Simultaneous Precise Solutions to the Visibility Problem of Sculptured Models

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1 Simulaneous Precise Soluions o he Visibiliy Problem of Sculpured Models Joon-Kyung Seong 1, Gershon Elber 2, and Elaine Cohen 1 1 Universiy of Uah, Sal Lake Ciy, UT84112, USA, seong@cs.uah.edu, cohen@cs.uah.edu 2 Technion, Haifa 32000, Israel gershon@cs.echnion.ac.il Absrac. We presen an efficien and robus algorihm for compuing coninuous visibiliy for wo- or hree-dimensional shapes whose boundaries are NURBS curves or surfaces by lifing he problem ino a higher dimensional parameer space. This higher dimensional formulaion enables solving for he visible regions over all view direcions in he domain simulaneously, herefore providing a reliable and fas compuaion of he visibiliy char, a srucure which simulaneously encodes he visible par of he shape s boundary from every view in he domain. In his framework, visible pars of planar curves are compued by solving wo polynomial equaions in hree variables ( and r for curve parameers and for a view direcion). Since one of he wo equaions is an inequaliy consrain, his formulaion yields wo-manifold surfaces as a zero-se in a 3-D parameer space. Considering a projecion of he wo-manifolds ono he -plane, a curve s locaion is invisible if is corresponding parameer belongs o he projeced region. The problem of compuing hidden curve removal is hen reduced o ha of compuing he projeced region of he zero-se in he -domain. We recas he problem of compuing boundary curves of he projeced regions ino ha of solving hree polynomial consrains in hree variables, one of which is an inequaliy consrain. A opological srucure of he visibiliy char is analyzed in he same framework, which provides a reliable soluion o he hidden curve removal problem. Our approach has also been exended o he surface case where we have wo degrees of freedom for a view direcion and wo for he model parameer. The effeciveness of our approach is demonsraed wih several experimenal resuls. 1 Inroducion A major par of rendering is relaed o he hidden surface removal problem, i.e., display only hose surfaces which should be visible. The main conribuion of his work can be summarized as follows: The exac boundary beween visible and hidden pars of planar curves or surfaces is compued by solving a se of polynomial equaions in he parameer space wihou any piecewise linear approximaions. All possible view direcions in he domain are considered, simulaneously, by lifing he problem ino a higher dimensional space and solving a coninuous visibiliy problem. This higher dimensional framework provides a reliable soluion o he compuaion of he visibiliy char.

2 2 The algorihm is easy o implemen and robus by mapping he problem in hand o a zero-se solving ha explois he convex hull and subdivision properies of NURBS. Topological analysis of he visibiliy char makes i easier o compue he global srucure of he visibiliy char. Research ino solving he hidden surface removal problem is one of he earlies areas of aciviy in compuer graphics, compuer-aided design and manufacuring, and many differen algorihms have been developed [24, 1, 9, 18, 14, 19]. Usually hey are developed for polygonal daa, so curved surfaces have radiionally been preprocessed and approximaed as large collecions of polygons [22, 17]. In his paper, we presen an algorihm for eliminaing hidden curves or surfaces direcly from freeform models wihou any polygonal approximaions. Visibiliy compuaions of sculpured models have various applicaions no only in he area of rendering bu also in such areas as mold design, robo accessibiliy, inspecion planning and securiy. Given a view direcion, he hidden surface removal problem refers o deermining which surfaces are occluded from ha view direcion. Mos of he earlier algorihms in he lieraure are for polygonal daa and hidden line removal [8, 20, 24]. In heir work, because he displayed edges of he polygons are linear edges, he displayed curves, such as he silhouees of an objec viewed from a view direcion, are no smooh. Curves can be displayed more smoohly