  The procedure of the holonomy-flux algebra construction along a piecewise
linear path, which consists of a countably infinite number of pieces, is
described in this article. The related construction approximates the continuous
distribution of the holonomy-flux algebra location along a smooth link
arbitrarily well. The presented method requires the densitized dreibein flux
and the corresponding operator redefinition. The derived result allows to
formulate the gravitational Hamiltonian constraint regularization by applying
the Thiemann technique adjusted to a piecewise linear lattice. By using the
improved Ashtekar connection holonomy representation, which is more accurate
than the one used in canonical loop quantum gravity, the corrections related to
the redefined densitized dreibein flux vanish. In this latter case, the Poisson
brackets of the continuously distributed holonomy-flux algebra along a link
between a pair of nodes are equal to the brackets for these smeared variables
located at the nodes.
