Invoking the optical extinction theorem (extended boundary condition) the conventional singular integral equation (for the density of reradiating sources existing in the surface of a totally reflecting body scattering monochromatic waves) is transformed into infinite sets of non-singular integral equations, called the null field equations. There is a set corresponding to each separable coordinate system (we say that we are using the 'elliptic', 'spheroidal', etc., null field method when we employ 'elliptic cylindrical', 'spheroidal', etc., coordinates). Each set can be used to compute the scattering from bodies of arbitrary shape, but each set is most appropriate for particular types of body shape, as our computational results confirm. We assert that when the improvements (reported here) are incorporated into it, Waterman's adaptation of the extinction theorem becomes a globally efficient computational approach. Shafai's use of conformal transformation for automatically accomodating singularities of the surface source density is incorporated into the cylindrical null field methods. Our approach permits us to use multipole expansions in a computationally convenient manner, for arbitrary numbers of separated, interacting bodies of arbitrary shape. We present examples of computed surface source densities induced on pairs of elliptical and square cylinders.