Mapping of field lines from the lower boundary to the vertical plane for several simple potential fields: a dipole, an arcade, and a dipole between the poles of an arcade. In each case, field lines were initiated at the lower boundary from the positive polarity, in the corners of a square grid (with spacing less than one pixel on the lower boundary). On the left, we plot four such field lines from the corners of one square at the lower boundary. If we define a flux tube to have a square cross section at the lower boundary with these field lines in the corners of the square, the cross section of this flux tube in the apex would correspond to the shaded shape on the middle plane, which is simply a rectangle with corners set by the intersection of the "corner" field lines with this plane. Note how different the cross sections are for three different cases—and how different they all are from the starting square at the lower boundary! The right column shows the mapping of many field lines initiated from square grid onto this middle plane (the nodes on this plot are where the field lines intersect this plane and the adjacent nodes are connected showing the cross sections of flux tubes with square bases at the lower boundary).