Non-Uniform Offsetting and Hollowing by Biarcs Fitting for RP

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  • 8/2/2019 Non-Uniform Offsetting and Hollowing by Biarcs Fitting for RP


    Non-uniform offsetting and hollowing objects by using biarcstting for rapid prototyping processes

    Bahattin Koc a, Yuan-Shin Lee b,*a Department of Industrial Engineering, State University of New York at Buffalo, Buffalo, NY 14260-2050, USA

    b Department of Industrial Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA

    Received 18 September 2001; accepted 18 September 2001


    This paper presents a new method of using non-uniform offsetting and biarcs tting to hollow out solid objects or thick walls to speed up the part building processes on rapid prototyping (RP) systems. Building a hollowed prototype instead of asolid part can signicantly reduce the material consumption and the build time. A rapid prototyped part with constant wallthickness is important for many different applications of rapid prototyping. To provide the correct offset wall thickness, wedevelop a non-uniform offsetting method and an averaged surface normals method to nd the correct offset contours of thestereolithography (STL) models. Detailed algorithms are presented to eliminate self-intersections, loops and irregularities of the offsetting contours. Biarcs tting is used to generate smooth cross-section boundaries and offset contours for RP processes.Implementation results show that the developed techniques can generate smoothed slicing contours with accuracy for rapidprototyping without suffering from handling the huge number of linear segments of the traditional methods. # 2002 ElsevierScience B.V. All rights reserved.

    Keywords: Rapid prototyping; Hollowing; Non-uniform offsetting; Biarcs tting; Contour smoothing

    1. Introduction

    Rapid prototyping (RP) refers to building partslayer-by-layer. Unlike the traditional material removalprocesses, most common rapid prototyping techniques

    build a part by gradually adding or solidifying materi-als layer-by-layer. Depositing material or tracing theliquid polymer with a laser over the cross-sectionalarea of the part is the most time consuming process. Toreduce the build time, the solid part can be hollowedout to speed up the rapid prototyping process [1].Since, the hollowing operation decreases the area that

    needs to be built, depositing or solidifying the materialon lesser area will not only reduce the build time, butalso reduce the material cost due to expensive RP buildmaterial. Building hollowed rapid prototypes ratherthan completely solids offers the signicant advantage

    of decreasing the time required in building the pro-totypes on the RP systems [2].Rapid prototyped parts can be used to create molds

    for different casting operations such as investmentcasting, die casting and sand casting. In casting opera-tions, the part fabricated by a rapid prototyping pro-cess can be used as a core to make the molds. A rapidprototyped part with uniform wall thickness is impor-tant for many different applications. For instance, arapid prototyped part can be used as a core enclosedby a ceramic shell in investment casting. A core with

    Computers in Industry 47 (2002) 123

    * Corresponding author. Tel.: 1-919-515-7195;fax: 1-919-515-5281.E-mail address : (Y.-S. Lee).

    0166-3615/02/$ see front matter # 2002 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 6 - 3 6 1 5 ( 0 1 ) 0 0 1 4 1 - 5

  • 8/2/2019 Non-Uniform Offsetting and Hollowing by Biarcs Fitting for RP


    non-constant wall thickness can result in non-evenshrinkage that may break the ceramic shell duringsolidication [3]. Molten material may also not ow

    uniformly into a mold created with a rapid prototypingpart with non-constant wall thickness. Therefore, uni-form wall thickness needs to be achieved when ahollowed part is used for a casting process.

    Before a part is fabricated in a RP system layer-by-layer, the STL model of the part needs to be sliced to

    obtain the cross-sectional contours. One would offsetthe cross-sectional contours by an offset distance t tocreate the hollowed part. However, this results in an

    inaccurate hollowed part when the part has free-formsurfaces or complex shapes. Fig. 1(a) shows a hol-lowed example part with a constant wall thickness t .Several planes are used to intersect with the examplepart. Fig. 1(b) shows the intersection contours onthe plane P 1 , of which the offset distance (at both

    Fig. 1. Cross-sectional contours of a hollowed object on different planes: (a) hollowed part with a constant thickness t ; (b) cross-sectionalcontours on plane P 1 (t 1 t ); (c) cross-sectional contours on plane P 2 (notice t 2 > t H2 > t ); (d) cross-sectional contour on plane P 3 (notice onoffset contour on plane P 3).

