Concept

Isometric — where it appears

4 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.

Also named here as orthographic — the same set of essays touches all of them, so they are one junction rather than several.

elevationx 1.000y 1.000z 0.000cavalierx 1.000y 1.000z 1.000cabinetx 1.000y 1.000z 0.500isometricx 0.816y 0.816z 0.816dimetricx 0.943y 0.943z 0.471axis scales measured from the drawingall 5 preserve midpoints

Parallel projection is not primitive perspective

Isometric and oblique drawing are not what people used before they worked perspective out. They are a different answer to a different question, and the difference is one measurable quantity — a parallel projection preserves the ratio in which a point divides a segment, and a perspective projection destroys it by 7% of the segment's drawn length at a comfortable depth, rising to 13% over the range the slider covers.

parallel · parallel
elevation0.0000two equal, orthographiccabinet0.5000two equal, obliquecavalier1.0000all three equal, obliquedimetric0.4714all three different, orthographicisometric0.8165all three equal, orthographic ←trimetric0.5479all three different, orthographicsmallest of the three axis scalesmeasured from each projection

What isometric actually means

The three axis scales are equal to each other. They are not equal to one. Every unit along every axis is drawn at 0.8165 of its true length, which is √(2/3), and a great deal of confusion about isometric drawing comes from the word promising something it does not deliver.

parallel · isometric
isometric — the limit2.5 m6 m20 m200 msame box, same drawn sizethe eye recedes

The eye taken to infinity

A parallel projection is a photograph from infinitely far away with the lens lengthened to match. That is not an analogy — it is the limit, it can be watched happening, and it explains why a long lens flattens a scene and why an isometric drawing has no viewing distance to state.

parallel · limit
2 — every orthographic projectionisometric · axonometric 0.816 · 0.816 · 0.8162.0000dimetric · axonometric 0.943 · 0.943 · 0.4712.0000trimetric · axonometric 0.876 · 0.966 · 0.5482.0000cavalier · oblique 1.000 · 1.000 · 1.0003.0000cabinet · oblique 1.000 · 1.000 · 0.5002.2500elevation · orthographic 1.000 · 1.000 · 0.0002.0000isometric's three equal scales are forced to √(2/3) = 0.816497not chosen — the identity leaves no other value

Which axis scales are possible

An orthographic projection's three foreshortening ratios always satisfy one identity — their squares sum to two. Isometric's famous 0.8165 is forced by it rather than chosen, and cavalier's 1, 1, 1 sums to three, which is the arithmetic saying cavalier is not the projection of anything.

parallel · axonometric

Named alongside it

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

OrthographicAxonometricPicture planeDimetricForeshorteningOblique projectionParallel projectionTrimetricAffine mapDepth compressionMidpointOrthographic limit

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