Area and Perimeter Derivatives of a Union of Disks

Transcription

1 Ara and rimtr Drivativs of a Union of Disks HoLun Chng and Hrbrt Edlsbrunnr Dpartmnt of Computr Scinc, ational Univrsity of Singapor, Singapor. Dpartmnt of Computr Scinc, Duk Univrsity, Durham, and Raindrop Gomagic, Rsarch Triangl ark, orth Carolina. Abstract. W giv analytic inclusionxclusion formulas for th ara and primtr drivativs of a union of finitly many disks in th plan. Kywords. Disks, Voronoi diagram, alpha complx, primtr, ara, drivativ. 1 Introduction A finit collction of disks covrs a portion of th plan, namly thir union. Its si can b xprssd by th ara or th primtr, which is th lngth of th boundary. W ar intrstd in how th two masurmnts chang as th disks vary. Spcifically, w considr smooth variations of th cntrs and th radii and study th drivativs of th masurmnts. W hav two applications that motivat this study. On is topology optimiation, which is an ara in mchanical nginring [1, 2]. Rcntly, w bgan to work towards dvloping a computational rprsntation of skin curvs and surfacs [7] that could b usd as changing shaps within a topology optimiing dsign cycl. art of this work is th computation of drivativs. Th rsults in this papr solv a subproblm of ths computations in th twodimnsional cas. Th othr motivating problm is th simulation of molcular motion in molcul dynamics []. Th stting is in thrdimnsional spac, and th goal is to simulat th natural motion of biomolculs with th computr. Th standard approach uss a forc fild and prdicts changs in tiny stps basd on wton s scond law of motion. Th surfac ara and its drivativ ar important for incorporating hydrophobic ffcts into th calculation [4]. Th main rsults of this papr ar inclusionxclusion formulas for th ara and th primtr drivativs of a finit st of disks. As it turns out, th ara drivativ is simplr to comput but th primtr drivativ is mor intrsting. Th major diffrnc btwn th two is that a rotational motion of on disk about anothr may hav a nonro contribution to th primtr drivativ whil it has no contribution to th ara drivativ. Outlin. Sction 2 introducs our approach to computing drivativs and stats th rsults. Sction provs th rsult on th drivativ of th primtr. Sction 4 provs th rsult on th drivativ of th ara. Sction 5 concluds th papr. Rsarch by both authors was partially supportd by SF undr grant DMS8745 and CCR00801.

2 ` = 2 Approach and Rsults In this sction, w xplain how w approach th problm of computing th drivativs of th ara and th primtr of a union of finitly many disks in th plan. Drivativs. W nd som notation and trminology from vctor calculus to talk about drivativs. W rfr to th booklt by Spivak [10] for an introduction to that topic. For is a linar map a diffrntiabl map, th drivativ at a point. Th gomtric intrprtation is as follows. Th graph of is an dimnsional linar subspac of. Th translation that movs th origin to th point on th graph of movs th subspac to th tangnt hyprplan at that point. Bing linar, can b writtn as th scalar product of th variabl vctor!" with a fixd vctor # known as th gradint of at $ :!%')(*#!+. Th drivativ maps ach, to, or quivalntly to th gradint # of at. In this papr, w call points in stats and us thm to rprsnt finit 7 sts of disks in. For.0/1, th stat rprsnts th st of disks 245 8:;, for >=?'= 1ACB, whr D E F 8E GIHKJ is th cntr and E ELM Q is th radius of 2. Th primtr and ara of th union of disks ar maps O8 E8R S. Thir drivativs at a stat " E8R ar linar maps T EUR V, and th goal of this papr is to giv a complt dscription of ths drivativs. Voronoi dcomposition. A basic tool in our study of drivativs is th Voronoi diagram, which dcomposs th union from 2Y as ZQ WQ[]\^W, U\. Th of disks into convx clls. To dscrib it, w dfin th powr distanc of a point WX disk thus contains all points with nonpositiv powr distanc, and th bounding circl consists of all points with ro powr distanc from 2_. Th bisctor of two disks is th lin of points with qual powr distanc to both. Givn a finit collction of disks, th (wightd) Voronoi cll of 2_ in this collction is th st of points W for which 24 minimis th powr distanc, Edc ba:wa Z WQ Zf WQ^Fghij Each Voronoi cll is th intrsction of finitly many closd halfspacs and thus a convx polygon. Th clls covr th ntir plan and hav pairwis disjoint intriors. Th (wightd) Voronoi diagram consists of all Voronoi clls, thir dgs and thir vrtics. If w rstrict th diagram to within th union of disks, w gt a dcomposition into convx clls. Figur 1 shows such a dcomposition ovrlayd with th sam aftr a small motion of th four disks. For th purpos of this papr, w may assum th disks ar in gnral position, which implis that ach Voronoi vrtx blongs to xactly thr Voronoi clls. Th Dlaunay triangulation is dual to th Voronoi diagram. It is obtaind by taking th disk cntrs as vrtics and drawing an dg and triangl btwn two and thr vrtics whos corrsponding Voronoi clls hav a nonmpty common intrsction. Th dual complx k of th disks is dfind by th sam rul, xcpt that th nonmpty common intrsctions ar dmandd of th Voronoi clls clippd to within thir corrsponding disks. For an xampl s Figur 1, which shows th dual complx

