Horizontal Curve External Distance

The external distance answers the right-of-way question: how far does the pavement stay from the corner the two tangents would have made? Measured from the PI along the bisector to the midpoint of the arc, E is the gap that has to clear whatever sits at the corner — a rock face, a building, a wetland boundary. Because the centre, the PI and the curve midpoint are collinear, R + E is the hypotenuse of the same half-angle triangle that produced the tangent length, giving the secant form.

A worked example: a 500 ft curve deflecting 60° has E = 500(sec 30° − 1) = 500(1.1547 − 1) = 77.35 ft. If the survey shows a building face 60 ft inside the PI on the bisector, that curve will not fit and either the radius has to shrink or the alignment has to shift. Watch the behaviour at large Δ: as the deflection approaches 180° the secant blows up and E runs away to infinity, which is the geometry's way of saying a hairpin needs a small radius, not a big one.