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

True length — where it appears

The length a segment draws at when the view is taken across it, which happens for a whole circle of directions rather than for one. That circle of directions is what an auxiliary view chooses from, and the choice is free in the geometry and narrow on the paper.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 the plan, with its four walls rotated outward about their base linesthe planwalls rotated about their own base lines16 assemblies

The pond with its trees laid flat

An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.

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

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.

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

The drawing that gives the solid back

One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.

parallel · Ruler the ramp's vanishing linethe ramp's own pointthe horizoncorrect from 16 cm, at 160 mm wide3.43 m out at the sixth tread

A measuring point for a ramp

Stepping true distances along a receding line needs a measuring point, and every printed rule puts it on the horizon. On a 1 in 6.0 ramp the ramp's own point lands every tread to 1.2e-13 pixels and the ground's puts the sixth one-metre tread at 2.57 metres instead of six. A halfway construction separates the two halves of the mistake, and the wrong radius costs 0.083 metres of the 3.43.

construction · Inclined

Named alongside it

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

Multiview drawingOrthographic projectionParallel projectionAuxiliary viewdegrees of freedomDescriptive geometryDrawing systemForeshorteningStraightedge constructionAffine mapAnisotropyArea scale

All concepts