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.

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

This site has said, for four rungs, 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
010200.1000.2000.3000.4001 / distance from the eye to the box (per metre)worst angle between edges the box has parallel (°)parallel projection: 0.000°55° at the near endthe line through the origin is the inverse law

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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