Dihedral projection: points, lines, planes, intersections and perpendicularity
The dihedral block carries heavy weight in the Andalusian paper; here the spatial reasoning is assessed through conventions. Every answer must state the two projections and respect the naming of traces and endpoints.
Points and lines
A point has two projections, A′ on the horizontal plane and A″ on the vertical one. Above the ground line is the cota (height over the horizontal plane); below it, the alejamiento (distance from the vertical plane). A line is defined by its traces: H, where it meets the horizontal plane, and V, where it meets the vertical plane; H″ and V′ lie on the ground line. Lines may be horizontal (parallel to the horizontal plane), frontal (parallel to the vertical plane), parallel to the ground line, vertical (perpendicular to the horizontal plane), perpendicular to the vertical plane, or profile lines, which need a third projection or a point to be fixed.
Planes
A plane is given by its two traces meeting on the ground line, or by two lines or three points, from which the traces are obtained. It may be horizontal, frontal, parallel to the ground line, projecting on either plane (perpendicular to one of them, with the corresponding trace perpendicular to the ground line) or a profile plane.
Intersections
- Line and plane: contain the line in a projecting plane, find the intersection line of that plane with the given one, and the piercing point is where that line meets the given line.
- Two planes: cut both with two auxiliary horizontal (or frontal) planes and join the two intersection points; if the traces meet outside the sheet, use a profile plane.
Parallelism and perpendicularity
A line is parallel to a plane when it is parallel to some line of that plane; two planes are parallel when they share the same direction, which normally shows as parallel traces. A line is perpendicular to a plane when its two projections are perpendicular to the corresponding traces of the plane; a plane is perpendicular to another when it contains a line perpendicular to it. True lengths come from the right triangle built on a projection and the difference of cotas or alejamientos, or from a change of plane.
Worked example
Through A, draw the line perpendicular to plane α and find the distance to the plane. Draw r′ through A′ perpendicular to α₁ and r″ through A″ perpendicular to α₂. Contain r in a projecting plane, intersect it with α to obtain the piercing point I, and the segment AI is the perpendicular; its true length is found by the right triangle on A′I′ and the difference of cotas.
Traps
Mixing second-dihedral sign conventions; giving a profile line only two projections; assuming perpendicularity can be read as 90° on the paper except through the trace rules; forgetting that H″ and V′ always fall on the ground line.