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  • 8/2/2019 Non-Uniform Offsetting and Hollowing by Biarcs Fitting for RP


    locations) t 1 t . Fig. 1(c) shows the intersectioncontours on the plane P 2 . Due to the change of partsurface normals, the offset distance on plane P 2 varies

    (notice t 2 T t H2 T t ), as shown in Fig. 1(c). Fig. 1(c)also shows the incorrect offset contour if the outer

    boundary is offset with a constant distance t (i.e.t 2 t ) on the same plane P 2 . Fig. 1(d) shows theintersection contour on the plane P 3 . Notice that thereis only one outer contour on plane P 3 due to theintersection location. Fig. 1(d) also shows the incor-rect offset contour when the outer boundary is offsetwith a constant distance t (i.e. t 3 t ) on plane P 3 .Therefore, the constant offsetting of the cross-sec-tional contours cannot be used directly to create thehollowed parts with varying surface normals (e.g.sculptured surface parts) for RP processes.

    To create hollowed objects for the rapid prototypingprocess, several methods have been proposed [24].These methods are classied into three categories: (i)spatial enumeration methods; (ii) constructive solidgeometry (CSG) offsetting methods; and (iii) curveoffsetting methods. Some researchers [5,6] used thespatial enumeration techniques to create hollowedobjects. Chiu and Tan [5] performed a one-dimen-sional Boolean operation between the ray representa-tions of the model and the voxel elements. Alexanderand Dutta [6] also used voxels to calculate the uniformwall thickness of the part. The use of the enumerationmethods causes the internal staircase effect. Theirmethods cannot be used if the accuracy of the internalboundary of the part is important such as in castingoperations. Lam et al. [4] and Yu and Li [1] used CSGtechniques to nd the thin-shell solid by subtractingthe original solid from its offset counterpart. However,their method can be applied only to CSG parts, whichare made from primitives. Their method cannot beapplied to parts in boundary representation (B-Rep) or

    other faceted approximation such as STL models.Thus, after a designed part is converted to an STLle for fabrication in rapid prototyping, their methodscannot be used to generate a hollowed part from theSTL model. Ganesan and Fadel [2] offset the slicedCAD model to create the hollowed part. They offsetcross-sectional contours with a constant offset dis-tance, which will cause a hollowed part with non-uniform hollowed parts as described earlier in Fig. 1.

    In this paper, we present a new method of usingbiarcs tting to hollow out the solid objects to speed up

    the part building processes on the RP systems. Detailsof the proposed techniques are presented in the fol-lowing sections. Section 2 details offsetting a part

    dened by a STL le using the averaged normal vectormethod at each vertex to create offset surface. Section3 presents, the slicing contours and the techniques of removing possible self-intersections, loops and irre-gularities from the contours. Section 4 presents aprocess of smoothing the cross-sectional contourswith biarcs tting. Section 5 shows the computerimplementation and illustrative examples of the devel-oped techniques. Finally, Section 6 concludes thepaper.

    2. Averaged surface normal method for vertexoffsetting

    The STL les are generated by tessellating theoutside skin of the CAD models. The tessellation(STL) is done by approximating the boundary of the CAD object with triangles. An STL le containscoordinates of the vertices and normals for each facet.To offset the STL model of the part, one can offseteach facet with a given offset distance in its corre-spondent normal direction as shown in Fig. 2. How-ever, this could result in intersections or gaps amongthe offset segments. As shown in Fig. 2(a), there is anintersection between the two offset segments of F 1 andF 2 , and there is a gap between the offset segments of F 2 and F 3 , as shown in Fig. 2(a). Finding all theintersections or lling the gaps is not an easy job [7].

    Since a STL le does not contain the geometricinformation of the vertex normal, the normal at eachvertex need to be calculated. There are several normalapproximation methods. In this paper, we use anaveraged normal vector method to offset each vertex

    with the corrected normal direction, as shown inFig. 2(b). An offset normal vector at a vertex iscalculated by averaging the normals of all the facetsthat are connected to the vertex. Since, the averagednormal vector method averages the normal vectors of the facets that are connected to the vertex, it approx-imates the original CAD model closely. However, theaccuracy of the method depends on the number of triangles used in the original STL model when approx-imating the CAD model of the designed part. Asshown in Fig. 3(a), a vertex normal ~ N V i at vertex V i,

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  • 8/2/2019 Non-Uniform Offsetting and Hollowing by Biarcs Fitting for RP