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

Development — where it appears

A curved surface laid flat, cut where it has to be cut and unrolled without stretching. Only a surface with no Gaussian curvature has one, and it is an isometry rather than a projection: every length on the surface survives it and nothing about the surface's position does.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 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

A bent plate gets two flat pictures on the same sheet and each is exact in what the other loses. The parallel drawing keeps the generators at one scale and stretches the arc over a factor; the development keeps every length on the surface and keeps nothing of the shape in space. Neither is the plate and the pair of them is.

parallel · platepatch

Named alongside it

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

DemonstrationDevelopable surfaceDrawing systemGaussian curvatureOrthographicTrue scaleArea scaleAuxiliary viewAxis scaleForeshorteningIsometrySurface normal

All concepts