Piercing point, true length and visibility on a roof panel
A roofing detail is drawn in dihedral projection with the ground line horizontal and all values in millimetres. Coordinates are written (x, y, z): x measured along the ground line, y the alejamiento (depth behind the vertical plane) and z the cota (height above the horizontal plane). A sloping roof panel lies in plane α, whose horizontal trace α1 passes through (0, 30, 0) and (60, 0, 0), and whose vertical trace α2 passes through (60, 0, 0) and (0, 0, 30). A conductor rod is drawn as the straight segment AC, with A(0, 0, 20) and C(30, 30, 5). The rod must be anchored exactly where it pierces the panel, and at least 20 mm of it must stand proud of the panel. Two colleagues disagree: one says the piercing point is simply where the plan and elevation of the rod cross each other; the other says the rod never reaches the panel, because A lies on the vertical plane where the panel also begins.
1. Determine the piercing point of the rod with the plane, using an auxiliary plane, and give its coordinates.
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Method: take an auxiliary projecting plane through the rod - for example the horizontal-projecting plane β that contains AC. β cuts α along a straight line i, and where i crosses the rod in plan, the rod pierces α; the cota is then carried up to the elevation. This works because β is seen edge-on in plan, so the intersection line i is obtained by a simple two-point construction with α1.
Algebraic verification: α cuts the axes at 60 on x, 30 on y and 30 on z, so x/60 + y/30 + z/30 = 1, i.e. x + 2y + 2z = 60. The rod is A + t(C − A) = (30t, 30t, 20 − 15t). Substituting: 30t + 60t + 40 − 30t = 60, so 60t = 20 and t = 1/3. The piercing point is P(10, 10, 15).
The crossing of the two projections of the rod is not the answer: that point only means the two lines of sight coincide, it lies on the rod's own projections and has nothing to do with α. Nor is A an obstruction: A has y = 0, which merely places it on the vertical plane.
2. Find the true length of the rod above the panel and the total true length AC.
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The segment AP has the vector (10, 10, −5), so its true length is √(100 + 100 + 25) = √225 = 15 mm. The whole rod AC has the vector (30, 30, −15), so |AC| = √(900 + 900 + 225) = √2025 = 45 mm. The proportion confirms it: t = 1/3 of 45 mm is 15 mm above the panel and 30 mm below it.
Note how the true length is obtained: the plan projection AP' is 14.14 mm long and the difference in cota is 5 mm, giving √(200 + 25) = 15 mm by the right-triangle method. The elevation alone does not give the true length, because the rod is oblique to both planes and is therefore foreshortened in both views. The 15 mm above the panel is less than the 20 mm the specification demands, so the rod must be re-positioned or lengthened; that is the practical conclusion the geometry delivers.
3. State, with reasons, which part of the rod is visible in plan and which in elevation.
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In elevation, visibility is decided by alejamiento: of two points whose elevations coincide, the one with the larger y is nearer the observer and therefore visible. Along the rod, y increases from 0 at A to 30 at C, so the portion PC (from the panel outward toward C) lies in front of the panel and is drawn as a continuous line, while AP is behind it and is drawn dashed up to P.
In plan, visibility is decided by cota: the higher point is the visible one. The cota falls from 20 at A to 5 at C, so AP is above the panel and continuous, and PC is hidden and dashed.
P is the only point common to both boundary decisions: it is the crossing of the rod with the surface, and it is where the line changes style in each view. The first colleague confused a projection crossing with a spatial intersection; the second confused y = 0 with a physical obstacle.