Academia.eduAcademia.edu

Figure 3 – uploaded by Helaman Pratt Ferguson

See full PDFdownloadDownload figure
Figure 3. Truncation of basic curves; (a) truncation of Palmer’s curve, (b) truncation of Moore’s curve Truncation is carried out by the creation of a new set of points that represent the original segment in the following manner:

Figure 3 Truncation of basic curves; (a) truncation of Palmer’s curve, (b) truncation of Moore’s curve Truncation is carried out by the creation of a new set of points that represent the original segment in the following manner:

Related Figures (17)

Figure 2 Expansion of basic shape to obtain space-filling pattern; (a) expansion, (b) level 1, 2, (c) level 2 Figure 1 Space-filling curves; (a) Peano, (b) Hilbert, (c) dragon
Figure 2 Expansion of basic shape to obtain space-filling pattern; (a) expansion, (b) level 1, 2, (c) level 2 Figure 1 Space-filling curves; (a) Peano, (b) Hilbert, (c) dragon
Figure 4 Effect of Bevel value on curve truncation; (a) bevel 0.25, (b) bevel 0.5, (c) bevel 0.75 The other curve investigated is a Moore’s curve®, which is a 9-fold repetition (see Figure 5). There are 9** (level) lines before truncation, and again the number doubles after truncation. The truncation of a Moore’s curve is shown in Figure 3b. The truncation sequence that is performed on Palmer’s curve (see Figure 4) is performed on the Moore’s curve in Figure 6.
Figure 4 Effect of Bevel value on curve truncation; (a) bevel 0.25, (b) bevel 0.5, (c) bevel 0.75 The other curve investigated is a Moore’s curve®, which is a 9-fold repetition (see Figure 5). There are 9** (level) lines before truncation, and again the number doubles after truncation. The truncation of a Moore’s curve is shown in Figure 3b. The truncation sequence that is performed on Palmer’s curve (see Figure 4) is performed on the Moore’s curve in Figure 6.
Figure § Expansion of Moore’s curve; (a) level 1, (b) level 2, (c) level 3
Figure § Expansion of Moore’s curve; (a) level 1, (b) level 2, (c) level 3
Figure 6 Truncation bevel values applied to Moore’s curve; (a) bevel 0.25, (b) bevel 0.5, (c) bevel 0.75 TOOL PATHS FOR NUMERICAL-CONTROL CUTTING
Figure 6 Truncation bevel values applied to Moore’s curve; (a) bevel 0.25, (b) bevel 0.5, (c) bevel 0.75 TOOL PATHS FOR NUMERICAL-CONTROL CUTTING
Figure 7 Distortion of sculptured surface caused by tool gouging
Figure 7 Distortion of sculptured surface caused by tool gouging
Figure 8 Mapping of domain to range; (a) domain, (b) range subset
Figure 8 Mapping of domain to range; (a) domain, (b) range subset
Figure 9 Mapping of tool path to range; (a) domain, (b) range subset
Figure 9 Mapping of tool path to range; (a) domain, (b) range subset
Figure 10 Tool paths mapped onto Bézier surface; (a) Moore’s path, (b) conventional path, (c) 2-directional conventional path The Moore’s curve is mapped onto a Bézier surface see Figure 10a) by choosing u and v values along the ool path, and mapping them to the surface. For comparison, a conventional tool path is also mapped onto the same Bézier surface (see Figure 10b). To generate a conventional tool path, the iteration of u and v must be defined. The values of u and v start at 0. The v value is held constant, while u is incremented from 0 to 1 by a set stepover rate. Then, u is held constant (u=1), while » is incremented by one stepover. Next, v is held constant, while u is incremented from 1 to 0, and the process is repeated. This creates a continuous path which travels pack and forth across the part surface. To have a surface finish that is similar to that of a space-filling curve tool path, the part must be machined twice with two sonventional tool paths offset by 90° (see Figure 10c).
Figure 10 Tool paths mapped onto Bézier surface; (a) Moore’s path, (b) conventional path, (c) 2-directional conventional path The Moore’s curve is mapped onto a Bézier surface see Figure 10a) by choosing u and v values along the ool path, and mapping them to the surface. For comparison, a conventional tool path is also mapped onto the same Bézier surface (see Figure 10b). To generate a conventional tool path, the iteration of u and v must be defined. The values of u and v start at 0. The v value is held constant, while u is incremented from 0 to 1 by a set stepover rate. Then, u is held constant (u=1), while » is incremented by one stepover. Next, v is held constant, while u is incremented from 1 to 0, and the process is repeated. This creates a continuous path which travels pack and forth across the part surface. To have a surface finish that is similar to that of a space-filling curve tool path, the part must be machined twice with two sonventional tool paths offset by 90° (see Figure 10c).
Figure 11. Palmer’s curve in machinable wax; (a) Palmer’s path on flat surface, (b) detail Figure 12 Moore’s curve in machinable wax; (a) Moore’s curve on Bézier surface, (b) detail
Figure 11. Palmer’s curve in machinable wax; (a) Palmer’s path on flat surface, (b) detail Figure 12 Moore’s curve in machinable wax; (a) Moore’s curve on Bézier surface, (b) detail
Colour Plate 1B NC machine in operation
Colour Plate 1B NC machine in operation
Colour Plate 1A Stereographic pair of tool path
Colour Plate 1A Stereographic pair of tool path
Table 1 Comparison of measurements for Moore’s curve, single- direction conventional path, and 2-directional conventional path
Table 1 Comparison of measurements for Moore’s curve, single- direction conventional path, and 2-directional conventional path
Figure 14 2-directional conventional tool path in machinable wax; (a) conventional 2-way path, (b) detail Figure 13. Conventional tool path in machinable wax; (a) conventional path on Bezier surface, (b) detail
Figure 14 2-directional conventional tool path in machinable wax; (a) conventional 2-way path, (b) detail Figure 13. Conventional tool path in machinable wax; (a) conventional path on Bezier surface, (b) detail
Related topics:
Mechanical EngineeringComputational GeometryNumerical ControlSurface Finish
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved