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

Auxiliary view — where it appears

A view taken along a direction chosen for what it makes true — a line at full length, a line as a point, a face at its own shape. Both the true-length view and the point view are the same identity read twice — the imaged length is the true length times the sine of the angle to the ray.

Named by 3 essays across one field — 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 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 auxiliary view is descriptive geometry's answer to a foreshortened plane — turn until the plane is parallel to the paper and it draws at true shape. A bent plate has no such direction and a dished one has none twice over: the best view of the first is out by 1 − cos w and the best of the second by 1 − cos²w, worse by exactly 1 + cos w, because its normals need two parameters rather than one.

parallel · Platepatch

Named alongside it

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

ForeshorteningArea scaleDescriptive geometryMultiview drawingOrthographic projectionTrue lengthAnisotropyDemonstrationDevelopable surfaceDevelopmentDrawing systemElevation

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