k-covering radius of a two-dimensional lattice
The k-covering radius of a lattice L is the smallest r such that every point of the plane lies within distance r of at least k lattice points. For k = 1 this is the usual covering radius.
The simulation below draws the lattice generated by two vectors v1 (green) and v2 (purple). Drag the arrow tips to change the basis. The k-covering radius is estimated by sampling N random points of the fundamental parallelogram and taking the largest distance to the k-th nearest lattice point. The red circles of that radius, centered at the lattice points, then cover the plane k times.