Let and define the lattice generated by and to be the set . Assume that and are linearly independent, which means they are nonzero and there is no real such that .

Suppose the point is in the generic lattice generated by 1 and (where is not real), so that . So if , , then , .

For a rhombic (centered) lattice, the lattice vectors are , and the lattice coordinates of are , .

For a rectangular lattice, the basis vectors are and , and the lattice coordinates of are , .

For a square lattice, the vectors are , , so that , .

For a hexagonal lattice, the vectors are , , and the lattice coordinates are , .

One way of visualizing a complex-valued function in the plane is to assign a unique color to each point of a certain part of the complex plane, for instance the color value on a picture of the point with coordinates and . The other way is to use RGB parameters that are functions of .

Since rendering these graphics might be slow, it is recommended that you first construct small pictures at low resolution using RGB colors, fix the lattice parameters and , and then use photos.