by increasing he number of polygons used for he approximaion, bu his resuls in memory and compuaional expense. Algorihms o resolve he hidden surface removal problem can be classified ino hose ha perform calculaions in objec-space, hose ha perform calculaions in imagespace, and hose ha work parly in boh, lis-prioriy [24]. Objec space echniques use geomeric ess on he objec descripions o deermine which objecs overlap and where. Iniiaed by Appel s edge-inersecion algorihm [1], he idea of quaniaive invisibiliy which deermines visible and invisible regions in advance was developed [9, 18, 11]. Image space approaches compue visibiliy only o he precision required o decide wha is visible a a paricular pixel, exemplified by [2]. Camull develops he deph-buffer or z-buffer image-precision algorihm which uses deph informaion [4]. Also, Weiler and Aheron [25] and Whied [26] develop ray racing algorihms which ransform he hidden surface removal problem ino ray-surface inersecion ess. Given a model composed of algebraic or parameric surfaces, i can be polygonized and hidden lines can be removed from he polygonized surfaces [22, 17]. However, he accuracy of he overall algorihm is limied by he accuracy of he polygonal approximaion. Furher, in boh mehods [22, 17], visibiliy is deermined for he endpoins of sraigh lines and hence, hey fail o deec invisibiliy occurring in he inerior region of a line when boh endpoins are visible. To remove hidden lines from curved surfaces wihou polygonal approximaion, Hornung e al. [11] exended he idea of quaniaive invisibiliy o bi-quadraic paches, and Newon s mehod was employed o solve for inersecions beween curves. Elber and Cohen [7] applied Hornung s echnique o nonuniform raional B-splines and exended i o rea rimmed surfaces. In paricular, Elber and Cohen [7] exrac he curves of ineres by considering boundary curves, silhouee curves, iso-parameric curves and curves along C 1 disconinuiy based on 2D curve-curve inersecions. Nishia e al. [21] used heir Bezier Clipping echnique for he hidden curve eliminaion. These mehods [11, 7, 21] are aimed a eliminaing

3 he hidden curves from line drawings of surfaces (no shaded drawings). Krishnan and Manocha [16] presened an algorihm for he eliminaion of hidden surfaces using a combinaion of symbolic echniques and resuls from numerical linear algebra. Elber e al. [6] presened an algorihm for compuing wo-dimensional visibiliy chars for planar curves. The visibiliy chars, however, are consruced by discreizing a coninuous se of view direcions [6]. Our algorihm is an exension of ha work ino he compuaion of coninuous visibiliy chars. Krishnan and Manocha [16] solves he hidden surface removal problem for a discree se of view direcions only. Our approach is unique in ha of solving he visibiliy problem for all view direcions in he domain, simulaneously. Summary of Our Approach We reduce he soluion o he visibiliy problem o he problem of finding he zeros of a se of polynomial equaions in he parameer space. For he curve case, visible curve locaions are compued by solving 2 polynomial equaions in 3 variables ( and r for curve parameers and for a view direcion). Since one of he wo equaions is an inequaliy consrain, his framework yields 2-manifold surfaces as a 0-se in a 3-D parameer space. A curve s locaion is invisible if is corresponding parameer belongs o he projeced region of he wo-manifolds ono he -plane. The problem for compuing hidden curve removal is hen reduced o ha of compuing he projeced region of he zero-se in he -domain. We recas his problem of compuing boundary curves of he projeced regions ino ha of solving hree polynomial consrains in hree variables, one of which is an inequaliy consrain. The presened approach for he hidden curve removal can be exended o he surface case where we have 2 degrees of freedom (dof) for a view direcion and wo for surface parameers. Similarly o he curve case, visible surface s locaions are compued by solving 3 polynomial equaions in 6 variables, one of which is an inequaliy consrain. Assuming a freeform surface S(u, v) is used o parameerize for all