  A point p is 1-well illuminated by a set F of n point lights if p lies in the
interior of the convex hull of F. This concept corresponds to triangle-guarding
or well-covering. In this paper we consider the illumination range of the light
sources as a parameter to be optimized. First, we solve the problem of
minimizing the light sources' illumination range to 1-well illuminate a given
point p. We also compute a minimal set of light sources that 1-well illuminates
p with minimum illumination range. Second, we solve the problem of minimizing
the light sources' illumination range to 1-well illuminate all the points of a
line segment with an O(n^2) algorithm. Finally, we give an O(n^2 log n)
algorithm for preprocessing the data so that one can obtain the illumination
range needed to 1-well illuminate a point of a line segment in O(log n) time.
These results can be applied to solve problems of 1-well illuminating a
trajectory by approaching it to a polygonal path.
