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

Pohlke — where it appears

Pohlke's theorem: three segments from a point, of any lengths and any directions not all on one line, are the parallel projection of three equal perpendicular ones. It means a drawing of three axes determines a cube once it is asserted to be one, so the assertion rather than the picture is doing the work.

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

what was drawnthe solid it depicts — reading 1 of 2xyzlooked at 45.00° off the normalcube edge 1.0000 of the drawn unitresidual 5e-16

Any three lines you draw are a cube

Four earlier essays said that cavalier projection is not the projection of anything — because its axis scales sum to three where every orthographic projection sums to two. That is true of orthographic projection and false of projection. Three lines from a point, drawn by hand, are a picture of an actual cube seen from an actual direction, and the cube and the direction come out of the drawing in closed form.

parallel · Pohlke elevation1.0000a circleisometric0.57741 : 1.732cabinet1.0000a circlecavalier1.0000a circlethe xy plane's drawn ellipseratio of the ellipse's axes, sampled

Oblique is a shear, and the shear is the whole system

Cavalier and cabinet are usually introduced as easy perspective for people with a set square. They are not a simplification of anything — they are the answer to a demand no orthographic projection can meet, which is a front face at true size and a depth axis at full length at the same time. The two are locked on a unit circle, and buying both costs exactly 45° of obliquity.

parallel · Oblique what was drawnthe solid it depicts — reading 1 of 2xyzlooked at 0.00° off the normalcube edge 1.0000 of the drawn unitresidual 1e-16

The drawing does not say which corner is nearer

The Necker cube is filed under optical illusion, as though the flipping were something the eye does. It is not: a parallel drawing of a cube is a drawing of exactly two cubes, mirror images of each other, and they project to the identical picture to the last bit. Perspective rules the second one out at a rate exactly inverse in the eye's distance, and never entirely.

parallel · Reversal

Named alongside it

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

Drawing systemParallel projectionAxonometricDemonstrationDepth reversalOblique projectionOrthographicPlanometricAnisotropyDegeneracyDepth cueForeshortening

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