  Three-dimensional convex bodies can be classified in terms of the number and
stability types of critical points on which they can balance at rest on a
horizontal plane. For typical bodies these are nondegenerate maxima, minima,
and saddle-points, the numbers of which provide a primary classification.
Secondary and tertiary classifications use graphs to describe orbits connecting
these critical points in the gradient vector field associated with each body.
In previous work it was shown that these classifications are complete in that
no class is empty. Here we construct 1- and 2-parameter families of convex
bodies connecting members of adjacent primary and secondary classes and show
that transitions between them can be realized by codimension 1 saddle-node and
saddle-saddle (heteroclinic) bifurcations in the gradient vector fields. Our
results indicate that all combinatorially possible transitions can be realized
in physical shape evolution processes, e.g. by abrasion of sedimentary
particles.
