Chapter 4. Non-Uniform Offsetting and Hollowing by Using Biarcs Fitting for Rapid Prototyping Processes

Transcription

1 Chapter 4 Non-Unform Offsettng and Hollowng by Usng Barcs Fttng for Rapd Prototypng Processes Ths chapter presents a new method of Non-Unform offsettng and usng barc fttngs to hollow out sold objects or thck walls to speed up the part buldng processes on RP systems. By buldng a hollowed prototype nstead of a sold part, materal consumpton and buld tme can be reduced sgnfcantly. A rapd prototyped part wth constant wall thckness s mportant for many dfferent applcatons of rapd prototypng. To provde constant wall thckness, an Averaged Surface Normals method s developed to fnd the correct normals to offset the vertces of the STL models. Detaled algorthms are presented to elmnate self-ntersectons, loops and rregulartes of offsettng contours. 4.1 Introducton Rapd Prototypng (RP) bulds parts layer by layer. Unlke the tradtonal materal removal processes, most common rapd prototypng technques buld a part by gradually addng or soldfyng materals layer-by-layer. Depostng materal or tracng lqud polymer wth a laser over the cross-sectonal area of the part s the most tme consumng process. To reduce buld tme, the sold part can be hollowed out to speed up the rapd prototypng process [Yu 95]. Snce the hollowng operaton wll decrease the area that needs to be bult, depostng or soldfyng the materal on less area wll not only reduce the buld tme but also reduce the materal cost due to expensve RP buld materal. By buldng hollow rapd prototypes rather than completely solds, there are sgnfcant advantages wth the decrease n tme requred n buldng the prototypes on the RP systems [Ganesan 94]. Rapd prototyped parts can be used to create molds for dfferent castng operatons such as nvestment castng, de castng and sand castng. In castng operatons, the fabrcated part by a rapd prototypng process can be used as a core to make the 40

2 molds. A rapd prototyped part wth constant wall thckness s mportant for many dfferent applcatons. For nstance, a rapd prototyped part can be used as a core enclosed by a ceramc shell n nvestment castng. A core wth non-constant wall thckness can result n non-even shrnkage that may break the ceramc shell durng soldfcaton [L 98]. The molten materal may also not flow unformly nto the mold created wth a rapd prototypng part wth non-constant wall thckness. Therefore, constant wall thckness needs to be acheved when a hollowed part s used for a castng process. Before a part s fabrcated layer by layer n a rapd prototypng system, the STL model of the part needs to be slced to obtan cross-sectonal contours. One would offset the cross-sectonal contours by an offset dstance t to create the hollowed part. However, ths results n an naccurate hollowed part. Fgure 4.1(a) shows an example hollow part wth a constant wall thckness t. Several planes are used to ntersect wth the example part. Fgure 4.1(b) shows the ntersecton contours on the plane P 1, whch the offset dstance t 1 = t. Fgure 4.1(c) shows the ntersecton contours on the plane P 2. Due to the change of part surface normals, the offset dstance on plane P 2 vares (.e., t 2 > t 2 > t) as shown n Fgure 4.1(c). Fgure 4.1(d) shows the ntersecton contour on the plane P 3. Notce that there s only one ntersecton contour on plane P 3 due to the ntersecton locaton. Fgure 4.2(a) shows the cross-sectons of another hollowed part wth a constant thckness t. As shown n Fgure 4.2(b) the offset dstance t 1 and t 2 on the same cuttng plane are not equal (t 1 t 2 t). Fgure 4.2(c) shows the ncorrect nner offset boundary when the outer boundary s offset wth a constant dstance t. Therefore, the constant offsettng of the cross-sectonal contours cannot be used to create the hollowed parts wth varyng surface normals for RP processes. To create hollow objects for the rapd prototypng process, several methods have been proposed [Ganesan 94, Yu 95, Lam 97, Chu 98, L 98, Alexander 00]. These methods are classfed nto three categores: (1) spatal enumeraton methods, (2) CSG (Constructve Sold Geometry) offsettng methods, and (3) Curve offsettng methods. Some researchers [Chu 98, Alexander 00] used the spatal enumeraton technques to create hollow objects. Chu and Tan [Chu 98] performed a one-dmensonal Boolean 41