possible view direcions V(, ϕ), he 0-se of he 3 equaions is consruced as four-manifolds in a 6-dimensional parameer space, and is projecion ino he uvϕ-domain prescribes he hidden pars of he surface S(u, v). A surface s locaion, S(u 0, v 0 ), is invisible from viewing direcion V( 0, ϕ 0 ) if is corresponding parameer, (u 0, v 0, 0, ϕ 0 ), belongs o he projeced region of he 0-se. The boundary of he projeced region is compued by inroducing one more equaion o he se of 3 equaions, herefore generaing 3- manifolds in he 4-dimensional parameer space. The visibiliy chars for he surface case are hen consruced using he 3-manifolds in he uvϕ-parameer domain. A paricular visibiliy query, which specifies and ϕ for a view direcion, is resolved by exracing one-manifold curves in he surface s uv-parameer domain. Those curves in he uv-domain rim away hidden surface regions and hus only he visible surfaces are rendered from ha view direcion. The opological srucure of he visibiliy char is furher analyzed in he same framework, which provides a reliable soluion o he compuaion of he visibiliy char. The number of conneced curve segmens ha delineae he hidden pars from he visible ones changes a criical poins where he global opology changes in he visibiliy char. Aspec graphs [3] are used in compuer vision o opologically analize he visibiliy problem. In his paper, algebraic consrains for hese criical poins are derived 3

4 4 as a se of 3 polynomial equaions in 3 variables for he curve case and precompued for he global analysis of he visibiliy char. Based on his opological informaion, i becomes easier o analyze he global arrangemen of he visibiliy char, avoiding he compuaion of complex combinaorial curve-curve inersecions. The res of his paper is organized as follows. In Secion 2, he hidden curves removal algorihm is discussed for planar curves. Secion 3 presens is exension o he eliminaion of hidden surfaces. Some examples are presened in Secion 4 and finally, in Secion 5, his paper is concluded. 2 Coninuous Visibiliy for Planar Curves Le V() be a one-parameer family of viewing direcions. The visibiliy for a planar curve C() is hen solved by lifing he problem ino a higher dimension, where he answer is represened using simulaneous soluion of wo polynomial equaions. Lemma 1. A planar curve poin C() is visible if and only if i saisfies he following wo polynomial equaions for all r, F(, r, ) = V() (C() C(r)) = 0, G 1 (, r, ) = V(), C() C(r) 0. Proof. Two equaions, F(, r, ) = 0 and G 1 (, r, ) 0, are saisfied only if C() is closer o he view source han C(r) while wo curve poins are on he same line o he view direcion V(). Therefore, here may be no oher curve poin C(r) ha blocks C() from V() if C() saisfies he above wo equaions for all r, which implies ha C() is visible from he viewing direcion. Figure 1 demonsraes Lemma 1. Given a viewing direcion V, wo curve poins C() and C(r) in Figure 1(a) saisfy he firs equaion F(, r, ) = 0. This means ha he vecor from C() o C(r) is parallel o he view direcion. The second condiion is saisfied only if C() is closer o he view source han C(r). Thus, he curve poin C() is visible for he view direcion V, while C(r) is no. For he curve poin C() o be visible, G 1 (, r, ) 0 should be saisfied for all r. This implies ha if here is any value of r such ha G 1 (, r, ) > 0, hen he curve poin C() is no visible. In Figure 1(b), C() is poenially visible from V if one considers he curve poin C(s) as is corresponding pair. The poin C(), however, is no visible since here exiss anoher curve poin C(r) ha fails a he second consrain of Lemma 1. Elber e al. [6] solves wo polynomial equaions in wo variables for a discree se of view direcions. If V is one such direcion, C () V = 0, (C() C(r)) V = 0. Soluion poins of hese wo equaions prescribe he visible porion of C for each V, providing only a discree soluion. In his paper, we solve he problem of compuing visible regions for all possible view direcions V() in he domain, simulaneously, providing a coninuous soluion o he visibiliy problem. For he clariy of explanaion, we consider invisible curve segmens insead.

