Plane at infinity — where it appears
Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.
One conic calibrates the camera
A focal length is usually recovered from two perpendicular vanishing points 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.
Seeing the scene fences in the plane at infinity
A reconstruction made without the calibration does not know which of its planes is infinitely far away. Requiring every point to lie in front of both cameras rules out every candidate that would tear the courtyard, and what is left is a convex region — 32 per cent of a generous slice — that always holds the true plane and never shrinks to it.
A drawing has three horizons
The triangle three vanishing points form is not a scaffold. Each of its sides is the vanishing line of one of the three coordinate planes — the floor's horizon and two walls' — and its orthocentre is the centre of the picture. So the horizon a perspective drawing is built on is one side of a triangle, and it is special only because the ground is where things stand.
A sky is carried as a direction
A triangle of sky has all three vertices at infinity, so the weight that lets a texture reach the horizon is zero everywhere and there is nothing to divide by. The attribute that belongs to such a triangle is the direction itself: carried over w like any other, it is every pixel's own ray to 5e-14 degrees. Carry the vertices' azimuth and elevation instead and a 60° triangle is 7.7° out, a 4096-texel sky needs triangles under ten degrees wide, and a triangle across the seam where azimuth wraps is painted with the opposite sky.
Ground and sky meet at the horizon without a crack
A ground of rate triangles and a sky of direction triangles share their vertices along the horizon, and rasterised as a graphics processor does it — positions snapped to a fraction of a pixel, every centre given to one triangle by the top-left rule — they lose no pixel and claim none twice, at any roll and any snapping. What the horizon does have is a sliver: a centre that falls within half a snapping step of it goes to whichever side the snapped edge puts it, and one that falls exactly on it, rolled one way, is given to the ground at a weight of zero. A ground stopped at a far plane leaves the crack the shared vertex never does: f·h/D rows.
A curved surface made of straight lines
A hyperboloid of one sheet carries two families of exactly straight lines, so a photograph of a cooling tower is full of straight lines bounding nothing flat — 28 of them here, drawn to 2.3e-13 px of straightness against a parallel of the same surface that bows 120 pixels. Both families' directions satisfy one asymptotic equation, so their vanishing points lie on one conic in the picture and not on two, to 1.6e-12 px with no fitting anywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Homogeneous coordinatesVanishing pointCamera matrixClip spaceConicDemonstrationHorizonOrthocentrepoint at infinityPrincipal pointProjective mapAbsolute conic