We study how a smooth irreducible algebraic variety $X$ of dimension $n$
embedded in $\mathbb{C} \mathbb{P}^{M}$ (with $m \geq n+2$), which degree is
$d$, can be recovered using two projections from unknown points onto unknown
hyperplanes. The centers and the hyperplanes of projection are unknown: the
only input is the defining equations of each projected varieties. We show how
both the projection operators and the variety in $\mathbb{C} \mathbb{P}^{M}$
can be recovered modulo some action of the group of projective transformations
of $\mathbb{C} \mathbb{P}^{M}$. This configuration generalizes results obtained
in the context of curves embedded in $\mathbb{C} \mathbb{P}^3$ and results
concerning surfaces embedded in $\mathbb{C} \mathbb{P}^4$.
We show how in a generic situation, a characteristic matrix of the pair of
projections can be recovered. In the process we address dimensional issues and
as a result establish a necessary condition, as well as a sufficient condition
to compute this characteristic matrix up to a finite-fold ambiguity. These
conditions are expressed as minimal values of the degree of the dual variety.
Then we use this matrix to recover the class of the couple of projections and
as a consequence to recover the variety. For a generic situation, two
projections define a variety with two irreducible components. One component has
degree $d(d-1)$ and the other has degree $d$, being the original variety.

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