1 Using Voxelization and Ray-Tracing to Identify Wall Thinness of ...

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$Henry T. Schatz Math or Science Education Fellowship Application$

2Table of Contents1. Introduction2. Voxels and Voxelization3. Ray Tracing4. X3D, Computer Graphics, and Triangles5. Octrees6. Algorithm Design and Implementationa. Triangle-cube intersections and the Separating Axis Theoremb. Ray-triangle intersections and the manifold propertyc. Classification of voxels using a flooding algorithm7. Results8. Conclusion9. Acknowledgments10. References1. IntroductionThere are several new models of three-dimensional printers currently on the market, and as thetechnology is refined the 3-D printing marketplace will only continue to grow. Though today'sprinters are not yet sophisticated enough to perfectly reproduce machinery at the smallest scale,they are sufficiently accurate and precise to print 3-D objects at a resolution that many artistic 3-D modelers require. It is very easy to imagine that as this technology advances, it could somedaybe used to replace industrial manufacturing. Printers could be used to generate pieces that are nolonger fabricated on a wide scale, perhaps used to replicate legacy parts, such as screws for avintage automobile.Three-dimensional printing is known by a variety of names including rapid prototyping,layered manufacturing, or solid free-form fabrication. The rapid prototyping methodology israther simple: first, a 3D input file is converted so that the printer can 'perceive' the input modelas a stack of very thin layers. The printer then builds the bottommost slice of the object, sprayinga powder or polymer that hardens into the first layer. Next, the printer builds the layer above that,and so on, spraying, and bonding material to existing material, aggregating the model slice byslice.Despite the usage of the word 'rapid', this fabrication process can take many hours.Nonetheless, the many hours it can take to print a prototype, or a model that has never beforebeen created, is actually much less time than would be used to print the same prototype with aconventional manufacturing technology, such as die casting. Though die casting or injectionmolding can be used to manufacture large quantities of parts quickly, the initial preparationneeded to build the necessary mold is high in terms of both cost and time. Rapid prototyping is
thus a relatively quick, competitive choice for production of 'one-off' models compared totraditional manufacturing methods.Rapid prototyping has other advantages as well. It can produce hollow shapes easily, aswell as more complicated usages of hollow space, including interlocking or nested parts a la aship-in-a-bottle or a set of Russian dolls. This technology, though readily available, is not exactlyat one's fingertips: unlike their ubiquitous two-dimensional counterparts, the cost of threedimensionalprinters puts ownership of one out of reach for most. However, people from all overthe world can make use of one by sending model data to a 3-D printing company. Thesecompanies will make a physical object out of your 3D model by converting the model into asuitable file format, sending the information to the printer and then mailing you a copy of yourpersonal creation.As is the case with any developing technology, rapid prototyping is not without itsimperfections. For instance, the materials used for 3D printing are relatively flexible andresistant to breakage, but they lack the strength of the materials used in conventional fabricationprocesses. As technology advances, metal alloys may eventually be used for rapid prototypingwhich would remedy this problem, yielding parts with greater strength. For now, objectsproduced by 3D printing are more suited for display than they are as machinery parts.Perhaps the biggest drawbacks to rapid prototyping are related to printer resolution andaccuracy. For one thing, the printed models are in some sense an accumulation of fused particles,and this means the finished product can have a somewhat granulated surface. More importantly,like their two-dimensional counterparts, 3-D printers have a minimum resolution below whichdetail is lost. The limiting factor is the size of the printer nozzle and the accuracy of the printerhead's movements. The printer simply cannot create details below a certain size. Furthermore,depending on the resolution of the printer and thickness of each slice, a staircase effect can beprominently visible. If significant portions of the input model were below the size threshold, theresulting printed model would be of indeterminate quality: it would suffer from loss of detail andit might be structurally unstable.Thus, it is valuable for 3-D printing companies to analyze model input before sending itto a 3-D printer. Clearly, it is advantageous to avoid any misprints; it is easier (at the very least, itis considerably less expensive) to analyze and then re-scale a polygonal model than it is to print apoor model, re-scale it, and print it again. It is important from a customer-satisfaction standpointthat the finished, printed model is as detailed as the customer expects, and not 'lossy'. It is alsoimportant from a financial standpoint not to waste any time or material on structurally unstableprints.This paper presents a novel application of voxelization and ray-tracing to identify areasof the model that fall below these minimum thickness requirements. The algorithm classifiesvoxels according to their wall thickness; this classification provides a visualization of the modelthat highlights areas that are too thin to print with accuracy or that risk structural collapse. Thevisualization can thus be used to evaluate the integrity of a model and determine if it is suitable3
Page 4: for printing. This technique can be
Page 8 and 9: y the curve and the exterior is unb
Page 10 and 11: The advantage of using XML-based X3
Page 12 and 13: 12Figure 7: X3D file containing a s
Page 14 and 15: and less 'boxy' the more polygons a
Page 16 and 17: 16Child 3 contains points where x i
Page 18 and 19: An algorithm that processes the inp
Page 20 and 21: The algorithm we used for the trian
Page 22 and 23: 22Figure 11: An odd number of inter
Page 24 and 25: 24Figure 13: Example of a model fai
Page 26 and 27: grid point is a zero dimensional ob
Page 28 and 29: 28Figure 15: A two-dimensional exam
Page 30 and 31: 7. ResultsFive values are calculate
Page 32 and 33: 32Figure 19: Many voxels are flagge
Page 34 and 35: 34Figure 21: Closeup of two columns
Page 36 and 37: 36Figure 24: Unshaded mesh geometry
Page 38 and 39: Table 2 Simple Sphere results with
Page 40 and 41: dimensions. Note that the current s
Page 42 and 43: 42Figure 28: Default visualization:
Page 44 and 45: At the original one millimeter reso
Page 46: Despite these successes, I predict

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