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

Orthographic projection — where it appears

A parallel projection along a direction perpendicular to the picture, so lengths across the ray draw at true size and the coordinate along it is dropped.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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

A stepped block on a six-cell grid draws a front, a top and a side view. So does a solid with a third of its material, and so does one with more than the block has — 192 cells against 64, every filled square in all three views identical. The drawing office's triple bounds a part between two solids and does not determine it, and the gap runs to a factor of n.

parallel · multiview 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

Descriptive geometry's first drill is to look at an edge from a direction perpendicular to it, so it draws at true length, and then along it, so it draws as a point. Both are the same identity — the imaged length is the true length times the sine of the angle to the ray — and the sine holds to 1e-12 across the whole sweep of directions.

parallel · auxiliary 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

Slide a box 3.2 m across the world and its parallel drawing is the same drawing translated 131.5 px — every edge the same length to 4e-14 px. The perspective drawing of the same move changes its edge lengths by 87.2%. One family's pictures depend on where a thing is; the other's do not, and almost everything the two families disagree about follows from that.

parallel · translation 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

A plane through a box makes a hexagon of 2.8996 m². The front view draws it at 1.9966 m² — the true area times the cosine, 0.6886 — and gets its corners wrong as well, the worst by 15.60°, because a foreshortening scales one direction and not the other. The area is recoverable with one number and the angles are not, which is why a section gets a view of its own.

parallel · auxiliary 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

Two perspective pictures give shape and no size. Two parallel pictures do not even give shape: the metric upgrade has six unknowns, two views supply six equations, and the system comes out rank five — a one-parameter family of solids that redraw both pictures to 7e-16 m. A third view closes it, and the mirror image survives every count.

parallel · parallelrecovery

Named alongside it

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

Parallel projectionForeshorteningMultiview drawingAffine mapAuxiliary viewDescriptive geometryElevationpoint at infinityreconstruction ambiguityTrue lengthAffine reconstructionAnisotropy

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