3 = Fig. 1: Two snapshots of a dcomposd union of four disks to th lft and th dual complx to th right. of four disks. Th notion of nighborhood is formalid by dfining th link of a vrtx "k as th st of vrtics and dg connctd to by dgs and triangls, ba $ c $ "k ij Similarly, th link of an dg is th st of vrtics connctd to th dg by triangls, ba c >k i. Sinc k is mbddd in th plan, an dg blongs to at most two triangls which implis that th link contains at most two vrtics. Masuring. W us fractions to xprss th si of various gomtric ntitis in th Voronoi dcomposition. For xampl, is th fraction of 2 containd in its Voronoi cll and is th fraction of th circl bounding 2 that blongs to th boundary of th union. Th ara and primtr of th union ar thrfor Z T and $Z %Tj W rfr to th intrsction points btwn circls as cornrs. Th cornr to th lft of th dirctd lin from to is W and th on to th right is W. ot that W W. W us Ca IB$i to indicat whthr or not W xists and lis on th boundary of th union. Th total numbr of cornrs on th boundary is thrfor. Finally, w dfin Q as th fraction of th chord W W that blongs to th corrsponding Voronoi dg. Givn th dual complx k of th disks, it is fairly straightforward to comput th T, T, Q, and T. For xampl, T B iff is an dg in k and if is a triangl in k thn lis to th right of th dirctd lin from to. W sktch inclusionxclusion formulas for th rmaining quantitis. roofs can b found in []. Dfin 2 a:w c Z WQ ZT W = i, which is th portion of 2Y of 2 s sid of th bisctor. Dfin and 2. Similarly, lt

4 Z = and b th lngths of th circl arcs in th boundaris of 2 QFMB T MB Z $Z : and 2 2. Thn whr in both quations th first sum rangs ovr all vrtics and th scond rangs ovr all in k. Finally, lt _ W %Wf b th chord dfind by 2 and 2 and dfin a:w" c f W Z WQ Zf WQ = i, which is th portion of on 2 s sid of th bisctors. Dfin and. Thn YMB whr th sum rangs ar ovr all $ in k. Th analytic formulas still rquird to comput th various aras and lngths can b found in [8], which also xplains how th inclusionxclusion formulas ar implmntd in th Alpha Shap softwar. Motion. Whn w talk about a motion, w allow all /1 dscribing paramtrs to vary: ach cntr can mov in and ach radius can grow or shrink. Whn this happns, th union changs and so dos th Voronoi diagram, as shown in Figur 1. In our approach to studying drivativs, w considr individual disks and look at how thir Voronoi clls chang. In othr words, w kp a disk 24 fixd and study how th motion affcts th portion of 2Y that forms th cll in th clippd Voronoi diagram. This ida is illustratd in Figur 2. This approach suggsts w undrstand th ntir chang as an accumulation Fig. 2: Two snapshots of ach disk clippd to within its Voronoi cll. Th clippd disks ar th sam as in Figur 1, xcpt that thy ar suprimposd with fixd cntr. of changs that happn to individual clippd disks, and w undrstand th chang of an individual clippd disk as th accumulation of changs causd by nighbors in th Voronoi diagram. A cntral stp in proving our rsults will thrfor b th dtaild analysis of th drivativ in th intraction of two disks.

5 ) ) Thorms. Th first rsult of this papr is a complt dscription of th drivativ of th primtr of a union of disks. LtI \ \ b th distanc btwn two cntrs. W writ for th unit vctor btwn th sam cntrs and for Q rotatd through an angl of 0 dgrs. ot that FM and $. ERIMETER DERIVATIVE THEOREM. Th drivativ of th primtr of a union of 1 disks with stat EUR is! E E (!%+, whr T E E B k $Z : ' % : FB ^ $ Y : % ^ is not an dg in thn T f :! j f. W can thrfor limit th sums in th is a full circl thn th primtr and its drivativ vanish. This is clar also from th formula bcaus TF and "#$ "% for all h. Th scond rsult is a complt dscription of th drivativ of th ara of a disk union. If rimtr Drivativ Thorm to all in th link of. If th link of in k AREA DERIVATIVE THEOREM. Th drivativ of th ara of a union of 1 disks with stat is! (('!+, whr ' E ' E *) 'E E' $Z Y $ j W can again limit th sum to all in th link of. If th link of in k thn th ara drivativ vanishs. Indd, ' E E bcaus T bcaus of th Minkowski thorm for convx polygons. is a full circl and Q rimtr Drivativ In this sction, w prov th rimtr Drivativ Thorm statd in Sction 2. W bgin by introducing som notation, continu by analying th cass of two and of 1 disks, and conclud by invstigating whn th drivativ is not continuous.

