Visual Plane
every picture states the point it is correct from
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The thread: Projected, not constructed

Every point in every figure here comes out of a camera with a focal length, not out of a line run to a vanishing point that was placed where it looked right. A picture no camera could produce fails to build rather than appearing with nothing to say so.
horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across What survives

What a projection destroys

A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.

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

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.

 the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide Where to stand

The point you have to stand at

A perspective picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm. Nobody stands there, and that single fact explains most of what gets called distortion.

 correct from 26 cm, at 160 mm wide34° across Light and mirrors

A shadow is a second projection

The construction that puts a shadow on the ground is the construction that puts the scene on the picture plane, with the lamp where the eye was. Shadow drawing is taught as a separate set of recipes and it is one operation with the centre moved, which is why the same code draws both.

 horizon — the image of the line at infinityvanishing point at x = 1464 — off the framecorrect from 26 cm, at 160 mm wide34° across What survives

Where parallel lines meet

They meet at a point that is not in the plane, and the horizon is the line made of all such points. Treating that as bookkeeping misses what it buys — a vanishing point becomes an ordinary point of the picture, findable from the drawn lines, checkable, and enterable into any calculation the others are.

 24VPcorrect from 26 cm, at 160 mm wide34° across Constructing a view

The measuring point, and the step the method leaves out

Laying out equal depths correctly needs a second vanishing point that most treatments never introduce — the one belonging to the diagonals. With it the construction lands on the projected divisions to eighty femtopixels. Without it, depth is placed by judgement and the drawing depicts something nobody chose.

 recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across What survives

Recovering the camera from the picture it drew

Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.

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