3 operaton between the ray representatons of the model and the voxel elements. Alexander and Dutta [Alexander 00] also used voxels to calculate the unform wall thckness of the part. The use of enumeraton methods causes the nternal starcase effect. Ther methods cannot be used f the accuracy of the nternal boundary of the part s mportant such as n castng operatons. Lam et al. [Lam 97, L 98] and Yu [Yu 95] used CSG technques to fnd the thn-shell sold by subtractng the orgnal sold from ts offset counterpart. However, ther methods can only be appled to CSG parts, whch are made from prmtves. Ther methods cannot be appled to parts n B-Rep (Boundary representaton) or other faceted approxmatons such as STL models. Thus, after a desgned part s converted to a STL fle for fabrcaton n rapd prototypng, ther methods cannot be used to generate a hollowed part from the STL model. Ganesan and Fadel [Ganesan 94] offset the slced CAD model to create the hollowed part. They offset cross-sectonal contours wth a constant offset dstance, whch wll cause a hollowed part wth non-constant wall as descrbed earler n Fgure 4.2. In ths research, we present a new method of usng Barcs fttng to hollow out the sold objects or thck walls to speed up the part buldng processes n the RP systems. Detals of the proposed technques are presented n the followng sectons. Secton 4.2 detals offsettng a part defned by a STL fle usng the Averaged Surface Normals method at each vertex to create an offset surface. Secton 4.3 presents the slcng contours and the technques of removng possble self-ntersectons, loops and rregulartes from the contours. Secton 4.4 concludes ths chapter. Computer mplementaton and llustratve examples of the developed technques wll be gven n Chapter Averaged Surface Normals Method for Vertex Offsettng The STL fles are generated by tessellatng the outsde skn of the CAD models. Tessellaton (STL) s done by approxmatng the boundary of the CAD object wth trangles. A STL fle contans coordnates of the vertces and normals for each facet. To offset the STL model of the part, one can offset each facet wth a gven offset dstance n ther correspondent normal drectons as shown n Fgure 4.3(a). However, ths could result n ntersectons or gaps among the offset segments, as shown n Fgure 4.3(a). 42

4 Fndng all the ntersectons or fllng the gaps s not an easy job [Cohen 96]. In ths report, we nstead offset each vertex n ther correspondent normal drectons as shown n Fgure 4.3(b). Snce a STL fle does not contan vertex normals, normals at each vertex need to be calculated. In ths report, we use an averaged normal vector method to offset each vertex wth the corrected normal drecton, as shown n Fgure 4.3(b). There are several normal approxmaton methods. In ths report, an offset normal vector at a vertex s calculated by averagng the normals of all the adjacent facets that are connected to the vertex. As shown n Fgure 4.4(a), a vertex normal Nv at vertex V, where there are n facets connected to, can be calculated as follows: n, j = j= 1 NV n j= 1 N N, j (4.1) where N, are the normals of the facets that are connected to the vertex V. j Although Equaton (4.1) can work for smooth surfaces, t may stll cause problems (for some specal cases) f t s used for vertces at sharp corners or flat surfaces. Dependng on the trangulatons generated n the STL fles, the same vertex may have dfferent sets of adjacent trangle facets connected to the vertex. A vertex on a flat surface or on an edge of the flat surface mght be connected to several faces wth the normals parallel to each other, as shown n Fgure 4.4(b). In Fgure 4.4(b), the two facet normals N and N are parallel (.e.,,1, 2 N,1 N,2 ). In Fgure 4.4(b), the normal vector Nv at vertex V s calculated as follows: + N + N N,1 + N,2 + N,3 + N,4 N = (4.2) V N,1,2,3 + N,4 43

