Distances

The idea

The distance between two elements is the true length of the shortest segment joining them. In dihedral: first build the segment, then put it in true length.

Point to plane

STEP 1 / 3

Setup

We have point P and the horizontal plane α at height 2. We look for the shortest distance: the perpendicular from P to the plane.

Point to line

STEP 1 / 3

Line and point

Line r is horizontal (constant height): it is seen in true length in the plan. Point P is outside it.

Tip: if the plane or the line are projecting, the distance is seen directly in the view where they look edge-on. You only need to construct it when they are not.

The missing cases

Besides point to plane and point to line, the exam asks for:

Line to plane (parallel): take any point of the line and measure its distance to the plane.
Between two parallel planes: draw a common perpendicular and measure the segment between the two planes.
Between two skew lines: draw the plane containing one and parallel to the other; the distance is that of a point of the second to that plane.
Point to line (general case): draw the plane perpendicular to the line through the point, find the foot I and measure PI in true length.

What is not seen directly

If neither the plane nor the line is projecting, the perpendicular does not appear in true length in any view: you must measure the segment with a rotation or a change of plane. That is the step most often forgotten in exams.

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