## 4 Slicing Operation  
### Approximation Slicing  
The approximation depends on the distance of facets to the real surface rather than on the layered cross-sections. This means that it is likely that the corner points of planar sections do not actually lie on the real surface but on approximating facets inside or outside the real body. The result is a slight warping of the intended shape, possibly not noticeable to the eye but significant in terms of manufacturing precision. The larger the layer thickness compared to the feature details, the greater the problem. The first step is to use the real geometry of the parts, rather than an approximation. This also leads to problems in obtaining the desired shape. The slicing problems are illustrated in Figures 5 and 6. A common characteristic in both Figures 5 and 6 is that the made shape lies inside the desired shape as the object expands and outside when the object is contracting, as indicated when comparing the red rings with the slices in both figures. In Figure 5, there is a representation of what might happen with a sphere. The right-hand side of the figure shows what might be the sections, the red ring indicates the shape intended. As with the approximated shape in Figure 5, the first section misses the body, since the intersection is at a single point. This means that the shape made is slightly higher than the desired shape. The yellow shape on the left represents a cross section through a sphere approximated by triangular facets. It is also necessary to point out that the accuracy of surface approximation with facets is commonly controlled by a chord-height parameter. This controls the maximum allowable discrepancy from the real surface. However, this is in the direction of the local surface normal.Single directional additive manufacturing adds slices in an arbitrary direction irrespective of the real surface normal. This means that there is no guarantee that the solid  
Figure 4: Planar slicing results between two curves with gradient distribution: (**a**) each slice and (**b**) each path. Figure 3: The notation of gradient distribution of two boundary curves. created will respect the specified chord height tolerance. The desired slicing would be like the Figure 7.  
This is simple geometric reasoning and fairly obvious. It is, however, important when considering additive manufacturing as part of a manufacturing process chain to make parts, where tolerance requirements are stricter than with visualisation. As indicated by the original use of additive processes, and the currently popular name of 3D printing, additive processes for visualisation have much lower requirements than for manufacturing and hence the need for a different control protocol. For manufacturing, additive processes should be seen as part of a multi-process chain. Because of the nature of additive processes, surface finish and precision are lower and so, for critical parts of a piece, it is necessary to finish by machining. This implies that the additive part should be slightly larger than the desired part all over. To achieve this, the part footprint in each layer should be determined to, in turn, determine the slice profile. Instead of considering a single slice, the object is sliced below and above the section to be produced and this fat section is then squashed. The result is shown in Figure 7, with the material around the red-cross section; in effect, an intersection with a rectangular block of thickness equal to the layer thickness. Figure 5: Naive slicing from a facetted model. Figure 6: Naive slicing. Figure 7: Exact slicing with thick sliced. Another example of problems with naive slicing is shown in Figure 8. The desired object is shown at the top of Figure 8. The naive slicing, with a single slice profile is shown underneath. On the third row you have the result of additive with these simple slices. Depending on the angle of the slope, these elements may not even join and the whole sloping element would disappear.