IS THE MINIMUM-TRACE DATUM DEFINITION THEORETICALLY CORRECT AS APPLIED IN COMPUTING 2D AND 3D DISPLACEMENTS?

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1 Procdgs, h FIG Symposum o Dformao asurms, Saor, Grc, 003. IS HE INIU-RACE DAU DEFINIION HEOREICALLY CORREC AS APPLIED IN COPUING D AND 3D DISPLACEENS? Wold Prószyńsk Isu of Appld Godsy, Warsaw Uvrsy of chology, Pl. Polchk, PL , Warszawa, Polad Asrac I s show ha h mmum-rac daum dfo as usd for compug D ad 3D posos ad dsplacm vcors as wll as hr accuracy characrscs dos o yld a physcally rpral daum wh rspc o orao. hs s xplad o h ass of h aalyss of muual rlaoshps w daum cosras h al o-lar Gauss-arkov modl (G) for a local work ad h corrspodg cosras h larzd sysm. A propry of mmum-rac daum for D works s provd whch, sp of h aov mod drawack, covcs o of h saf us of hs daum pracc. h propry s llusrad o a smpl praccal xampl, g a D lar-agular moorg work. o kp h cos of h papr wh h lms s y h Symposum orgasrs, h rlva proofs, xcp for o, ar gv a rf oul oly.. Iroduco A physcally rpral daum s a prcpal lm of vry procss of drmg h posos or dsplacm vcors. I addo o lmag h dfc of h moorg work, such a daum should gv a ral ss oh o h compud quas ad hr accuracy characrscs. A quso arss whhr h frquly usd mmum-rac daum dfo fulfls h scod rqurm. h daum orgad h sxs as a cor cocp of fr adjusm ad has sc xsvly vsgad y may rsarchrs such as ssl (96), rmyr (97), Plzr (97), Grafard, Schaffr (974), Caspary (988), Papo, Prlmur (989) o mo u a fw. hr rsarch cocrad o lar (larzd) modls, gomrcal rprao of h daum cosras ad h proprs of h las squars smaors. From h aalyss of hos fdgs, whch ar o coclusv as rgards h orao of D ad 3D works, sms ha h quso of physcal rpraly of h mmum-rac daum ds a spcal ram. Followg hs osrvao, h prs papr focuss o h quso how hs daum dfs h orao of h work, ad hc, h orao of h dsplacm vcors. h physcally-ord approach appld hs papr s rflcd h followg wll-kow prcpls: - a co-orda sysm coms dfal a physcal raly y aachg uquly o a group of physcal pos (.. work moums). h rqurm for uqu aachm follows from h fac ha h drmd posos of hs pos ar sochasc quas; - h covarac marx for a vcor of posos (or poso chags) mus hav a physcally dfal zro-varac as. I a local work h zro-varac as s dfd o a spcfd sus of h work pos;

2 h ass of h aalyss wll h rlaoshps w h cosras h al o-lar G ad hos s larzd form. I s h formr modl ha cosus a propr mahmacal modl of h posog or moorg ask.. Daum-dfg cosras a o-lar G ad s larzd form L us rcall oh a paramrc o-lar modl wh mmum cosras ad s larzd form F(X) + = y ; (, C ) ===> Ax + = y y ; ( 0, C ) () - 0 ap - G(X) = c ===> Sx = 0... whr: X, x = h u vcors of paramrs, F(X), G(X) = h ad d fucoal vcors, such ha d[ F( X)] d[ G( X)] A = dx X, S = ap dx X ; A ( u)marx; rak(a) = u d ap (d h work dfc); S (d u), rak(s) = d; rak[ A S ] = u; X ap c y y ap = h u vcor of approxma co-ordas = h d vcor of cosas; = h x vcor of ukow radom rrors; = h vcor of osrvaos; = h vcor of h approxma osrvao valus; C = h covarac marx of osrvaos (pos. df). or spcfcally, h cosras whch cosu h dfo of h daum for a local work should wr as G ( X ) = c ad S x = 0 o dca ha hy ar dfd o a chos sus P of h work pos. h al o-lar G, wh usd for posog, ca rprd as a dscrpo of h followg ask (s Fg.a for D cas): - cosruc fgur Fg (y) ad loca uquly h co-orda sysm,.. accordg o h spcfd valus of h locao paramrs G(X), hr Xo = Xo ( X), Yo = Yo ( X), αo = αo ( X). l 0 l 0 α 0 a) posos ) dsplacms α 0 G(X) Fg( y) Fg( X ) G(X) Fg( y) Fg( X ) X P 0 P 0 G(X) Fg( y) Fg( X) 0 0 X 0 0 (X, Y ) (X, Y ) Y Fg. Spcfcao of h posog ask (a) ad h moorg ask () Y

3 I h cas of moorg h dsplacms h dscrpo of h ask wll as follows (s Fg. ): - cosruc fgur Fg( y ) ad fgur Fg( y ), corrspodg o wo dffr m moms, ad loca ach uquly h co-orda sysm,.. accordg o h spcfd valus of locao paramrs G(X), hr Xo = X o ( X), Y o = Y o ( X ), α o = αo ( X). No ha w oa h suprposo of h fgurs Fg( y ), Fg( y ) h drc way,.. hrough locag ach h sam daum. Sc X o = 0; Yo = 0; αo = 0, h po Po ad h l l o dfd o h posos of h rfrc pos cosu also h zro-varac as for h accuracy characrscs of compud posos or dsplacm vcors. From h aalyss prsd so far follows ha h mmum cosras G(X) = c should sasfy h followg rqurms:. physcal rpraly ach cosra should corrspod o a spcfd dgr of frdom of Fg( y) a co-orda sysm;. dffraly h possly of oprag wh a lar modl Sx = 0. O hs ass w may formula h corrspodg rqurms for lar cosras Sx = 0 wh hy ar crad for a larzd modl whou pror spcfyg G(X) = c: Ax + = y y ; ( 0, C ) (s ()) ap -. graly h possly of oag h rlaoshps of h yp G(X) = c. physcal rpraly ach rlaoshp oad from grao should dfal as a cosra corrspodg o a spcfd dgr of frdom of Fg( y) a co-orda sysm. 3. Shorcomgs of h mmum-rac daum dfo I s a spcfc cas of daum for a local work, logg o h class of h mmumcosras daums. I uss S = (or mor spcfcally S = S ]) such ha S [, 0 AS = 0, whch alog wh v C v = m ylds xˆ = m (ad hc r( Cx ˆ, ) = m). Also ach of h marcs S = BS, whr B s o-sgular, ca usd quvally. h cosras ar formulad hr a a lvl of h larzd modl so h rlaoshps corrspodg o () ca wr as follows F(X) + = y ; ( 0, C ) ====> Ax + = y y ; ( 0, C ) - ap - () G(X) = c <=? == S x 0 = Accordg o Sco, h lar cosras S x = 0 should sasfy h rqurms of graly ad physcal rpraly. I s asy o s ha such s h cas wh posoal cosras, whr w oa afr grao X C = cos. ; Y C = cos. ; Z C = cos.,c g

4 h gravy cr of h group of rfrc pos. Drc grag of h orao ad scal cosras ylds h corrspodg cosras for h al G h form G ( X, Xap ) = c. hy do o fulfl h rqurm of g Xap - dpd,.. G(X) = c. Skg such a form w shall cosdr h cosras S x = 0 wh h approxma coordas g rplacd y h corrspodg varals. h, o grag h scal cosra w oa mmdaly ρ,c = cos., whch y aalogy o cocps usd mchacs (Lyko 997), may calld a polar mom of ra for h group of rfrc pos wh rspc o s gravy cr C. As rgards h orao cosras S x = 0, h proof gv Appdx B allows o o draw h followg coclusos: 3D works - for 3 h orao cosras ar o gral; D works - for 3 h orao cosra s o gral. Summg up h aov, w may df fr adjusm as solvg h al fucoal modl as () wh h cosras as follows: G p ( X ) = c p ; G α ( X, X ap ) = cα ; G s ( X) = cs, whr p, α, s doo posoal, orao ad scal cosras rspcvly. For a D work whou scal dfc h modl ca rprd as h dscrpo of h followg ask (s Fg.): Fg( y) - cosruc fgur ad loca uquly h co-orda sysm accordg o h spcfd valus of h locao paramrs G p ( X),.. X C, Y C, ad h zro valu of h rlav orao paramr G α ( X, X ap ),.. A( X, Xap ) - g h algrac sum of h hachd aras. Fg. h lack of XOY-rlad orao paramr for a D work Fg( X) For 3 h orao of h rsulg fgur cao dfd drcly rlav o h axs of h co-orda sysm,.. hrough G α (X). Covrsly, sc h l l o cao dfd w cao rproduc h orao of h X0Y sysm ad w do o hav h zrovarac as for work orao. h possly of mprovg h accuracy of X ap h succssv raos, v ll Xˆ X ap G α ( X). coms glgly small, dos o lad o fdg h rqurd orao paramr

