  Computing the closed convex envelope or biconjugate is the core operation
that bridges the domain of nonconvex with convex analysis. We focus here on
computing the conjugate of a bivariate piecewise quadratic function defined
over a polytope. First, we compute the convex envelope of each piece, which is
characterized by a polyhedral subdivision such that over each member of the
subdivision, it has a rational form (square of a linear function over a linear
function). Then we compute the conjugate of all such rational functions. It is
observed that the conjugate has a parabolic subdivision such that over each
member of its subdivision, it has a fractional form (linear function over
square root of a linear function). This computation of the conjugate is
performed with a worst-case linear time complexity algorithm. Our results are
an important step toward computing the conjugate of a piecewise quadratic
function, and further in obtaining explicit formulas for the convex envelope of
piecewise rational functions.
