[en] The transport theory of the beam phase spaces with noncentrosymmetric polygonal configuration x + bix' - ai = 0 (i = 1, 2,...,n) as well as noncentrosymmetric smooth convex close curve configuration f(x, x') = 0 through linear field is developed. The formulas of the beam envelope of such beam phase spaces through linear field are derived. Detailed calculations are worked out for typical examples to show the practival application of the theory