## 6 Case Studies
### Squashing Operation for Curved Part  
The goal of the second test problem is to show how different slice geometries affect the accuracy of the created object. In both cases the squashing algorithm is used to create slices to ensure that there is no missing material. For comparison, the traditional slicing process for the object is shown in Figure 14.  
The input geometry is a curved beam. The geometry of this model is shown in Figure 15a. The slicing algorithm was applied to each layer, with the result shown in Figure 15b. The slice lines due to intersection with the layer planes are shown in side view in Figure 15c. The object which would result is shown in side view in Figure 15d, with the target shape in darker grey and the layers as they would be in lighter grey. Comparing Figure 14 with Figure 15d the difference between classic slicing (Figure 14) and thick slicing and squashing (Figure 15d) can be clearly seen. Except at the top, where the slice extends outside the object, the result material is inside the desired shape with traditional slicing whereas the desired shape is surrounded by deposited material with the slicing-and-squashing method. The squashing operation is important because it prevents missing material volumes and is consistent in adding material compared with traditional slicing. The squashing operation achieves consistency by combining the top and bottom sections together with silhouette geometry to calculate the material 'footprint' as the section profile. Tool path generation differentiates between two methods of offsetting each slice: direct offsetting and modified offsetting after each slice is squashed. The particular layers, such as bottom layer and top layer, seen in this comparison were chosen as they exhibit the highest missing volume of the slices. A comparison of the results produced by the legacy slicing operation  
Figure 13: Gradient distribution of NURBS function: (**a**) A tower shape including top-side gradient, (**b**) A tower shape including right top-side gradient, (**c**) A curved bean discretized gradient, (**d**) A curved beam including top-side gradient. is shown in Table 2. Table 2 describes that legacy slicing method could not describe the original shape perfectly since it looses the original shapes volume except for layer 5. The difference of volumes of the original shape and the outputs of both algorithms is shown in Table 3. As shown in Table 3, output of both algorithms(squash slicing, cylindrical slicing) has additional volume compared to the base(original shape), on the other hand legacy slicing algorithms has smaller volume which means it has lost its volume. For more specific result, Table 4 describes the volume of the each sliced layer and its summation on each different slicing algorithm.In Figure 15d, it is clear that the proposed algorithm produces an output that is close to original shape and does not show any missing volume. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{5}{c}{**Layer Volume**} & \multirow{2}{*}{\(\Sigma\)} \\ \cline{2-2} \cline{4-7}  & **L1** & **L2** & **L3** & **L4** & **L5** & **L6** \\ \hline Legacy slicing (Figure 14) & - & 21.06 & 27.52 & 34.69 & 35 & - & 118..27 \\ \hline Squashed-plane slicing (Figure 16) & 21 & 27.61 & 34.88 & 40 & 37.62 & - & 161.11 \\ \hline Squashed-cylindrical slicing (Figure 15) & 19.3 & 24.32 & 29.22 & 34.68 & 41.08 & 5.36 & 153.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of Slicing method. Figure 14: A result of legacy slicing in side view (**a**) and isometric view (**b**). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Layer Number**} & \multirow{2}{*}{**Original Shape**} & \multirow{2}{*}{**Legacy Slicing**} & \multirow{2}{*}{**Missing(\(-\))/Additional(\(+\))**} \\ \hline
1 & 11.46 & 0 & \(-\)11.46 \\
2 & 24.26 & 21.06 & \(-\)3.2 \\
3 & 31.17 & 27.52 & \(-\)3.65 \\
4 & 36.92 & 34.69 & \(-\)2.23 \\
5 & 22.03 & 35 & +12.97 \\ \hline \hline \end{tabular}
\end{table}