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

The other systems

Isometric, oblique, elevation. Not perspective done badly by people who had not worked it out yet, but a different answer to a different question — and the difference is measurable.
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

The measurable difference between parallel and perspective projection, what each preserves, and why neither is a defective version of the other.

 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

Isometric's axis scale of 0.8165, why it is √(2/3), and the three different things that all get called isometric.

 halfway along the drawn linethe actual midpointcorrect from 26 cm, at 160 mm wide7 px apart

The eye taken to infinity

Parallel projection as the limit of perspective as the eye recedes, why a long lens flattens, and what it means for a drawing to have no station point.

 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 · 0.0002.0000military · planometric 1.000 · 1.000 · 1.0003.0000isometric's three equal scales are forced to √(2/3) = 0.816497not chosen — the identity leaves no other value

Which axis scales are possible

The sphere the achievable axis-scale triples lie on, why isometric's equal scales are forced to the square root of two-thirds, and how the identity separates the axonometric systems from the oblique ones.

 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

Pohlke's theorem as a closed-form recovery — the cube and the projection direction behind any three drawn axes, with the reprojection residual at 2e-16.

 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

Frontal fidelity and depth length obey r² + s² = 1 under orthographic projection; the oblique systems escape it by paying an obliquity the recovery can name.

 isometric0.57741 : 1.732dimetric0.88191 : 1.134cavalier1.0000a circlemilitary0.57741 : 1.732the xy plane's drawn ellipseratio of the ellipse's axes, sampled

A ruler on an isometric drawing

The anisotropy of a coordinate plane is what decides whether a ruler works — √3 on every isometric plane, exactly 1 on a military plan.

 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 depth reversal counted as solutions to a recovery — two cubes per parallel drawing, and how fast a perspective picture eliminates the second.

 true ellipse — axis ratio 0.5774four arcs — 5.72% short along the major axisthe four tangent points are exactworst departure 5.72% of the semi-major axis

The ellipse the drawing office draws

The four-centre approximation to an isometric circle, carried out and measured against the conic it stands in for.

 front view · 32 filledtop view · 36 filledside view · 36 filledthe largest solid with these views — 192 cellsand a solid with the same views — 64 cells6 × 6 × 6 cellsthe three views are identical; the solids differ by 128 cells

Three views do not fix the solid

What a front-top-side triple of orthographic views determines about a solid, measured as the gap between the largest and smallest solids that draw the same three pictures.

 true length1.3928 mpoint viewthe edge is one pointtrue shape1.3232 m²one face, three ray directionseach step is a choice of direction, not a construction on the paper

The view that makes a line a point

The auxiliary view as a choice of projection direction rather than a construction on the paper, with the sine law that makes true length and point view two readings of one identity.

 horizonisometric — the same drawing, movedperspective — a different drawingcorrect from 23 cm, at 160 mm wideparallel: 4e-14 px · perspective: 87.2%

Nothing moves when the object does

Translation invariance as the defining behaviour of parallel projection, what it buys, what it costs, and the size–distance signal it deletes.

 90°106°front view · 1.997 m²its own view · 2.900 m²area × 0.6886, the cosineworst corner out by 15.60°

The true shape of a cut

What a standard view does to a cut through a solid, why the area is recoverable and the angles are not, and what the section view is for.

 2468-1-0.50000.5001position along the family the two pictures leave freehow far the solid is from the scene's own shape (stretch ratio)the sceneevery member redraws both pictures to 7e-16 mthe ambiguity is a family of solids, not a tolerance

What two parallel views leave free

The affine factorisation of parallel views, the one-parameter relief family two views leave, and the reflection no number of parallel views removes.

 isometricdimetrictrimetricelevationx scalez scalea curve, not a squarethe third scale is not free

The dimetric the set square draws

The axonometric systems as a two-parameter family on the sphere of axis scales, with the set-square dimetric measured against the direction it stands in for.

 plan: two different solidsthe picture, unmoved to 0e+0 pxcabinetkernel (0.32, 0.32, -0.89)

