Optimized Coverage of Rectangles with Circles and Its Use in the Analysis of Lighting Applications, Transport Safety Aspects

A set of circles cover a rectangle if the union of the circles contains the rectangle. We search for the smallest radius of k congruent discs that cover a fixed rectangle and the arrangement of the circles. Up to now the known results are about the common radii of the circles if the number of circles is less than or equal to 7 or the number of the circles is large, the rectangle is enough long. In this paper we present new result if the number of circles is large but the rectangle is not very long. In this paper we also describe the methods and steps of a lighting design project and found three cases where the developed mathematical model could be used for better optimisation. We also mention the transport safety aspects.