  We use the octahedron recurrence, which generalizes the quadratic recurrence
found by Kuo for standard Aztec diamonds, in order to compute boundary
one-refined and two-refined partition functions for two-periodic Aztec
diamonds. In a first approach, the geometric tangent method allows to derive
the parametric form of the arctic curve, separating the solid and liquid
phases. This is done by using the recently reformulation of the tangent method
and the one-refined partition functions without extension of the domain. In a
second part, we use the two-refined tangent method to rederive the arctic curve
from the boundary two-refined partition functions, which we compute exactly on
the lattice. The curve satisfies the known algebraic equation of degree 8, of
which either tangent method gives an explicit parametrization.