What one oblique drawing shows

The kernel of a parallel projection, the shear family every single drawing is consistent with, and what a cabinet drawing's half-length depth axis is and is not a claim about.

 the face's normal, drawnratio 0.8112 against the template's 0.5774tilt 0°major axis ⟂ the drawn normal

A circle off the coordinate planes

The image of a circle under a parallel projection, for a plane that is not a coordinate plane, and what a drawing-office template is actually a statement about.

 paper construction and projected shadow agree to 1e-16isometricone angle: 41.76°

The shadow rules that hold here

The paper construction of a shadow, exact in every parallel system because the drawing is a linear map, and the one system in which a ruler also reads the sun's altitude.

 plan: the apex has movedfree along the kernelisometricrecovered to 1e-16

The drawing that gives the solid back

Recovering a solid from a single axonometric drawing by walking its edge graph, what the walk cannot reach, and why the correspondence rather than the picture is doing the work.

 orthographicaspect 1.0000cavalieraspect 1.4142a camera, 37° off axisaspect 1.2593correct from 7 cm, at 160 mm widecentres 4e-14 / 3e-13 / 2.17 px

The ball a drawing does not draw round

The outline of a quadric under a parallel projection, computed from the same three matrix products a camera uses, with the exact aspect an oblique drawing gives a sphere and the centre every parallel drawing keeps.

 a flat plate · best view0a bent plate · best view0.1224a dished plate · best view0.2298a bent plate · along its generators0patches wrapping 29°dished / bent = 1.878

No view draws a curved plate true

The true-shape view searched for over the whole sphere of directions on three patches, and the family of curves a developable has that a doubly curved surface does not.

 00.2500.5000.7501-1-0.50000.5001across the plate, from one edge to the otherdrawn length of a unit of surfacealong the generatorsround the benda plate bent through 57°one ruler in one direction

The drawing and the development

What a parallel drawing of a developable surface preserves and what its development preserves, measured on the same plate, and why the two flat pictures are complementary rather than redundant.

 one platethe otherthe hingedrawn at 94.3° · reads as 100.0° or 86.6°an orthographic drawing of a foldtwo readings, 13.4° apart

A drawn fold has a phantom

The two dihedral angles one orthographic drawing of a hinged pair of plates is consistent with, the map between the true one and its phantom, and the one dihedral that is its own.

 frontierthe transfer joins points 1.414 R apartone ball, two outlinesmeeting in exactly two points

Two outlines are two curves

The contour generator of a curved surface under parallel projection, why it slides continuously where a polyhedron's silhouette jumps, and what that costs the transfer rule of multiview drawing.

 no single viewpoint — the rays miss by no distance at all — the rays are parallel13 of 21 faces

A picture with no eye

What a design carried along a direction rather than from a point can reach on a faceted object, why the map onto each face becomes affine, and why no centre of projection still does not mean nothing hidden.

 no single viewpoint — the rays miss by no distance at all — the rays are parallel3e-14 px of movement

Nothing moves along the direction

The exact invariance that makes exploded and cutaway drawings possible, why the free direction is the kernel of the projection rather than the picture plane's normal, and what the same slide costs in perspective.

 face 15.75×1.00× parallelface 21.41×1.00× parallelface 31.34×1.00× parallelface 41.13×1.00× parallelface 51.05×1.00× parallelface 61.03×1.00× parallelface 71.00×1.00× parallelface 81.00×1.00× parallelthe spread of scale within one faceparallel: 1.000000×

One face, one scale

The two singular values of the map from a design onto a face, why a parallel projection makes them constants of the face, and what a single number was hiding.

 -1.50-1-0.50000.50011.5022.50how far back the eye stands, in object radii (log₁₀)how far a mark lands from where the direction puts it, in metres (log₁₀)one design samplethe faces settlethe marks settlea seating rake, from a point and from a direction3 radii against 34

Two distances to infinity

How far back an eye has to stand before a design behaves like a parallel one, measured separately for the set of faces reached and for the places the marks land.

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