5 Drectly averagng these normals (Fgure 4.4(b)) to calculate the vertex normal may result n a normal vector shfted towards the faces wth parallel facet normals (.e., N,1 N,2 ). As shown n Fgure 4.4(b), the averaged surface normal Nv at the vertex V could result n a vector that s closer to the faces F,1 and F,2 due to the fact that these two adjacent faces have the same parallel normals ( N,1 N,2 ). Fgure 4.4(c) shows the corrected normal vector Nv found by elmnatng the duplcated parallel normals n the calculaton of averaged normal. In Fgure 4.4(c), the corrected surface normal vector Nv at the vertex V s calculated by averagng all the adjacent facet normals wthout the duplcated parallel normal as follows: + N N,1 + N,3 + N,4 N = (4.3) V N,1,3 + N,4 After the corrected normal vectors Nv at each vertex V are found, the offset vertces V can be calculated by offsettng the vertces n ther normal drectons wth a gven offset dstance t as follows: V = V ± t Nv (4.4) In Equaton (4.4), the +- sgn depends on whether t s offset outward or nward from the orgnal part surface. The algorthm for calculatng the averaged normal vectors and the offset vertces are shown as follows: Algorthm 4.1: Calculatng Averaged Normals and the Offset Vertces INPUT: STV={V }: a set of the vertces from a STL model, 1 num_vertex, num_vertex: total number of vertces. STF={ F,j }: a set of faces that are connected to vertex V, 1 j num_face, num_face : total number of faces that connect to vertex V, 44

6 STFN={ N,j }: normal vector of facet F j, t : vertex offset dstance. OUTPUT: STV ={V }: Offset vertces, 1 num_vertex Intalze 1, j 1, k 1, normal_check 0, WHILE ( num_vertex) { *** man loop of procedure *** Nv N 1 ; FOR (j = 2 to num_face, j++): { FOR (k = j-1 to 1; j--): { } END IF (Ν j.ν k ==1) *** Ν j and Ν k are parallel to each other*** THEN { normal_check 1; k 0; } ELSE {normal_check 0; } *** Ν j and Ν k are not parallel *** } *** End of for loop *** IF (normal_check == 0) THEN {Nv Nv + N j ; } } *** End of for loop *** Normalze Nv ; Calculate the offset vertex V usng Equaton (4.4); STV STV {V }; + 1; Algorthm 4.1 can be used to construct the slcng contours and offset contours for RP processes. Detals of the procedures are dscussed n the next secton. 4.3 Slce Contour Generaton and Self-ntersectonLoop Removal To generate cross-sectonal curves, a tessellated part s ntersected wth a set of planes perpendcular to the buld drecton of the STL fle. The dstance between the ntersectng planes can be a constant n unform slcng or can be varyng n adaptve slcng [Dolenc 94, Kulkarn 96, Tyberg98]. Gven a plane and ts heght (n z drecton), all the facets are searched to fnd a startng ntersecton pont. After an ntal ntersecton pont s found, the followng ntersecton ponts are traced usng the nformaton of neghborng facets. The ntersecton ponts are then connected to form the cross-sectonal contours [Lee 92]. Ths procedure s appled to both the orgnal surface and the offset surface untl all the facets are slced. 45

7 After the cross-sectonal contours are generated, the self-ntersectons and loops need to be removed to ensure that offset contours are smple and closed as mentoned earler. As shown n Fgure 4.5(a), f the offset dstance t s greater than the mnmum radus n the concave regons, the offset surfaces may cause self-ntersectons [Maekawa 99]. The offset surface mght also ntersect tself, whch causes loops as shown n Fgure 4.5(a). Self-ntersectons and loops can be removed from the offset surfaces by frst detectng the ntersectons and loops. The self-ntersected surfaces are then trmmed from the offset surfaces. However, ths requres tme-consumng surface-to-surface ntersecton calculatons, complex self-ntersecton detectons, and trmmng operatons to form the closed offset surfaces. Even after the closed offset surface s generated, the orgnal part and ts offset surface have to be slced to buld the hollowed part layer by layer. Therefore, we can remove the self-ntersectons and loops from the cross-sectonal curves after the slcng operaton s carred out. Complex 3D ntersectons and trmmng operatons can be reduced to 2D curve-to-curve ntersectons and trmmng operatons. Fgure 4.5(b) shows the cross-sectonal contours from the orgnal part surface and ts offset surface shown n Fgure 4.5(a). As shown n Fgure 4.5(b), the self-ntersecton and loop stll exst but now on the 2D cross-sectonal plane. Fgure 4.5(c) shows the same cross-sectonal contours after the self-ntersecton and loop are removed from the offset contour. To elmnate the loops, the self-ntersecton ponts need to be detected frst. An ntersecton between two segments can be detected by checkng whether there s an ntersecton pont between the current segment and other segments along the offset crosssectonal contours. Fgure 4.6 shows an offset contour wth a self-ntersecton. The ntersecton pont P * between two segments P P + and P 1 k P k+ 1 on Fgure 4.6 can be calculated as follows [Lee 92]: P * = P + t V1 = Pk + s V2 (4.5) where = P P V and V P P k 2 = k + 1 be found as follows [Lee 92]:. By solvng the Equaton (4.5), parameter t and s can 46

