Convergence of piecewise-linear envelope curves in transportation design

Many transportation design problems involve the demarcation of safe zones surrounding the intended guideway, either to prevent physical collisions or to provide clearance for other requirements such as sight distance. Typically, an instantaneous solution to one of these problems can be represented by a line in the plane; the entire solution is then the envelope of this infinite family of lines. In this paper the theory of envelope curves is applied to show how these infrastructure design problems can be solved in closed form. When the solution cannot be expressed in terms of familiar functions, an intuitive practical solution might be to use a finite set of lines to produce an approximate piecewise-linear solution. We show that under general conditions, such discrete solutions converge uniformly to a single continuous curve. In many cases the approximate solutions are perfectly acceptable with regard to error bounds and computation time; our results affirm the reasonableness of these approximate solutions by proving their convergence to the real solution.