Concept

Circular points — where it appears

The two complex points at (1, i, 0) and (1, -i, 0) that every circle in a plane passes through, and that every similarity of the plane fixes.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Also named here as metric rectification — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

ConicDemonstrationline at infinityMetric rectificationAbsolute conicHomogeneous coordinatesLaguerre formulaProjective stratificationRectificationSimilarityCamera calibrationCamera matrix

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