5 5 C(r) C() C(s) C() C(r) V (a) V (b) Fig. 1. (a) Given a viewing direcion V, a planar curve poin C() is visible while C(r) is no. (b) A poin C() has anoher curve poin C(r) which makes i invisible from he view direcion V. Corollary 1. A planar curve poin C() is invisible if and only if here exiss anoher curve poin C(r) such ha he following wo polynomial equaions hold F(, r, ) = V() (C() C(r)) = 0, (1) G 2 (, r, ) = V(), C() C(r) > 0. (2) Now, any r for which G 2 (, r, ) > 0 holds renders curve poin C() invisible. As his second equaion, G 2 (, r, ) > 0, is an inequaliy consrain, he soluion of boh consrains is a 2-manifold in 3-D parameer space. Furhermore, he soluion is symmeric wih respec o he = r plane so, we can consider one more inequaliy consrain, > r, o speed up he equaion-solving process by purging half he soluion domain. Denoe by M he soluion of Equaions (1) and (2) ha deermines he hidden pars of he planar curve C(). The projecion of M ino he -plane characerizes he regions where he curve is no visible. Tha is, if a parameer (, ) falls ino he projeced region of M, hen he corresponding curve poin C() is no visible for he viewing direcion V(). Is complemen, he uncovered region (under his projecion) in he plane, deermines all he visible secions of C along coninuously varying view direcions. Figure 2 shows an example of such a visibiliy char. Gray regions in Figure 2(a) represens he 2D projecion of M for he planar curve C(). Given a viewing direcion V, one can exrac a se of visible curve segmens from he uncovered (whie) regions (see Figure 2(b)). As one can see from Figure 2(b), visibiliy queries are resolved by exracing corresponding whie regions from he visibiliy char. Thus, solving he visibiliy problem for planar curves can be reduced o ha of finding boundary curves of he projeced regions of M in he parameer space. Since he projecion is performed o he -plane, he boundary of he projeced region under his projecion occurs eiher a he boundaries of he zero-se M or a is local exrema. Since M is coninuous and closed, i has no boundary and hence, he visibiliy problem reduces o finding r-exrema of he zero-se M which are he r-direcional silhouees of M. Definiion 1. Given a one-parameer family of viewing direcions V(), a C 1 -coninuous planar curve C, and he soluion manifold M of Equaions (1) and (2) for C;

6 6 C( 1 ) C() (a) V C( 3 ) C( 2 ) C( 4 ) (b) Fig. 2. (a) Given a planar curve C(), he gray region in he -plane represens hidden curves of C. (b) Visible curve segmens can be exraced from he uncovered (whie) regions. 1. The r-direcional silhouee curves, S r, comprise he se of poins on M whose r-direcional parial derivaive vanishes (bold lines in Figure 3(a) shows he projecion of S r in he -plane). 2. Denoe by S r I Sr he se of poins ha falls in he inerior of he projecion of M, among he se of r-direcional silhouees S r (see doed line segmens in Figure 3(b)). Then, he sough boundary of M, M, ha delineaes he visible segmens of C from all possible views, can be compued using he wo ses S r and S r I as: M = S r S r I. Figure 3(c) presens M in bold lines and M as a shaded region. The r-direcional silhouee curves, S r, of M can be compued by finding he simulaneous soluion of Equaions (1), (2) and (3), where F (, r, ) = 0. (3) r Having wo equaliy equaions in hree variables, soluions of he hree equaions are curves in he r-parameer space. As F and G 2 are piecewise raional funcions, he soluion can be consruced by exploiing he convex hull and subdivision properies of NURBS, yielding a highly robus divide-and-conquer compuaion [5]. The solver [5] recursively subdivides raional funcions along all parameer direcions unil a given maximum deph of subdivision or some oher erminaion crieria is reached. A he end of he subdivision