5 4. Advaagous proprs of h mmum-rac daum dfo I ca provd (s Appdx C) ha for ach po of D work h followg propry holds (s Fg.3): - h agal compo of h soluo vcor oad wh mmum rac daum s a wghd ma of h agal compos (j =,,...,) of soluo vcors oad w wh h daums, ach g po C fxd, h l (for h formula ad h proof s Appdx C). (j) CP j fxd ; h wghs g p (j) = L j C P m (j) () max W C,max,max P L,, L Fg.3 Spcfc propry of h mmum-rac daum for D works hs mpls h followg proprs: * g a wghd ma, s always sd h rval < m, max > ad hus sd h rval oad wh physcally rpral daums; * for as a wghd ma, w hav, <, max. h aov proprs ar coss wh hos formulad for mmum-cosras daums wh h gravy cr fxd (s rlaoshps (A), Appdx A). No also h proprs of mmum-rac daum rlad o h rfrc pos (quals (A5) ad (A6), Appdx A). Praccal xampl. Fgur shows h agal compos ad j =,,...,6 of dsplacm vcors for som pos of h agular-lar moorg work. W ca s ha h arragm of h valus s smlar for all h hr work pos. (j) 5. Cocludg rmarks O h ass of h aalyss carrd ou hs papr o may formula h followg coclusos: - for 3 h mmum-rac daum dfo dos o spcfy a physcally rpral rfrc as for h orao of D ad 3D works, ad hc for h dsplacm vcors ad hr accuracy characrscs; - s a advaagous propry of h mmum-rac daum dfo for D works ha h rsulg soluo vcors for ach work po fall w hos oad wh physcally rpral daums. O may xpc ha h aalogous proprs ca foud for 3D works;

6 Fg.4 h agal compos of dsplacm vcors a moorg work - vw of praccal applcaos h advaagous propry as aov compsas for h lack of physcally rpral rfrc as for orao; - h accuracy of h approxma co-ordas, whch ca mprovd succssv rao sps, s hr h caus or adds o h shorcomgs of h mmum-rac daum dfo. Rfrcs Baarda W. (973). S-rasformaos ad crro marcs. Nh. Comm., Pul. O Godsy, Nw Srs 5, No., Dlf. Caspary W.F. (988). Cocps of work ad dformao aalyss. oograph, School of Survyg, h Uvrsy of Nw Souh Wals, Ksgo, N.S.W., Ausrala. Chr S.S., Ch W.H., Lam K.S. (999). Lcurs o dffral gomry. Srs o Uvrsy ahmacs,. World Scfc Pulshg Co., Ic., Rvr Edg, NJ. Domrz W. (00). Chck o graly of orao cosras mmum-rac daum dfo (ral rpor). Faculy of ahmacs ad Iformao Scc, Warsaw Uvrsy of chology. Grafard E., Schaffr B. (974). Uasd fr adjusm. Survy Rvw, XXII, 7. Lyko J. (997). Gral mchacs Dyamcs ( Polsh). PWN, Warszawa. ssl P. (96). D Ir Gaugk s Pukhaufs. Os. Zfv, 50 (5):59-65; 50(6): rmayr E. (97). Zur Ausglchug frr Nz. Zfv 97(): Papo H., Prlmur A. (989). Daum of a dyamcal work. auscrpa Godaca 4: Plzr H. (97). Zur Aalys Godaschr Dformaosmssug. DGK, uch, Rh C, Hf 64. Prószyƒsk W. (980). Gomrcal faurs of a s of pos ad accuracy of hr drmao o h ass of grg survys ( Polsh). Godzja Karografa, z.3/4: Rao C.R. (973). Lar sascal frc ad s applcaos, Wly, Nw York. Appdx A: Proprs of h mmum-cosras daums wh fxd gravy cr (D ad 3D works) L us cosdr h followg wo opos of mmum-cosras daum for a work whou scal dfc: S x = 0 S x = 0 mmum-rac daum N daum wh fxd gravy cr ad orao cosras ohr ha hos aov,