8 t = s = P Pk V1 V2 V1 V2 V2 V1 V2 P Pk V1 V1 V2 V1 V2 V1 V2 (4.6) (4.7) If P P + and P 1 k P k+ 1 ntersect each other, both the parameters t and s have to be between [0,1]. The ntersecton pont P * can be calculated by substtutng s or t n Equaton (4.5). After the ntersecton pont P* s calculated, the ntersecton pont P* can be used to separate the curve nto two loops. As shown n Fgure 4.6, ntersecton pont P * dvdes the curve nto two loops, [P *, P +1 -P k, P * ] and [P *, P k+1 -P n, P 1 -P, P * ] (for < k). By usng the procedure, the self-ntersectons can be deleted from the offset contours. Snce we do not know whch loop s the correct offset contour and whch one s the self-ntersecton, a procedure s needed to determne the self-ntersecton. There are several self-ntersecton detecton methods n the lterature [Tller 84]: (1) Consder the number of segments n each loop and remove the loop wth the least segments (2) Consder the total length of the each loop and remove the shortest loop, (3) Consder loop nestng and remove the loop whch s nsde another loop. These methods mght not work f the orgnal loop s relatvely smaller than the loop that needs to be removed. For nstance, the loops LP 1 and LP 2 n Fgure 4.7(b) would be removed f one of the above methods were used. But then the orgnal loops would be removed ncorrectly. As shown n Fgure 4.7(c), the loop LP 2 should be removed and the loops LP 1 and LP 3 should be kept. To fnd the loops that need to be removed, the surface normals on the offset contours are used. The normals of each facet on the orgnal surface can be obtaned from the STL fle of the part. However, these normals cannot be used for the offset facets because the offset facets change the shape due to offsettng the vertces along ther 47

9 normal drecton. Therefore, normals on the offset facets are not the same as the normals on the orgnal facets. Gven a facet F and ts vertces V,1, V,2 and V,3 ordered n the counter-clockwse sequence, the offset facet F and ts vertces V,1, V,2 and V,3 can be calculated by usng Algorthm 4.1. Note that the offset vertces are also n the sequence along the counter-clockwse drecton. Then the normal vector N of the offset facet F can be calculated by takng the cross product of the two vectors N v, 1N v, 2 and N v, 2N v,3 as: N = N v N v,1,1 N v N v,2,2 N v N v,2,2 N v N v,3,3 (4.8) To calculate the projected normals on the cross-sectonal contours, the surface normals N need to be projected onto the cuttng plane. Gven the heght of cuttng plane z cut, the projected normals [ yp, zp,1] found as follows: where xp n the homogeneous coordnate system can be N, N N xp z N N N N N N cut (4.9) z N [, yp, zp,1] = [ x, y, z,1], are the x-y-z elements of the offset facet normals N. Fgure 4.7(b) x N yn, zn shows the surface normals projected on a cuttng plane. The normals pont outward on the outsde boundary whereas the normals pont nward n the nsde offset boundary of the STL model. Therefore, the loops whose normals pont outward along the ntersecton loops are the self-ntersectons, as the loop LP 2 shown n Fgure 4.7(b). In Fgure 4.7(b), the offset loops LP 1 and LP 3 have ther normals pontng nward whch ndcates they are the correct offset loops. However, the normals on the loop LP 2 pont nward whch ndcates LP 2 s an self-ntersecton loop and needs to be removed. Fgure 4.7(c) shows 48