sep, a discree se of poins are numerically improved ino a highly precise soluions using a mulivariae Newon-Raphson ieraive sage. Finally, hese discree poins are conneced ino a se of piecewise linear curves in he parameer space (See [23] for more deails). An enire curve segmen or any porion of he curve segmen in S r can fall inside he projeced region of M (see Figure 3(a)). We need o rim away S r I from Sr since hey correspond o inerior curve segmens. An efficien and robus algorihm for purging S r I away is presened in his secion and is based on he analysis of a opological change in he visibiliy chars. Given a coninuous one parameer family of view direcions

7 7 C() a b c d b d a c (a) (b) (c) Fig. 3. (a) r-direcion silhouee curves S r projeced ino he -plane. (b) Doed line segmens represen S r I and (c) M = S r S r I is shown in bold. Criical poins are compued using a opological analysis and shown in (b). Their corresponding curve poins and view direcions are also shown in (a). V(), a opological change (i.e. a change in he number of conneced componens) can occur eiher globally or locally. Global opological changes occur where he viewing direcion is parallel o a bi-angen line segmen of C connecing wo (or more) poins. Topological changes occur locally where he viewing direcion is parallel o he angen direcion of C, a an inflecion poin. The bi-angen line segmen of C ouches angenially he curve a wo or more differen poins. Bi-angen direcions can be compued by simulaneously solving he following hree equaions, in hree variables: F(, r, ) = 0, F (, r, ) = V(), N() = 0, (4) F (, r, ) = V(), N(r) = 0. r (5) Equaions (4) and (5) consrain he viewing direcion V() o ouch C angenially a wo differen poins C() and C(r), respecively. The bi-angen direcion of C iself can be compued using wo polynomial equaions in wo variables. In his conex, however, he viewing direcion V(), which is parallel o he bi-angen direcion, mus be compued for furher processing. Inflecion poins of a planar curve occur a poins where he sign of he curvaure, a raional form if C is raional, changes. Soluion poins of = r clearly saisfy all he above equaions and mus be purged away. Le T be a se of poins (, r, ) in he r-parameer space ha correspond o eiher bi-angens or inflecion poins. We consrain poin (, r, ) T o be ouside he projeced region. The black bold dos in Figure 3(b) represens hese criical poins, a which he opological srucure of he visibiliy char changes. Thus, he r-direcional silhouee curves, S r, are rimmed a such criical poins (, r, ) T. The curve segmens SI r (Doed line segmens in Figure 3(b)) can be deermined using a simple visibiliy check of a single poin, esing wheher he segmen falls inside he projeced region of M or no. Figure 3(c) shows he visible boundaries M of he projeced regions as a se of piecewise curves.

8 8 3 Coninuous Visibiliy for Freeform Surfaces The presened algorihm for compuing visibiliy of planar curves can be exended for compuing he hidden surfaces. Given wo-parameers family of viewing direcions V(, ϕ), he visibiliy problem for he surface case is solved in a six-dimensional parameer space, (u, v, s,,, ϕ). Much like he curve case, his higher dimensional formulaion simulaneously considers all view direcions in he domain, and provides a reliable soluion o a paricular visibiliy query. We firs presen a se of condiions for deermining wheher a surface locaion S(u, v) is visible or no. Lemma 2. A surface poin S(u, v) is invisible if and only if here exiss anoher surface poin S(s, ) such ha F(u, v, s,,, ϕ) = S(u, v) S(s, ), V (, ϕ) = 0, (6) G(u, v, s,,, ϕ) = S(u, v) S(s, ), V (, ϕ) = 0, (7) ϕ H(u, v, s,,, ϕ) = S(u, v) S(s, ), V(, ϕ) > 0, (8) where V(, ϕ) is a polynomial approximaion o he sphere ha spans all possible viewing