Determination of gradient and curvature constrained optimal paths

This article provides an analysis of gradient and curvature constraints on path form and length, with particular reference to road, rail, and pipeline route selection. Initially, we examine the case of a single (global) gradient constraint and a planar surface, with or without boundaries and obstacles. This leads to a consideration of surface representation using rectangular lattices and procedures for determining shortest gradient-constrained paths across such surfaces. Gradient-constrained distance transforms are introduced as a new procedure to enable such optimal paths to be computed, and examples are provided for a range of landform profiles and gradients. Horizontal and vertical curvature constraints are then analyzed and incorporated into final solution paths at subsequent stages of the optimization process. Such paths may then be used as preanalyzed input to detailed cost and engineering models to speed up, and where possible improve, the quality and cost-effectiveness of route selection.