## 3 Types of Scanning Patterns
### Square Spiral Scanning
#### 3.4.2 Mean Acquisition Time for Hexagon Scan  
If the uncertainty area spans from \(-u\) to \(+u\) in both the horizontal (\(x\)) and vertical (\(y\)) directions, then for the hexagonal scan, it can be divided into seven regions as shown in Figure 6. The hexagonal scan will create different loops \(P\). For any point with coordinates \((x,y)\) in a particular loop and on a specific path segment \(\ell\), the cumulative length up to that point from the starting point can be calculated as [36]  
\[l_{P}=n\Big{(}3P^{2}+3P\Big{)}, \tag{46}\]  
\[\left\{\begin{array}{l}P_{1}=-\frac{v+\sqrt{3}x/3}{n}\\ l_{1}(x,y)=l_{P_{1}}+\frac{2\sqrt{3}x}{3}\end{array}\right., \tag{47}\]  
\[\left\{\begin{array}{l}P_{2}=\frac{\sqrt{3}x/3-y}{n}-\frac{1}{2}\\ l_{2}(x,y)=l_{P_{2}-1}+n+\frac{2\sqrt{3}(x-\sqrt{3}n/2)}{3}\end{array}\right., \tag{48}\]  
\[\left\{\begin{array}{l}P_{3}=\frac{x}{n}\\ l_{3}(x,y)=l_{P_{3}-1}+\frac{3}{2}nP_{3}+y\end{array}\right., \tag{49}\]  
\[\left\{\begin{array}{l}P_{4}=\frac{y+\sqrt{3}x/3}{n}\\ l_{4}(x,y)=l_{P_{4}-1}+3 nP_{4}-\frac{2\sqrt{3}x}{3}\end{array}\right., \tag{50}\]  
\[\left\{\begin{array}{l}P_{5}=\frac{y-\sqrt{3}x/3}{n}\\ l_{5}(x,y)=l_{P_{5}-1}+3 nP_{5}-\frac{2\sqrt{3}x}{3}\end{array}\right., \tag{51}\]  
\[\left\{\begin{array}{l}P_{6}=-\frac{x}{n}\\ l_{6}(x,y)=l_{P_{6}-1}+\frac{9}{2}nP_{6}-y\end{array}\right., \tag{52}\]  
\[\left\{\begin{array}{l}P_{7}=-\frac{y+\sqrt{3}x/3}{n}\\ l_{7}(x,y)=l_{P_{7}}+\frac{2\sqrt{3}x}{3}\end{array}\right.. \tag{53}\]  
Similar to other scanning methods, when using the hexagonal scan method, the total scan time for a single scan should be averaged using the probability density function of the receiver satellite position in both the vertical and horizontal directions. Therefore, the  
Figure 5: Hexagon scan path covering the uncertainty area. (**a**) Hexagon scan path with an overlap factor = 0. (**b**) Hexagon scan path with an overlap factor = 0.2.  
expression for the mean acquisition time \(E[T_{\rm h}]\), after decomposing the uncertainty area domain \(\Omega_{u}\) into a union of subdomains, can be written as  
\[E[T_{\rm h}]=\sum_{i=1}^{7}D_{i}, \tag{54}\]  
where  
\[D_{i}=\iint_{(i)}T_{\rm h}(x,y)\,f(x,y)\,dx\,dy, \tag{55}\]  
and  
\[T_{\rm h}(x,y)=\frac{l_{i}(x,y)}{n}\times\Delta t. \tag{56}\]  
For domain \(D_{1}\):  
\[D_{1}=\int_{0}^{\sqrt{3}n/2}\int_{\sqrt{3}x/3-2\sqrt{3}u/3}^{-\sqrt{3}x/3}\frac {l_{1}(x,y)}{n}\times\Delta t\,f(x,y)\,dx\,dy. \tag{57}\]  
For domain \(D_{2}\):  
\[D_{2}=\int_{\sqrt{3}n/2}^{u}\int_{\sqrt{3}x/3-2\sqrt{3}u/3}^{-\sqrt{3}x/3}\frac {l_{2}(x,y)}{n}\times\Delta t\,f(x,y)\,dx\,dy. \tag{58}\]  
For domain \(D_{3}\):  
\[D_{3}=\int_{0}^{u}\int_{-\sqrt{3}x/3}^{\sqrt{3}x/3}\frac{l_{3}(x,y)}{n}\times \Delta t\,f(x,y)\,dx\,dy. \tag{59}\]  
For domain \(D_{4}\):  
\[D_{4}=\int_{0}^{u}\int_{\sqrt{3}x/3}^{-\sqrt{3}x/3+2\sqrt{3}u/3}\frac{l_{4}(x,y)}{n}\times\Delta t\,f(x,y)\,dx\,dy. \tag{60}\]  
Figure 6: Decomposition of the uncertainty area domain \(\Omega_{u}\) into subdomains \(D_{1}\) to \(D_{7}\).  
For domain \(D_{5}\):  
\[D_{5}=\int_{-u}^{0}\int_{-\sqrt{3}x/3}^{\sqrt{3}x/3+2\sqrt{3}u/3} ^{\sqrt{3}x/3+2\sqrt{3}u/3}\frac{l_{5}(x,y)}{n}\times\Delta t\,f(x,y)\,dx\,dy. \tag{61}\]  
For domain \(D_{6}\):  
\[D_{6}=\int_{-u}^{0}\int_{\sqrt{3}x/3}^{-\sqrt{3}x/3}\frac{l_{6}(x,y)}{n}\times \Delta t\,f(x,y)\,dx\,dy. \tag{62}\]  
For domain \(D_{7}\):  
\[D_{7}=\int_{-u}^{0}\int_{-\sqrt{3}x/3-2\sqrt{3}u/3}^{\sqrt{3}x/3} \frac{l_{7}(x,y)}{n}\times\Delta t\,f(x,y)\,dx\,dy. \tag{63}\]  
Therefore, the mean acquisition time for the hexagonal scan can be expressed as  
\[E[T_{\text{h}}]=D_{1}+D_{2}+D_{3}+D_{4}+D_{5}+D_{6}+D_{7}. \tag{64}\]  
The above equation can be analyzed numerically.