10 the self-ntersecton loop LP 2 beng removed from the offset contour. Gven a segment P orented n the counter-clockwse drecton and ts normal vector N, the P + 1 locaton of the normal vector N can be determned as follows: = P P + N NC 1 (4.10) If the z-value of the vector NC s negatve, then the normal s at the rght sde of the segment P P + 1 otherwse t s at the left sde of the segment. For each loop on the offset contour, locaton of ther normals can be determned usng Equaton (4.10). If the normals are on the left sde of the segments then the offset loop s vald, otherwse t s nvald and needs to be removed. There mght be some nested ntersectons on the offset contours. Therefore, the self-ntersecton removal process needs to contnue untl all the ntersectons are removed. Offsettng surfaces can cause other rregularty problems on the offset contours besdes self-ntersectons and loops problems. These rregulartes nclude jagged lnes, collnear ponts and nflecton ponts, as shown n Fgures 4.8(a), (b) and (c) respectvely. In order to generate the correct cross-sectonal contours, these sngulartes need to be detected and corrected. As shown n Fgure 4.8(a), the jagged lnes can be detected by checkng the normal vector of each segment. If the change between the normal vectors of two consecutve segments s equal or close to π then these two segments are consdered to be jagged lne segments as shown n Fgure 4.8(a). In Fgure 4.8(b), the collnear ponts occur when the normal vectors of three or more ponts are parallel or close to parallel to each other. In Fgure 4.8(c), the nflecton ponts result from the dscontnuous tangents (.e. T, 1 and T, 2 ) at a pont P. Jagged lnes and nflecton ponts can be smoothed by the 2 nd -dfference farng technque [Cho 98]. As shown n Fgures 4.8(a) and (c), the corrected pont P can be calculated as follows [Cho 98]: d 1 d + 1 P P = d d P (4.11) 2 49

11 where d -1, d 0, and d +1 are 2 nd -dfferences and they are defned as follows: d = P 1 P, d + 1 = P + 1 P, d 0 = ( d 1 d 1) 2 (4.12) Collnear ponts can be easly removed except for the begnnng and the end ponts of the collnear segments. As shown n Fgure 4.8(b), only ponts P -1 and P +2 are saved whle the ponts P and P +1 are removed from the contour. After the ntersectons and the rregulartes are removed, the next step s to smooth the contours wth barc curves. Barc curve fttng s appled to both the orgnal part boundary and the offset contour. Therefore, smooth cross sectonal segments can be generated. Detals of the barc curve fttng are gven n Chapter 3. The Max-ft barcs fttng algorthm progresses through the STL slcng data ponts and offsettng contour ponts to fnd the most effcent barc curve fttng, whle mantanng the requred tolerance [Koc 00b]. After the smooth cross-sectonal contours are generated, the data can be sent to the RP systems for preprocessng and prototypng. Detals of the procedures of generatng offset contours for RP processes are shown as follows: Algorthm 4.2: Generatng Cross-sectonal Contours INPUT: STV={V }: a set of vertces 1 num_vertex, num_vertex: total number of vertces, STF={ F,j }: a set of faces (trangles) that connect to vertex V, 1 j num_face, num_face : total number of faces that connect to offset vertex V, STV ={V }: a set of offset vertces from Algorthm num_vertex, STF ={ F,j }: a set of faces (trangles) that connect to offset vertex V, 1 j num_face, layer_thcknes: layer thckness, OUTPUT: STI={ I l,k }: a set of ntersecton ponts of the orgnal contour at the cuttng plane k, 1 k num_layer, 1 l num_pont k, STI ={ I l,k }: a set of ntersecton ponts of the offset contour 50