6 otation. For th 7 cas of two disks, w us th notation shown in Figur. Th two disks ar 2, 8 and W assum that th two bounding circls intrsct in two cornrs, W and W. Lt b half th distanc btwn th two cornrs. Thn is th distanc btwn and th bisctor, and similarly, is th distanc btwn and th bisctor. If and li on diffrnt sids of th bisctor thnx is th distanc btwn th cntrs. W hav and thrfor ^F B (1)? for IB. If th two cntrs li on th sam sid of th bisctor, thn " is th distanc btwn th cntrs. W hav and again Equation (1) for and. Lt b th angl W at, and similarly dfin θ 0 + θ 1 0 r x 0 1 θ r 0 01 θ1 ζ ζ 0 1 x r ψ 1 u v Fig. : Two disks boundd by intrscting circls and th various lngths and angls thy dfin. W W. Thn (2)? for :B, and w not that is th angl formd at th two cornrs. Th contributions of ach disk to th primtr and th ara of 2 2 ar for? F Z>; %8 () Z> (4) IB. Th primtr of th union is C, and th ara is. Motion. W study th drivativ of undr motion by fixing and moving th othr cntr along a smooth curv, with d. At th vlocity vctor of th motion is. Lt X and b th unit vctor obtaind by rotating through an angl of 0 dgrs. W dcompos th motion into a slop prsrving and a distanc prsrving componnt, M( ^ + ( ^ f+. W comput th two partial

7 B drivativs with rspct to th distanc and an angular motion. W us Equations (1), to comput th drivativ of with rspct to th cntr distanc, (2), and () for? By symmtry, B. Th drivativ of is th sum of th two drivativs, and thrfor B j B B j (5) To prsrv distanc w rotat around and lt dnot th angl dfind by th vctor. During th rotation th primtr dos of cours not chang. Th rason is that w loos or gain th sam amount of lngth at th lading cornr, W, as w gain or loos at th trailing cornr, W. Sinc w hav to dal with situations whr on cornr is xposd and th othr is covrd by othr disks, w ar intrstd in th drivativ of. W hav a gain on th boundary of th contribution nar W, which w dnot by 2 minus a loss on th boundary of 2, namly j () As mntiond abov, th changs at th two cornrs cancl ach othr, or quivalntly,. Growth. W grow or? shrink th disk 2 by changing its radius,. Using Equations (1), (2), and () for as bfor, w gt Z Z bcaus and. Th computation of th drivativ of is mor straightforward bcaus? and both rmain constant as changs. Using Equations (1), (2), and () for B, w gt B I j

8 ? ot that is qual to. Th drivativ of th primtr, which is th sum of th two drivativs, is thrfor Z B j (7) Th first trm on th right in Equation (7) is th rat of growth if w scal th ntir disk union. Th scond trm accounts for th angl at which th two circls intrsct. It is not difficult to show that this trm is qual to, which is gomtrically th obvious dpndnc of th drivativ on th angl btwn th two circls, as can b sn in Figur. Assmbly of rlations. Lt b th primtr of th union of disks 24 =?[=, for 1L B. By linarity, w can dcompos th drivativ along a curv with vlocity vctor!5 E8R? into componnts. Th th triplt of coordinats dscribs th chang for 2. Th first two of th thr coordinats giv th vlocity vctor of th cntr. For ach othr disk 2, w dcompos that vctor into a slop and a distanc prsrving componnt, M( %+ ( +. Th drivativ of th primtr along th slop prsrving dirction is givn by Equation (5). Th lngth of th corrsponding vctor in th thorm is this drivativ tims th fractional numbr of boundary cornrs dfind by 2[ and 2, which is " ". Th drivativ along th distanc prsrving dirction is givn by Equation (). Th lngth of th corrsponding vctor in th thorm is that drivativ tims, sinc w gain primtr at th cornr W and loos at W (or vic vrsa, if ( 8 $ + ). Th drivativ with rspct to th radius is givn in Equation (7). Th first trm of that quation is th angl of 24 s contribution to th primtr, which in th cas of 1 disks is Z T. Th scond trm accounts for th angls at th two cornrs. It contributs to th drivativ only for cornrs that blong to th boundary of th union. W thus multiply th corrsponding trm in th thorm by th fractional numbr of boundary cornrs. This complts th proof of th rimtr Drivativ Thorm. 4 Ara Drivativ In this sction, w prov th Ara Drivativ Thorm statd in Sction 2. W us th sam notation as in Sction, which is illustratd in Figur. Motion. As bfor w 7 considr two disks 2 and 2 *, w kp fixd, and w mov along a curv with vlocity vctor at. Th unit vctors and ar dfind as bfor, and th motion is again dcomposd into a slop and a distanc prsrving componnt, ( ^ + ( ^ f+. Th distanc prsrving componnt dos not chang th ara and has ro contribution to th ara drivativ. To comput th drivativ with rspct to th slop prsrving motion, w us Equations (2) and (4) for to gt th drivativ of with rspct to, j

