Computational geometry and the design of aperture-synthesis telescopes I. A measure of the quality of the uv-plane coverage

When designing an aperture-synthesis telescope for use in astronomy it is essential to find a layout of the antennas on the site that provides good coverage of the uv-plane. Large holes in the distribution of baseline points in that plane are particularly undesirable, and so an algorithm was devised for locating all the holes in any given design and assessing whether they are unacceptably large. It finds, in each region of the uv-plane, the largest circle that does not contain any of the baseline points and then takes the properties of that empty circle, such as the radius and the position of the centre, as measures of the properties of the local hole. The algorithm is based on the Delaunay triangulation of the set of baseline points and makes use of the result that the circumscribed circle of any Delaunay triangle is always empty. The largest empty circles in the uv-plane are readily found in this way, and the algorithm selects from these circles a non-overlapping set that is a good match to the intuitive concept of the largest holes in the distribution of baseline points. Modified algorithms are also presented that rank each circle not by its absolute radius but by the radius normalized to the mean radius of the circles in the same neighbourhood of the uv-plane; these versions make proper allowance for the large-scale variation of the density of baselines caused by the taper and are therefore of greater use to the designers of a telescope. The algorithms are efficient, requiring O(NB log NB) operations where NB is the number of baselines, and may be used in several ways. One is to take the radius of the largest empty circle as a figure of demerit so that, of a set of trial configurations, the one with the smallest value may be regarded as having the best uv-plane coverage. Another is to take the baselines on the circumference of the largest circle to indicate which antennas are the most suitable for repositioning in order to improve the quality of the uv-plane coverage of a trial configuration.