12 at the cuttng plane k, 1 k num_layer, 1 l num_pont k, num_layer: number of layers, num_pont k : number of ntersecton ponts at layer k, NP, k : projected normal vectors of each segment at the cuttng plane k, 1 num_pont k -1. Intalze k 0, z mn mn {z V }, z max max {z V }; *** z V s the z-value of the vertex V *** num_layer ( z max - z mn )layer_thckness; WHILE (k num_layer) { *** man loop of procedure *** k k +1; z cut z mn + k*layer_thckness ; fnd ntal ntersecton pont at z cut by searchng all the facets calculate the ntersecton ponts I l,k, I l,k on the orgnal boundary surface and the offset surface by ntersectng cuttng plane at z cut and F j and F j ; calculate the projected normals NP usng Equatons (4.8) and (4.9); STI STI {I l,k }; STI STI {I l,k }; END Algorthm 4.3: Generatng Unform Offsettng for RP processes INPUT: STI={ I l,k }: a set of ntersecton ponts of the orgnal contour at the cuttng plane k, 1 k num_layer, 1 l num_pont k, STI ={ I l,k }: a set of ntersecton ponts of the offset contour at the cuttng plane k, 1 k num_layer, 1 l num_pont k, num_layer: number of layers, num_pont k : number of ntersecton ponts at layer k, 51

13 NP, k : projected normal vectors of each segment at the cuttng plane k, 1 num_pont k -1. OUTPUT: Barc(m k ): A set of barc curves of the orgnal contour at the layer k, 1 k num_layer, m k : number of barcs ftted to the orgnal contour at layer k. Barc (n k ): A set of barc curves of the offset contour at the layer k, 1 k num_layer, n k : number of barcs ftted to the offset contour at layer k. Intalze 1, j 1, k 0, current_segment { }, check_segment { }, vald_loop { }, loop 1 { }, loop 2 { }, FOR (each layer k and STI and STI ) { *** man loop of procedure *** WHILE ( ntersect_check == 1){ *** self ntersecton elmnaton *** FOR (each I l,k vald_loop) { current_segment I l, k I l+ 1, k ; FOR (each I sk where l < s < num_pont k ) { *search for the ntersecton * check_segment I s, k I s+ 1, k ; calculate the s and t between current_ segment and check_segment usng Equatons (4.6) and (4.7); IF ( 0 s $1' t THEN { *** curent_ segment and segment ntersects*** ntersect_check 1; calculate ntersecton pont P * usng Equaton (4.5); save P * as a nflecton pont; loop 1 {P *, I l+1k -I sk, P * }; loop 2 {P *, I s+1k -I num_pontk,k,i 1k -I lk, P * }; FOR (each loop, = 1,2) { Check f loop s vald usng Equaton (4.10); IF (loop s vald) THEN{ vald_loop loop }; 52

14 ELSE { ntersect_check 0; } } *** End of for loop *** } *** End of for loop *** } *** End of whle loop *** FOR (each I l,k vald_loop) { *** removng rregulartes ** current_segment check_segment I l, k I l+ 1, k ; I l+ 1, k I l+ 2, k ; N current N check l k NP, ; *** projected normal vector of current_segment *** NP l+ 1, k ; *** projected normal vector of check_segment *** IF ( the angle between N current and N check s close to π or I l,k s nflecton pont ) THEN{ calculate new pont I l,k usng Equatons (4.11) and (4.12) }; IF ( the angle between N current and N check s close to zero) THEN{ *** collnear ponts *** remove I l+1,k from vald_loop}; } *** End of for loop *** Barc(m k ) barc ft the vald_loop usng Algorthm 3.1; Barc (n k ) barc ft the vald_loop usng Algorthm 3.1; END These algorthms are used to construct and fnd the smoothed slce contours of the hollowed object for RP process. Computer mplementatons and llustratve examples wll be gven n Chapter Summary A new method of usng barc fttngs to hollow out thck walls or sold objects has been proposed. The developed algorthms provde hollowng wth constant wall thckness whch can avod non-constant shrnkage and warpage after the RP part s bult [Koc 00b]. To create the hollowed object, the vertces are frst offset wth a gven offset dstance along ther correspondent calculated normals. The resulted surfaces (both the orgnal and 53

