Computing Steiner points for gradient-constrained minimum networks

Abstract

Let Tg 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 m. 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 Tg can be approached as an unconstrained problem where embedded edges can be considered as straight but measured according to their gradients. All edges in Tg are labelled by their gradients, being m, in the gradient metric space. Computing S..