  Let V be a semialgebraic set parameterized by quadratic polynomials over a
quadratic set T. This paper studies semidefinite representation of its convex
hull by projections of spectrahedra (defined by linear matrix inequalities).
When T is defined by a single quadratic constraint, we prove that its convex
hull is equal to the first order moment type semidefinite relaxation of $V$, up
to taking closures. Similar results hold when every quadratic polynomial is
homogeneous and T is defined by two homogeneous quadratic constraints,or V is
defined by rational quadratic parameterizations.