15 the offset surfaces) are then slced to generate cross-sectonal curves. The selfntersecton and rregulartes are removed from the offset contours to form smple and closed contours. The fnal contours are then smoothed out by barc fttng. The resulted cross sectonal contours can be sent to a RP machne to generate support structure and fnally, to be fabrcated. Computer mplementaton and examples are presented n Chapter 6. 54

16 P 3 t P 2 P 1 (a) Hollowed part wth a constant thckness t P 1 P 2 t 1 t 1 t 2 t 2 (b) Cross-sectonal contours on plane P (t=t) 1 t=t 1 1 t>t >t 2 2 (c) Cross-sectonal contours on plane P (t >t >t) P 3 (d) Cross-sectonal contour on plane P 3 Fgure 4.1 Cross-sectonal contours of a hollowed object on dfferent planes 55

17 Cuttng plane t 1 t 2 t t t=t=t 1 2 (a) Hollowed part wth a constant thckness t t 1 t 1 t 2 t 2 t=t=t 1 2 (b) Actual cross-sectonal contour of the hollowed part t=t=t 1 2 (c ) Incorrect offset contour f a constant offset dstance s used on the cuttng plane Fgure 4.2 Cross-sectonal contours of a hollowed object 56

18 Intersecton t.n 2 F 2 F 3 F 1 F 2 t.n 1 t.n 2 t.n 3 t.n 4 t.n 3 F 3 F 4 F 1 F 4 t.n 1 Gap t.n 4 (a) Offsettng surfaces along the segments normal drecton v 1 v 3 F 2 F 3 F 4 F 1 t.nv1 v 1 F 1 F 2 v 2 t.nv2 F 3 v 2 v 3 t.nv3 F 4 (b) Offsettng vertces n the corrected surface normal drectons Fgure 4.3 Offsettng surfaces and vertces 57

19 Nv N, 4 N, 3 N, 5 F, 4 F, 3 F V, 5 F, 2 N, 2 n j 1 NV n j 1 N, j N, j 1 j 5 F, 1 N, 1 (a) Calculatng the vertex normal Nv by averagng the facet normals N, 3 N, 3 Nv Nv F, 2 F, 3 V F, 4 N, 4 F, 2 F, 3 V F, 4 N, 4 N, 2 N, 1 F, 1 N, 2 N, 1 N N N, 2 N, 1 F, 1, 2, 1 (b) Incorrect normal vector Nv shfted towards faces F and F, 1, 2 (c) Corrected normal vector Nv by deletng the parallel facet normals Fgure 4.4 Calculatng the averaged normal vectors Nv at a vertex 58

20 Self-ntersecton Part Surface Offset surface Loop Cuttng plane (a) Part surface and ts offset surface wth a self-ntersecton and a loop Self-ntersecton Orgnal contour Offset contour Loop (b) Cross-sectonal contours of the part surface and ts offset surface Orgnal contour Offset contour (c) The self-ntersecton and the loop are removed from the same offset contour Fgure 4.5 Self-ntersectons and loops of the offset surfaces 59

21 P k+1 V 2 P +1 V 1 P * P k P Fgure 4.6 Detecton of a self-ntersecton on the offset cross-sectonal contour 60

22 Orgnal surface Offset surface (a) A secton of the part surface and ts offset surface wth an ntersecton Orgnal contour Offset contour Surface normals LP LP LP (b) Cross sectonal contours of the orgnal and offset surfaces wth projected surface normals LP LP 1 3 (c) Intersecton s removed from the offset contour Fgure 4.7 The orgnal and offset surfaces and ther cross-sectonal contours before and after the ntersecton s removed from the offset contour 61

23 P P +1 P +2 T, 1 T, 2 P -1 P P -1 P P +1 P +2 P -1 P +1 P P +1 P +2 P P -1 P +2 P P -1 P +1 P P -1 (a) Jagged lnes (b) Collnear ponts (c) Inflecton ponts Fgure 4.8 Irregulartes and ther correctons 62