The algorithm described above was implemented with C/C++ in the OpenGL environment. Three case studies are presented here to illustrate the efficacy of the algorithm for constructing a direct RP model. The first two case studies are based on simulated data sets in which the original cloud data are generated by mathematical equations, so that the theoretical shape errors can be obtained accurately and compared directly. The third is an actual case, and its cloud data were obtained with a laser scanner. The results after processing are input to an SLS machine for fabrication.
Figure 6.7. Estimation of the bandwidth of the 2-D data points
In the first case study, a sphere is selected by taking the advantage of its known geometry, so that the shape error of the actual slicing can be compared accurately with the theoretical. Although its geometry is simple, slicing it using a constant layer thickness will produce varying staircase errors. Therefore, it is a good case to test the adaptive slicing algorithm. The equation of a sphere with a radius of 2 is given as follows (random error is incorporated in the equations to simulate noise in the point cloud):
x = 2cosβ cos α +τ , τis randomly distributed in [–0.01, +0.01].
y = 2cos β sin α +τ , β = [–π /2, + π /2].
z = 2sin β , α = [0, 2π ].
We use a sampling increment of 0.01 and 0.02 to sample β and α, respectively, to obtain the cloud data for the sphere. There are totally 98,721 points generated.
The original cloud data are shown in Figure 6.8a.
For data processing, the initial layer thickness is set at 0.04, and the initial N-circle radius is 0.1. ρlow and ρhigh are set at 0.85 and 0.9, respectively. Employing a shape-error tolerance of 0.08, the direct RP model of the sphere shown in Fig- ure 6.8b is obtained. This model contains 11,812 vertices distributed in 74 layers.
Figure 6.9a shows the maximum shape error of each layer in the RP model. It can be seen that the maximum shape errors of all the layers are very close to 0.08. The sphere with a radius of 2 is then sliced into 74 layers according to the layer thickness in the generated model. The theoretical shape errors of all corre- sponding layers are shown in Figure 6.9b. It can be seen that the theoretical errors are close to 0.08 too (except for those close to the two pole areas).
The second case study uses an object composed of four spherical patches (see Figure 6.10a). Compared with the first case, this case poses a more difficult problem for adaptive slicing, which is caused by the combination of the four spherical patches of different diameters and at different positions in space. The parameter of the larger sphere is the same as that in the first case study, and the equations of the three smaller half-spheres are based on the following basic form:
x = cosβ cos α +τ, τis randomly distributed in [–0.001, +0.001].
y = cos β sin α +τ , β = [0, +π /2].
z = sin β , α = [0, 2π ].
The three half-spheres are then formed by transforming the basic form as fol- lows:
(1) Translate the basic form along the z-axis by 1.732 to obtain the first half- sphere.
(2) Rotate the basic form clockwise by 60° around the y-axis, and translate it along the z-axis by 1.732 to form the second half-sphere.
(3) Rotate the basic form counterclockwise by 120° around the y-axis, and translate it along the z-axis by 1.732 to form the third half-sphere.
136 6 Relationship Between Reverse Engineering and Rapid Prototyping
Figure 6.8. The original cloud data and the RP model in case study one. (a) The original cloud data. (b) The RP model. Reprinted from Computer-Aided Design, Vol 36, Wu Y, Wong Y, Loh H, Zhang Y., Modelling cloud data using an adaptive slicing approach, pp. 232–239, Copyright (2004) with permission from Elsevier.
A color reproduction of this figure can be seen in the Color Section (pages 219–230).
Figure 6.9. Shape-error comparison in case study one. (a) Maximum shape error in each layer. (b) Theo- retical maximum shape error. Reprinted from Computer-Aided Design, Vol 36, Wu Y, Wong Y, Loh H, Zhang Y., Modelling cloud data using an adaptive slicing approach, pp. 232–239, Copyright (2004) with permission from Elsevier.
We use sampling increments of 0.02 and 0.05 to sample β and α, respectively, to obtain the cloud data for the object. There are totally 46,057 points, which are shown in Figure 6.10a.