9 j $. Th drivativ of th ara with rspct to th dis, which is Symmtrically, w gt tanc btwn th cntrs is (8) bcaus. This rsult is obvious gomtrically, bcaus to th first ordr th ara gaind is th rctangl with width and hight obtaind by thickning th portion of th sparating Voronoi dg.? Z Growth. Using Equation (4) for and B, w gt Th right hand sid consists of four trms of which th fourth vanishs bcaus. Th third trm quals Th scond and third trms cancl ach othr bcaus. Th scond trm is O.. Hnc, Z> j () This quation is again obvious gomtrically bcaus to th first ordr th gaind ara is th fraction of th annulus of width and lngth d thickning th boundary arc contributd by 2. Z obtaind by Assmbly of rlations. Lt b th ara of th union of disks 24, for =X?= 1'dB. W dcompos th drivativ into trms, as bfor. Th drivativ along th slop prsrving dirction is givn by Equation (8). Th lngth of th corrsponding vctor ) in th thorm is this drivativ tims th fractional chord lngth, which is. Th drivativ with rspct to th radius is givn by Equation (). It is qual to th contribution of 2 to th primtr, which in th cas of 1 disks is Z. This complts th proof of th Ara Drivativ Thorm. 5 Discussion Considr a finit collction of disks in th plan. W call a motion that dos not chang radii and that at no tim dcrass th distanc btwn any two cntrs a continuous xpansion. Th Ara Drivativ Thorm implis that th drivativ along a continuous xpansion is always nonngativ. Th ara is thrfor monotonously nondcrasing. This is not nw and has bn provd for gnral dimnsions in 18 by

10 Csikós [5]. Th mor rstrictd vrsion of this rsult for unitdisks in th plan has bn known sinc 18. Bollobás proof uss th fact that for unit disks th primtr is also monotonously nondcrasing along continuous xpansions []. rhaps surprisingly, this is not tru if th disks in th collction hav diffrnt radii. Th critical trm that spoils th monotonicity is contributd by th rotational motion of on disk about anothr. That contribution can b nonro if xactly on of th two cornrs dfind by th two circls blongs to th boundary of th union. Continuous xpansions that dcras th primtr ar thrfor possibl, and on is shown in Figur 4. Fig. 4: Moving th small disk vrtically downward dos not dcras any distancs but dos dcras th primtr. Rfrncs 1. M.. BEDSØE. Optimiation of Structural Topology, Shap, and Matrial. Springr Vrlag, Brlin, Grmany, M.. BEDSØE AD C. A. MOTA SOARES (EDS.) Topology Dsign of Structurs. Kluwr, Dordrcht, Th thrlands, 1.. B. BOLLOBÁS. Ara of th union of disks. Elm. Math. 2 (18), R. BRYAT, H. EDELSBRUER,. KOEHL AD M. LEVITT. Th ara drivativ of a spacfilling diagram. Manuscript, B. CSIKÓS. On th volum of th union of balls. Discrt Comput. Gom. 20 (18), H. EDELSBRUER. Th union of balls and its dual shap. Discrt Comput. Gom. 1 (15), H. EDELSBRUER. Dformabl smooth surfac dsign. Discrt Comput. Gom. 21 (1), H. EDELSBRUER AD. FU. Masuring spac filling diagrams and voids. Rpt. UIUC BIMB401, Bckman Inst., Univ. Illinois, Urbana, Illinois, 14.. M. LEVITT AD A. WARSHEL. Computr simulation of protin folding. atur 25 (175), M. SIVAK. Calculus on Manifolds. AddisonWsly, Rading, Massachustts, 15.