Perturbing graphs
by Steve Sigur
xy = 1 and y = x are the blue graphs perturbed by larger and larger numbers.
Perturbing Graphs
Introduction
If
the graphs of functions f = 0 and g = 0 are know, the graph of f · g
= 0 is both graphs displayed simultaneously. If these two graphs intersect,
we can play a nice game, that, in effect, makes 
each
graph “repel” the other at these intersection points. This
is a little known way to produce dramatic and interesting graphs.
A simple example
We use the example where f is a circle of radius 5 and g is a straight line. The first picture shows f g = 0 which is composed of the two graphs.
We
now perturb this graph,
by
setting f g to be a small value, 1 in this case. For most points
on fg this has little effect, but at the intersection of the two graphs,
where each function is trying to be zero, the fact that their product
is not zero causes real trouble, so the graphs are distorted around these
intersection points. This is clearly shown in the second picture.
We continue the process by taking this second graph (the perturbed one) and perturb it again, this time using the line y + x = 0. This is shown on the third graph.
On the following page we have continued several more steps, perturbing the graph with the two lines xy = 0.
A final picture in the sequence
The blue graph in this picture is the result of perturbing a circle with four straight lines.
This process demonstrates that graphs with as many closed loops as desired can be generated.
