprincipal pointv_zorthocentre: 812.7691 px · vᵀωu = 0: 812.7691 pxconjugacy residual 5.9e-10 in focal-length unitscorrect from 19 cm, at 160 mm wide46° across

One conic calibrates the camera

This site has recovered a focal length from two perpendicular vanishing points since its first phase, by an orthocentre construction with a square root in it. There is a second derivation with no construction and no square root — two vanishing points of perpendicular directions must be conjugate with respect to one conic in the picture — and the two agree to the last bit. They are not two methods. The conic is what a calibrated camera is.

foundations · absoluteconic circlethe picture isthe pair on the horizona 37° world angle readsradius 3 mellipse345.000 ± 0.000i37.000000°radius 5 mellipse345.000 ± 0.000i37.000000°radius 8 mellipse345.000 ± 0.000i37.000000°radius 10 mhyperbola345.000 ± 0.000i37.000000°radius 14 mhyperbola345.000 ± 0.000i37.000000°radius 20 mhyperbola345.000 ± 0.000i37.000000°a 25% ellipse read as a circleellipsea different pair31.083°one camera, one ground, 6 circlesthe pair drifts 1e-13 px · the angle is out by 7e-14°

Two lines at infinity

A picture of a plane has two of them and they are not the same line. One is the horizon, where the plane's own infinity went; the other is where the picture's coordinates run out. The words ellipse and hyperbola are about the second, and every scrap of metric information is on the first — so a circle whose photograph is a hyperbola calibrates exactly as well as one whose photograph is an oval, to 7e-14 of a degree.

foundations · circularpoints

Named alongside it

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

Circular pointsConicline at infinityAbsolute conicDemonstrationHomogeneous coordinatesHorizonLaguerre formulapoint at infinityProjective stratificationRectificationSimilarity

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