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 non-degenerate 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.