direcions. Proof. By Equaions (6) and (7), he wo surface poins S(u, v) and S(s, ) are on he same line wih he same direcion o he view direcion V(, ϕ). By saisfying Equaion (8), S(s, ) is closer o he view source han S(u, v), which makes S(u, v) invisible for ha view direcion. Since Equaion (8) is an inequaliy consrain, he simulaneous zeros of he hree Equaions (6) (8) are 4-manifolds in a six-dimensional parameer space. Le M be he 4-manifold zero-se of Equaions (6) (8). Then, similarly o he curve case, he projecion of he zero-se ino he uvϕ-domain prescribes he hidden pars of he surface S(u, v). If (u, v,, ϕ) falls ino he inerior of he projeced region of M, hen he corresponding surface locaion, S(u, v), is no visible from viewing direcion V(, ϕ). In oher words, he uncovered region (under his projecion), in he uvϕ-domain, deermines all he visible secions of S(u, v) along coninuously varying viewing direcions. In Figure 4(a), a shaded region depics he projecion of he zero-se, M, ino he uvϕ-parameer space. A parameer (u 1, v 1, 1, ϕ 1 ) falls ino he projeced region in Figure 4(a) and hus, is corresponding surface poin S(u 1, v 1 ) is invisible for viewing direcion V( 1, ϕ 1 ) (see Figure 4(b)). On he oher hand, poin S(u 2, v 2 ) is visible since parameer (u 2, v 2, 1, ϕ 1 ) is locaed ouside he projeced region. Projeced ino he uvϕ four-dimensional space, he boundaries of he projecion of he zero-se M can be deermined as he s-direcional silhouees of M, by finding all he simulaneous zeros of Equaions (6) (9), where I(u, v, s,,, ϕ) = V(, ϕ), N(s, ) = 0, (9) and N(s, ) is a normal vecor field of S(s, ). The common zero-se of Equaions (6) (9) is now a 3-manifold in a six-dimensional space, which is he boundary of he

9 9 ϕ (u 2, v 2, 1, ϕ 1) (u 1, v 1, 1, ϕ 1) S(u 2, v 2) u (a) v V( 1, ϕ 1 ) S(u 1, v 1) (b) Fig. 4. (a) A shaded volume depics a projecion of he soluion M ino he uvϕ-parameer space. (b) S(u 1, v 1 ) is invisible for a viewing direcion V( 1, ϕ 1 ) since (u 1, v 1, 1, ϕ 1 ) falls ino he projeced volume. Compare i wih S(u 2, v 2). projeced volume of M. Given a paricular viewing query V( 0, ϕ 0 ), wo of he soluion space s remaining degrees-of-freedom are fixed and we can exrac 1-manifold soluion curves from he projeced region of M. These curves in he parameer space correspond o curves ha delineae he hidden surfaces from he visible ones. I is quie difficul o eiher visualize or conour 3-manifolds in a six-dimensional space. By fixing a paricular viewing direcion, 1-manifold curves in a six-dimensional space resul. So i is possible o use he algorihm presened by Seong e al [23] o exrac all he visible pars of S(u, v). Figure 5(a) shows a surface S wih a viewing direcion V. The boundary curves of visible secions in he uv-domain are compued using our approach (see Figure 5(b)). In Figure 5(c), gray-colored rimming surfaces represen hidden surfaces of he original surface and he bold ones are visible secions for he viewing direcion. Shaded regions in he parameer domain (Figure 5(b)) correspond o he hidden surfaces in Euclidean space (Figure 5(c)). 4 Experimenal Resuls We now presen examples of compuing a visibiliy char in a coninuous domain for boh planar curves and 3D surfaces. For all he figures, he gray-colored region represens he projecion of he zero-se of he corresponding se of polynomial equaions in he parameer space and characerizes hidden pars of planar curves or surfaces. Bold lines in curves or surfaces represens visible pars from he given view direcion. Figure 6 shows a planar curve and is visibiliy chars in a coninuous domain. Bold lines in Figure 6(a) represen a se of r-direcional silhouees of he zero-se manifold. The boundary curves of he projeced region are compued based on a opological analysis of he visibiliy chars and shown in Figure 6(b). In Figures 7, (a) and (c) show wo planar curves and (b) and (d) are he visibiliy chars for all viewing direcions. For a paricular viewing direcion, V, a se of visible curve segmens are shown in bold lines in Figures 7(a) and (c). Figures 7(b) and (d)

