The line

To define a line we need 2 points. In dihedral, we join their matching projections.

Traces are the points where the line cuts the projection planes. There is nothing to calculate: they are read from the projections.

Horizontal trace H: where r″ crosses the ground line, drop a perpendicular to r′. That point is H′ (which is H itself, because it lies on the PH). Its vertical projection H″ falls on the ground line, on the same vertical: H′ and H″ are not the same point.
Vertical trace V: where r′ crosses the ground line, raise a perpendicular to r″. That point is V″ (V itself). Its horizontal projection V′ falls on the ground line.
A line parallel to a plane has no trace with that plane: a horizontal line does not cut the PH and a frontal line does not cut the PV.
A profile line needs the third view to locate its traces without ambiguity.

Checked example: A(20, 10, 30) and B(60, 40, 10). Horizontal trace H(80, 55, 0); vertical trace V(6.67, 0, 36.67).

Most common

It cuts both planes. Its projections are oblique to the ground line.

// to PH

Parallel to the horizontal plane. Vertical projection parallel to the ground line.

// to PV

Parallel to the vertical plane. Horizontal projection parallel to the ground line.

// to both

Parallel to both planes. Both projections parallel to the ground line.

⊥ to PH

Perpendicular to the PH. Its horizontal projection is a point.

⊥ to PV

Perpendicular to the PV. Its vertical projection is a point.

Needs the third view

Projections perpendicular to the ground line. We see a single vertical line.

PASO 1 / 3

Two points define a line. We join their elevations (r″) and their plans (r′).