  We present two algorithms that compute the Newton polytope of a polynomial
defining a hypersurface H in C^n using numerical computation. The first
algorithm assumes that we may only compute values of f - this may occur if f is
given as a straight-line program, as a determinant, or as an oracle. The second
algorithm assumes that H is represented numerically via a witness set. That is,
it computes the Newton polytope of H using only the ability to compute
numerical representatives of its intersections with lines. Such witness set
representations are readily obtained when H is the image of a map or is a
discriminant. We use the second algorithm to compute a face of the Newton
polytope of the L\"uroth invariant, as well as its restriction to that face.
