Setting Out Curves

Choosing the radius determines everything else: tangent length R·tan(Δ/2), curve length R·Δ, and both tangent point positions. The radius comes from design speed, permitted superelevation and side friction — so the geometry is dictated by traffic speed rather than by what fits the site.

At the tangent point the curvature steps from zero to 1/R instantly, which is a step change in lateral acceleration. Nobody can turn a steering wheel that fast, so drivers cut the corner and drift while correcting.

It is the same failure as a cam's constant-velocity motion law: the discontinuity is in the derivative, not in the shape. A transition curve has curvature growing linearly with distance, so L·R is constant and the wheel turns at a steady rate — matching the path a driver takes anyway.

The transition also provides the length over which superelevation is applied gradually, which is the second and equally important reason it exists.

Rankine's method pegs the whole curve from one setup, each deflection being half the angle the chord subtends at the centre. The coordinate method handles obstructions and moved setups as ordinary cases, and crucially makes each peg independent — so one error affects one point rather than everything after it.

A vertical curve is parabolic, because that makes the rate of change of gradient constant. Its length is set by sight distance over a crest, while a sag curve is governed by headlight reach at night and by comfort under the additional vertical acceleration.

Setting out is where a survey error becomes physical, and the cost of the same mistake rises by roughly an order of magnitude at each stage — drawing, peg-out, pour. That is why it is checked independently: a different person, a different method, arriving at the same coordinates.