## 6 Analysis of Scan Patterns Dimensions: Rules-of-Thumb for Scanner Designs
### Blind (Inner) Zone of the Scan Patterns  
This aspect of the central blind spot is particularly important in practice, for applications such as laser processing. There are two methods to produce a blind spot in the center of the FOV. The first and most reliable one implies using identical Risley prisms, as the expressions of the \(\Delta_{j},\ j=a,\ b,\ c,\ d\) radiuses deduced in Section 3 are not functions of the distance \(L\) to the target, as can also be observed from Figure 4. This is important in applications where this distance \(L\) is variable and cannot be controlled (imaging, for biomedical or industrial purposes, Remote Sensing, CAD/CAM with laser scanners, etc.).  
From the study related to Figure 11 in Section 5.2, the \(\Delta_{j},\ j=a,\ b,\ c,\ d\) radiuses can be decreased by decreasing \(e,\ n,\) and \(b,\) the latter excepting configuration (d). One can also see from Figure 11, column (1), that configuration (b) provides the smallest values of \(\Delta,\) then (in this order) configurations (a), (c), and (d). Thus, depending on the application, one can chose between one or the other configuration, for example (b) for laser manufacturing of contact lenses and (d) for scanning the entire FOV in biomedical apparatuses or machine vision.  
A second way of tackling the inner blind zone is by considering different \(k\
eq 1\) cases. For \(k=2,\) 3, and 5 in Figure 5b-d, the minimum linear deviation given, according to Equation (8), by the sum of Equations (29) and (31), becomes negative, as the tip of the loops 'passes' beyond the O.A. A disc untouched by the laser beam of a radius that increases with \(k\) is thus generated in the center of the pattern. This may look like a similar effect, but it has completely different causes than in the \(k=1\) case. In laser manufacturing or metrology, for example, the distance \(L\) between the scanner and the screen/plane of work can be precisely adjusted; therefore, one could use a scanner with a certain \(k\) and use the equations of the linear deviations in Section 3 to adjust the diameter of the blind inner spot by adjusting \(L\).  
There are applications where the entire disc corresponding to \(D_{j}^{max}\) (and \(y_{j}^{max}\)) must be covered in the most convenient way, for example with the inner part of the loops passing through the O.A. to minimize the time of scan without increasing the fill factor in a non-useful way for the application (e.g., Remote Sensing) for spotting targets of certain sizes in a certain FOV. One can obtain easily using the graphical method a distance \(L\) for a certain value of \(e\) for which this condition is fulfilled [49]. Reciprocal, for a certain distance \(L\) to the scanned plane, distance \(e\) between the prisms can be considered in this respect. This graphical procedure replaces for \(k\
eq 1\) cases the necessity to solve the equation \(y_{j}^{min}=0\), where Equation (8) is utilized with the values of the two segments, \(y_{jP}^{min}\) and \(y_{jS}^{min}\), deduced in Section 3 for \(j=a,\ b,\ c,\ d\). For \(k=1\), this would mean solving the equation \(\Delta_{j}=0\), where \(\Delta_{j}\) is provided in Section 3 for each of the four configurations. Although this may not always give satisfactory results because of the scanner-specific parameters, one can see from the examples in Figure 4 that a situation can be obtained when even for \(k=1\) the patterns (i.e., the tips of the loops) are very close to the FOV center (in that case, at 1 mm for a FOV with a radius of 71.2 mm obtained for \(L=1\) m). Thus, covering the entire FOV can be efficiently done.