The initial layer thickness is set at 0.04, and the initial N-circle radius is 0.1.
ρlow and ρhigh are set at 0.85 and 0.9, respectively. Employing a shape-error toler- ance of 0.06, the direct RP model of the sphere shown in Figure 6.8b is obtained.
This model contains 21,306 vertices distributed in 88 layers. Figure 6.11a shows the maximum shape errors in each layer. It can be seen that the shape errors of all layers are very close to 0.06. Figure 6.11b shows the theoretical maximum
shape error of each layer when directly slicing the object surface using the same corresponding layer thickness as in the RP model. Most of the shape errors in each layer (theoretical) are close to 0.06, although it is not as uniform as in the one in the case study. This may be due to the complexity of the object. On the other hand, in both case studies one and two, the theoretical shape errors in the two pole areas are much larger than the given shape-error tolerance because of the zero radius at the poles. Similarly, pole problems exist for the three smaller half-spheres, which can be seen from Figure 6.11b.
Figure 6.10. The original cloud data and the RP model in case study two. (a) The original cloud data.
(b) The RP model. Reprinted from Computer-Aided Design, Vol 36, Wu Y, Wong Y, Loh H, Zhang Y., Model- ling cloud data using an adaptive slicing approach, pp. 232–239, Copyright (2004) with permission from Elsevier.
Figure 6.11. Shape-error comparison in case study two. (a) Maximum shape error in each layer. (b) Theo- retical maximum shape error. Reprinted from Computer-Aided Design, Vol 36, Wu Y, Wong Y, Loh H, Zhang Y., Modelling cloud data using an adaptive slicing approach, pp. 232–239, Copyright (2004) with permis- sion from Elsevier.
138 6 Relationship Between Reverse Engineering and Rapid Prototyping
Figure 6.12. Case study three: (a) The original cloud data; (b) the generated RP model; (c) a zoom-in view at the head area on the RP model; (d) shape errors at each layer in the RP model; (e) the toy cow fabricated by using a SLS machine. Reprinted from Computer-Aided Design, Vol 36, Wu Y, Wong Y, Loh H, Zhang Y., Modelling cloud data using an adaptive slicing approach, pp. 232–239, Copyright (2004) with permission from Elsevier.
The object used in the third case study is a toy cow shown in Figure 6.12a. It has a very complex geometry, in particular, around the head. The original object occupies a volume of 150 × 120 × 90 mm and was digitized with a laser scanner, Minolta VIVID-900. The data sets were obtained from different view angles and then merged to produce a cloud data set of 1,098,753 points. The adaptive slicing algorithm was then applied to the cloud data employing an error tolerance of 0.7 mm, initial layer thickness of 0.2 mm, and an initial N-circle radius of 0.2 mm. ρlow and ρhigh were set at 0.85 and 0.9, respectively. This resulted in the direct RP model shown in Figure 6.12b with 115 layers and 59,686 points. In the model construction process, the head area (ears and horns) has a very complex shape and poses a multiple-loop problem. This can be seen clearly in Fig- ure 6.12c, in which the multiple loops are separated successfully and the corre- sponding layers are generated. The shape error of each layer in the generated
model is shown in Figure 6.12d,which clearly shows that the shape errors are within 0.7 mm.
It took about 30 minutes for the adaptive slicing algorithm to generate the RP model using a PC of 1.5 GHz (Pentium III) CPU. The direct RP model was then converted to a layer-based RP slice-data file in CLI format and fed to a SLS ma- chine. For this RP machine, a uniform thickness of layers is required, and hence the direct RP model of 115 layers was further sliced into 535 layers, with a layer thickness of 0.2 mm (the thinnest layer in the model). It took about 6 hours to complete the fabrication. Figure 6.12e shows the workpiece fabricated by the RP machine based on the direct RP model. However, if a RP machine that can de- posit material with changing layer thickness could be used, the time saving could be as much as 78.5%.