Orthographic projection — where it appears
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
The dimetric the set square draws
An orthographic direction has two parameters and produces three axis scales, so the achievable triples are a surface rather than a list. The drawing office's dimetric — one axis at 1 in 8, the other at 7 in 8 — has the right three scales exactly and the wrong two angles, and the picture it makes is an oblique projection of a cube rather than an orthographic one.
A circle off the coordinate planes
An ellipse template is cut at one ratio, and the ratio is the cosine of one angle: between a coordinate plane's normal and the direction of projection. A face tilted out of that plane needs a different ratio and — the half that gets drawn wrong even when the ratio is close — a major axis pointing somewhere else, perpendicular to the drawn normal rather than to any edge.
The shadow rules that hold here
Drop the foot, run a line from the top at forty-five degrees, take the intersection: the drawing manual's shadow construction is exact in a parallel drawing, to arithmetic noise, at every point of the picture and with one set square. It is where the rule came from, and carrying it into a perspective picture is what broke it.
No solid casts an aspective figure
Fitting the best single rigid view to an aspective figure — head and legs in profile, eye and shoulders turned square — misses its own marks by 2.6% of the drawn height, and no yaw does better than 3.0% in a full sweep. A genuine single-view drawing of the same body fits to 7.6e-13 pixels, and the five rotations recovered from the marks alone match the convention's own list to 0.0e+0°.
Named alongside it
The objects these essays reach for when they reach for this one.
ForeshorteningParallel projectionAffine mapdegrees of freedomDrawing systemMultiview drawingStraightedge constructionTrue lengthAnisotropyAuxiliary viewDescriptive geometryElevation