7 S p, 0 Sp, 0 x whr: S = [ S, 0] = ; S N = [ S N, 0] = ; S β, 0 x = S α, 0 xc h dcs ad c do hos work pos whch form h rfrc as () ad hos whch do o log o hs as (c); p sads for posos, α ad β - sad for orao. L Lˆ do h lgh of h l jog h gravy cr C ad a po P ad - h Lˆ, sadard dvao of Lˆ oad from h LS smao PROPERIES. For ay work po oh Lˆ cosras,.. ad ar vara o h choc of orao Lˆ Lˆ ( S N ) = Lˆ ( S ), ( S N ) = ( S ) =,,..., (A) Lˆ, Lˆ, PROOF. Applyg a wll-kow smlary rasformao (hr somry rasformao) s Baarda (973), w g whr G = I S x ˆ = Gx (A) N ˆ * S N S * ) ( S S p, Sp,c N ; S* = ; Sβ, Sβ, c xˆ xˆ = x ˆ,,c ; ˆx N xˆ = x ˆ N, N,c ad carryg ou a squc of smpl opraos (cludg h rducos of muually orhogoal facors), w oa whr Q xˆ N = U 0 xˆ,, (A3) Ic xˆ,c Q = I Sβ, ( Sα, Sβ, ) Sα, U = Sβ, c ( Sα, Sβ, ) Sα,. I a lar modl () w ca rprs Lˆ as Lˆ = Lap + gxˆ, whr g = [0,0,0,..., X, Y, Z,...,0,0,0] L I s asy o chck ha g,sβ, = 0 ad g,cs β, c = 0, ad hc N,, ap g x ˆ = g xˆ ad g x ˆ = g xˆ, (A4), whch ylds mmdaly,,c N, c,c Lˆ ( S N ) = Lˆ ( S ) for =,,...,., c From (A4) follows ha xˆ,n g C g = g C xˆ, g ad hus ( S N ) = ( S ) for =,,..., (Q.E.D.) Lˆ, Lˆ,

8 Fgur 5 llusras hs proprs for a D work. h soluo vcor x N,o ad h rror ar B N,o corrspod o a daum havg a orao cosra dα = 0, whr α s a arg of h l CP ad h sadard llpsods.. h aalogous ouds u h form of plas xs for 3D soluo vcors L P ap, x N x x N,o E N P E B N,o C CP Fg.5 h dsac (a) ad s sadard dvao orao cosra h proprs (A) ca xdd upo h works wh scal dfccy. C Lˆ L () as vara o h choc of,,, L r,, do rspcvly h radal ad h agal compos of h soluo vcor x for h h work po. Sc x < x ad r( C ) < r( C ), ad accordg o (A), N, xˆ, xˆ N, r ( S ) = r ( S ) ad S ) = ( S ), w oa mmdaly N r, ( N r, L ad ( () ' " ' " ) + ( ) < ( N ) + ( N ) () () () + (), (), < (), N + (), N (A5) ad for D works < N ad, <, N (A6) () () () () Appdx B: Igrag h cosras quaos mmum-rac daum dfo L (x,y,z) = 3 R whr ( x,..., x, y,..., y, z,..., z ) 3. L D do h followg Pfaffa sysm o dy = 0 ; y dz zdy = 0 dz = 0 ; z dx xdz = 0 dx = 0 ; x dy y dx = 0 do h sadard co-ordas sysm o 3 R :

9 PROPOSIION D s o gral. Proof. L us do I s asy o s ha ( ) = dy = y dz zdy 3 = dz 3 = z dx xdz α = dx ; β = x dy y dx α ; β α ; β ( ) α = d x ; α = d y ; α3 = d ( z ) ; hrfor d α = 0, =,,3 ad d = dx dy. W df γ s a 8-form o β γ = d β, 3 R. By drc calculao w oa γ( α α α3 β β β3,,,,,,, x y x y z x3 z z3 ) = = (x3y y3 xy3 x3y y3 + x y3 + xyy3 x yy3 x 3y + xyy3 x yy3 - xyy + x y + x yy3 + x3y x yy3 xy + x yy ) whch s a polyomal of h 3 rd dgr. As γ dos o vash o ay op sus ofr, y Frous horm (s Chr, Ch ad Lam 999) D s o gral (Q.E.D.). 3 Appdx C: A spcfc propry of h mmum-rac daum dfo for D works L x N( j) j =,,..., do h soluo vcors oad wh mmum-cosras daums ach wh h gravy cr fxd ad h orao cosra dα j = 0 ad x - h soluo vcor oad wh mmum-rac daum. L N( j) = x N( j) j =,,..., ad = x, whr s h ( ) roao marx, h rspcv agal compos of h soluo vcors.

10 PROPERY L j p( j) N( j) = j =,,...,, whr p ( j) = (C) j= L j j= PROOF Sc x N( j) = G N( j) x (s A) h rlaoshp (C) aks h form p( j) G N( j) x = Τx j= (C) I ca show ha (C) holds ru, f ( p ( j) G N( j) Τ) Go = 0 (C3) j= o ) whr G = I S ( S S S. akg us of (A3) w ca wr h codo (C3) h form Q 0 Q 0 I 0 p ( ) G = 0 U I p() U I 0 I o c c c ad furhr o p( j) Q p j ( j) I U j 0 G 0 o = 0 Vrfyg ha hs codo s sasfd compls h proof.