  Two dimensional steepest descent curves (SDC) for a quasi convex family are
considered; the problem of their extensions (with constraints) outside of a
convex body $K$ is studied. It is shown that possible extensions are
constrained to lie inside of suitable bounding regions depending on $K$. These
regions are bounded by arcs of involutes of the boundary of $K$ and satisfy
many inclusions properties. The involutes of the boundary of an arbitrary plane
convex body are defined and written by their support function. Extensions SDC
of minimal length are constructed. Self contracting sets (with opposite
orientation) are considered, necessary and/or sufficients conditions for them
to be subsets of a SDC are proved.