10 10 v V S (a) u (b) (c) Fig. 5. (a) A surface S wih a viewing direcion V. (b) A se of rimming curves in he uvparameer domain. (c) Visible pars of S are shown for he given view direcion. C() (a) (b) Fig. 6. (a) Given a planar curve C(), he projeced region of M and projeced r-direcional silhouee curves S r are shown in gray and bold lines, respecively. (b) A se of visible segmens, Svr, is shown in bold lines. show he corresponding parameer domain in hick lines. The compuaion ime for generaing he visibiliy chars over all possible view direcions for he curve case vary according o he curve s complexiy, aking from 1.3 o 6 seconds on a Penium IV 2GHz deskop machine. Figure 8(a) shows an envelope surface generaed by sweeping a scalable ellipsoid along a space rajecory. A se of rimming curves is shown in Figure 8(b), which is he resul of solving Equaions (6) (9) afer fixing a viewing direcion. Each rimmed surface sub-region is esed for visibiliy using a simple ray-surface inersecion mehod. Figure 8(c) draws visible surface paches only. The original surfaces in Figure 9(a) and (d) are bi-quaric NURBS having abou 250 conrol poins and shown wih differen view direcions. Figure 9(b) and (e) show a se of rimming curves which are boundaries beween visible pars and hidden surfaces in he uv-parameer domain. Figure 9(c) and (f) show visible surface paches only along a specified viewing direcion. On a 2GHz Penium IV machine, compuing he rimming curves in he uv-domain for Figures 8 10 ook abou 13 o 45 seconds. The eapo in Figure 10 is represened by four bi-cubic NURBSs surfaces which are open (Figure 10(a)). Each of he four surface paches can be hidden by any of he oher ones according o he viewing direcion. In Figure 10(a), par of he body is blocked by boh a handle and a cap for he given viewing direcion (a figure is generaed along he viewing direcion). Furhermore, i blocks iself and makes shadow regions. Fig-

11 11 C() V (a) (b) C() V (c) (d) Fig. 7. (a), (c) A planar curve C() and he visible curve segmens ha are shown in bold lines. (b), (d) A coninuous visibiliy chars compued by solving Equaions (1) (3). ure 10(b) shows he rimming curves in he parameer domain of he body. They are comprised of hree se of curves. Trimming curves generaed due o a cap are represened by gray-colored lines in Figure 10(b) and four open curve segmens locaed in he middle par of he domain are generaed by he handle. Since he surface pach of he handle is no closed, he rimming curves are also open. Thus, he geomeric inersecion curve beween he handle and he body is needed for a proper rimming. All he oher rimming curves in Figure 10(b) sems from he body iself. Figure 10(c) (e) show a se of rimming curves for he handle, spou and he cap, respecively. Finally, Figure 10(f) draws all he visible pars. 5 Conclusion and Fuure Work We have presened a robus and efficien scheme for compuing hidden curve/surface removal, in he coninuous domain. The approach is based on he derivaion of a se of algebraic consrains ha deermine he visibiliy of curve s or surface s locaions. All view direcions in he domain are considered simulaneously, and he algorihm provides a coninuous char for he visibiliy from all possible views. By simulaneously solving 2 polynomial equaions for a curve case and 3 polynomial equaions for a surface case, in he parameer space, he presened approach can deec all he hidden pars of he sculpured model for coninuously varying view direcions. The zero-se of he polynomial equaions prescribes he hidden pars of he model and we consruc a visibiliy char by projecing he zero-se ino an appropriae parameer space. Furhermore,

