Extremal point queries with lines and line segments and related problems

We address a number of extremal point query problems when P is a set of n points in Rd, d⩾3 a constant, including the computation of the farthest point from a query line and the computation of the farthest point from each of the lines spanned by the points in P. In R3, we give a data structure of size O(n1+ɛ), that can be constructed in O(n1+ɛ) time and can report the farthest point of P from a query line segment in O(n2/3+ɛ) time, where ɛ>0 is an arbitrarily small constant. Applications of our results also include: (1) Sub-cubic time algorithms for fitting a polygonal chain through an indexed set of points in Rd, d⩾3 a constant, and (2) A sub-quadratic time and space algorithm that, given P and an anchor point q, computes the minimum (maximum) area triangle defined by q with P∖{q}.