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

three-point perspective — where it appears

The case where the picture plane is oblique to all three of a box's axes, so all three families of edges have finite vanishing points. The third vanishing point is usually far off the page, which is why it is the one draughtsmen place by eye and the one that costs the most when misplaced.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px

One, two and three point are one construction

The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity.

construction · Pointcount horizoncentre of the picture0.58° of lean, invisiblethe horizontals are unaffected

The third point put where it looks right

Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.

wrong · Pointcount correct from 20 cm, at 160 mm wideverticals converge 3.59° · horizon 213 px off centre

The plane is a choice

A projection has a centre and a surface, and they move independently. Keep the eye and turn the picture plane and every point of any scene lands where one 3×3 matrix says, to 2.5e-13 px. Move the eye instead and the matrix fitted to four points is exact at those four and out by 32.0 px everywhere else. The first is a homography of the picture; the second is parallax, and nothing about the picture can undo it.

construction · Planechoice correct from 16 cm, at 160 mm wide4 divisions per run · worst 5.7e-13 px

Three-point, laid out with a straightedge

Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.

construction · Threepoint

Named alongside it

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

HorizonPrincipal pointVanishing pointCamera tiltFocal lengthOrthocentrePicture planeCamera calibrationcentre of projectionConditioningDemonstrationDrawing system

All concepts