Descriptive geometry: point, line and plane; intersections and distances
Dihedral system (sistema diédrico) questions in the PAU ask for intersections, true magnitudes, angles and distances. The reference planes are the vertical plane (PV) and the horizontal plane (PH), meeting at the línea de tierra (LT). A point is fixed by its alejamiento (distance to PV) and its cota (distance to PH); the exam rewards correct coordinates as much as correct drawing.
Core rules
- Line: defined by two projections; its traces are where it meets PH (traza horizontal) and PV (traza vertical). A recta de perfil (perpendicular to LT in both projections) needs a third projection or an auxiliary construction.
- True length: you never take a projection as a true length. Use the right triangle (one leg equal to a projection, the other equal to the difference of cotas) or an abatimiento.
- Plane: by its two traces, or by two lines, or by three points. Horizontal plane: α₂ parallel to LT; frontal plane: α₁ parallel to LT; plane parallel to LT: both traces parallel to LT.
- Plane–plane intersection: the traces of the two planes meet pairwise, giving two points of the intersection line.
- Line–plane intersection: contain the line in an auxiliary projecting plane, intersect that plane with the given one, and cross the resulting line with the original line.
- Distances: point–plane is measured along the perpendicular (projections perpendicular to the corresponding traces); point–line uses a perpendicular plane through the point.
Worked example: distance from point P to plane α
Draw through P the line perpendicular to α: its horizontal projection is perpendicular to α₁ and its vertical projection perpendicular to α₂. Cut this perpendicular with α using an auxiliary projecting plane, obtaining point I. The distance is the true length of PI: if the horizontal projections P₁ and I₁ are 25 mm apart and the difference of cotas is 30 mm, then PI = √(25² + 30²) ≈ 39.1 mm. Do not report the projected 25 mm.
Traps that cost marks
Swapping alejamiento and cota; forgetting that a point in the second quadrant has negative alejamiento; extending traces too little and losing the intersection outside the drawing; drawing the perpendicular to only one trace; quoting a projection as a distance; not labelling the auxiliary plane.