  Any totally positive $(k+m)\times n$ matrix induces a map $\pi_+$ from the
positive Grassmannian ${\rm Gr}_+(k,n)$ to the Grassmannian ${\rm Gr}(k,k+m)$,
whose image is the amplituhedron $\mathcal{A}_{n,k,m}$ and is endowed with a
top-degree form called the canonical form ${\bf\Omega}(\mathcal{A}_{n,k,m})$.
This construction was introduced by Arkani-Hamed and Trnka, where they showed
that ${\bf\Omega}(\mathcal{A}_{n,k,4})$ encodes scattering amplitudes in
$\mathcal{N}=4$ super Yang-Mills theory. Moreover, the computation of
${\bf\Omega}(\mathcal{A}_{n,k,m})$ is reduced to finding the triangulations of
$\mathcal{A}_{n,k,m}$. However, while triangulations of polytopes are fully
captured by their secondary polytopes, the study of triangulations of objects
beyond polytopes is still underdeveloped.
  We initiate the geometric study of subdivisions of $\mathcal{A}_{n,k,m}$ and
provide a concrete birational parametrization of fibers of $\pi: {\rm
Gr}(k,n)\dashrightarrow {\rm Gr}(k,k+m)$. We then use this to explicitly
describe a rational top-degree form $\omega_{n,k,m}$ (with simple poles) on the
fibers and compute ${\bf\Omega}(\mathcal{A}_{n,k,m})$ as a summation of certain
residues of $\omega_{n,k,m}$. As main application of our approach, we develop a
well-structured notion of secondary amplituhedra for conjugate to polytopes,
i.e. when $n-k-1=m$ (even). We show that, in this case, each fiber of $\pi$ is
parametrized by a projective space and its volume form $\omega_{n,k,m}$ has
only poles on a hyperplane arrangement. Using such linear structures, for
amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that
the Jeffrey-Kirwan residue computes ${\bf\Omega}(\mathcal{A}_{n,k,m})$ from
$\omega_{n,k,m}$. Finally, we propose a more general framework of fiber
positive geometries and analyze new families of examples such as fiber
polytopes and Grassmann polytopes.
