Computing Steiner points for gradient-constrained minimum networks

Let TgTg be a gradient-constrained minimum network  , that is, a minimum length network spanning a given point set in 3-dimensional space with edges that are constrained to have gradients no more than an upper bound mm. Such networks occur in underground mines where the slope of the declines (tunnels) cannot be too steep due to haulage constraints. Typically the gradient is less than 1/7. By defining a new metric, the gradient metric, the problem of finding TgTg can be approached as an unconstrained problem where embedded edges can be considered as straight but measured according to their gradients. All edges in TgTg are labelled   by their gradients, being m>m, in the gradient metric space. Computing Steiner points plays a central role in constructing locally minimum networks, where the topology is fixed. A degree-3 Steiner point is labelled minimal if the total length of the three adjacent edges is minimized for a given labelling. In this paper we derive the formulae for computing labelled minimal Steiner points. Then we develop an algorithm for computing locally minimal Steiner points based on information from the labellings of adjacent edges. We have tested this algorithm on uniformly distributed sets of points; our results help in finding gradient-constrained minimum networks.