12 12 S(u, v) v (a) (b) u (c) Fig. 8. (a) An envelope surface generaed by sweeping a scalable ellipsoid along a space rajecory is shown. (b) A se of rimming curves in he uv-parameer domain is presened in bold lines. (c) Visible pars of he surface are shown for he given viewing direcion. he opological srucure of he visibiliy char is analyzed in he same framework, providing a reliable soluion o he compuaion of he visibiliy char. The presened approach can be applied o rimmed models as well. The original rimming curves need o be considered in he compuaion of he boundary curves beween visible and invisible pars in he case of rimmed models. Visibiliy compuaions for perspecive views are desirable exensions o he mehod presened. To his end, we need o deal wih even higher-dimensional soluion spaces. Acknowledgmens All he algorihms and figures presened in his paper were implemened and generaed using he IRIT solid modeling sysem [12] developed a he Technion, Israel. This work was suppored in par by NSF IIS All opinions, findings, conclusions or recommendaions expressed in his documen are hose of he auhor and do no necessarily reflec he views of he sponsoring agencies. References 1. A. Appel. The Noion of quaniaive Invisibiliy and he Machine Rendering of Solids. Proceedings ACM Naional Conference, W.J. Bouknigh. A procedure for Generaion of Three-Dimensional Half-oned Compuer Graphics Represenaions. CACM, Vol. 13, No. 9, K. Bowyer and C. Dyer. Aspec graphs: An inroducion and survey of recen resuls. Proc. SPIE Conf. on Close Range Phoogrammery Mees Machine Vision, 1395, pp , E. Camull. A Subdivision Algorihm for Compuer Display of Curved Surfaces. Ph.D Thesis, Repor UTEC-CSc , Compuer Science Deparmen, Universiy of Uah, Sal Lake Ciy, UT, G. Elber and M. S. Kim. Geomeric Consrain Solver Using Mulivariae Raional Spline Funcions. Proc. of Inernaional Conference on Shape Modeling and Applicaions, pp , MIT, USA, June 15-17, 2005.

13 13 v V S (a) (b) u (c) V v S (d) (e) u (f) Fig. 9. (a), (d) A surface S is shown wih a view direcion. (b), (e) A se of rimming curves in he uv-parameer domain is presened in bold lines. (c), (f) Visible pars of he surface are shown for he given viewing direcion. 6. G. Elber, R. Sayegh, G. Bareque and R. Marin. Two-Dimensional Visibiliy Chars for Coninuous Curves. Proc. of ACM Symposium on Solid Modeling and Applicaions, Ann Arbor, MI, June 4-8, G. Elber and E. Cohen. Hidden curve removal for free form surfaces. Compuer Graphics, Vol. 24, No. 4, pp , J. Foley, A. Van Dam, J. Hughes, and S. Feiner. Compuer Graphics: Principles and Pracice. Addison Wesley, Reading, Mass., R. Galimberi and U. Monanari. An algorihm for Hidden Line Eliminaion. CACM, Vol. 12, No. 4, pp , C. Hornung. A Mehod for Solving he Visibiliy Problem. CG&A, pp , July C. Hornung, W. Lellek, P. Pehwald, and W. Srasser. An Area-Oriened Analyical Visibiliy Mehod for Displaying Parameerically Defined Tensor-Produc Surfaces. Compuer Aided Geomeric Design, Vol. 2, pp , IRIT 9.0 User s Manual, Ocober 2000, Technion. hp:// iri. 13. T. Ju, F. Losasso, S. Schaefer, and J. Warren. Dual Conouring of Hermie Daa. In Proceedings of SIGGRAPH 2002, pp , T. Kamada and S. Kawai. An Enhanced Treamen of Hidden Lines. ACM Transacion on Graphics, Vol. 6, No. 4, pp , M. S. Kim and G. Elber. Problem Reducion o Parameer Space. The Mahemaics of Surface IX (Proc. of he Ninh IMA Conference), London, pp 82 98, S. Krishnan and D. Manocha. Global Visibiliy and Hidden Surface Algorihms for Free Form Surfaces. Technical Repor: TR94-063, Universiy of Norh Carolina, 1994

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