 Circular-arc graph for distinct distances

Around each of $n$ points draw its at most $t$ distinct-distance circles. On every circle containing at least three points, join cyclically consecutive points by arcs. The resulting loopless multigraph has at least $n(n-1)-2nt$ edges and a drawing with $O(n^2{t}^2)$ crossings. A pair joined by several arcs has every corresponding centre on its perpendicular bisector, which is the key to bounding multiplicity.