For the following discussion of moiré elimination, we focus on the locations of the fundamental halftone frequency vectors and their 2D harmonics and ignore the amplitude and phase of each component. For example, the three halftone patterns shown in Fig. 1 demonstrate the outputs of halftoning the C, M, and K separations where the halftone dots provide approximately 30% area coverage. The example illustrates the results from screens constructed of rotated square halftone cells at 75, 15, and $45deg$, respectively, in the classical three-color moiré-free configuration. The frequency representations, or the Fourier transforms, of the three 2D periodic halftone patterns of Fig. 1 are illustrated by the three plots of Fig. 2, where we use circular dots to indicate locations of all discrete frequency components including the fundamentals and all high-order harmonics within the frequency limits specified by the plots. Subscript 1 and subscript 2 notations are used as a convention to refer to frequency vectors that are above (0 to $90deg$) or below (270 to $360deg$, or equivalently $\u221290$ to $0deg$) the 0-deg axis, respectively. Unless otherwise noted, we use subscripts $c$, $m$, $y$, and $k$, to aid in teaching the presently described halftoning processes due to the common practice of four-color printing with cyan, magenta, yellow, and black. Even though we are teaching using that notation, the concepts are general so that other colorants may be used. For example, we may use the notation $fm1$ and use examples that refer to it as a frequency vector for the magenta screen, but it is to be understood that we intend it to generally imply a frequency vector in the first quadrant for some available colorant. Further, we note that colorants for particular screen geometries are interchangeable. For example, we may have yellow halftoned with a screen of a first geometry, and black halftoned with a screen of a second geometry, but it is practical and reasonable to assume that the screens may be interchanged and yellow may be halftoned with the screen of the second geometry and black with the first.