COORDINATE TRANSFORMATIONS FOR CADASTRAL SURVEYING

Transcription

1 COORDINATE TRANSFORMATIONS FOR CADASTRAL SURVEYING R. E. Deaki Shool of Mathematial ad Geospatial Siees, RMIT Uiersit Marh 007 ABSTRACT A two-dimesioal (D) oformal trasformatio, (that preseres shape ad hee agles), is a useful tool for pratiig adastral sureors. It a be used as a aid to re-establishmet where oupatio boudar orers of allotmets, sureed i the field o a arbitrar sure oordiate sstem, a be trasformed to the adastral title oordiate sstem ad oupatio/title omparisos made. The trasformatio proess osists of two parts. The first part is the determiatio of the trasformatio parameters; sale s, rotatio θ ad traslatios, t. This requires a miimum of two otrol poits haig oordiates i both the title ad sure oordiate sstem. If there are three or more otrol poits, the the trasformatio parameters are determied b a least squares proess ad a weightig sheme a be emploed. The seod part is to use the trasformatio parameters (determied from the otrol poits) to trasform the other sureed poits oto the title sstem. This paper sets out the eessar theor of D oformal trasformatios ad the determiatio of trasformatio parameters usig least squares. I additio, weightig shemes are disussed as well as trasformatios that presere sale (i.e., a sale s = ). t ad INTRODUCTION Coformal oordiate trasformatios, are widel used i the sureig professio. For istae, i geodes, 3D oformal trasformatios a be used to oert oordiates related to the Australia Geodeti Datum (AGD66, AGD84) to the

2 Geoetri Datum of Australia (GDA94), i egieerig sureig the form part of moitorig ad otrol sstems used i large projets suh as the ostrutio of eleated freewas ad tuels (Deaki 998), ad i photogrammetr the are used i the orietatio (iterior ad eterior) of aerial digital images. I D form, trasformatios are used i adastral sure re-establishmets (Bebb 98, Sprott 983 ad Bird 984), mathig digitized adastral maps (Shmutter ad Dotsher 99) ad "sewig together" the edges of strips of digital images (Bellma, Deaki ad Rolligs 99). I geeral, the effet of a D trasformatio o a polgo (a plae multi-sided figure) will ar from a simple hage of loatio ad orietatio (with o hage i shape or size) to a uiform hage i sale (o hage i shape) ad fiall to hages of shape ad size of differet degrees of oliearit (Mikhail 976). The most ommo trasformatios i sureig appliatios, ad the ol tpe dealt with i this paper, are oformal, i.e., trasformatios that presere agles ad thus the shape of objets. Theor ad appliatios of other oordiate trasformatios, suh as affie, polomial, projetie et. a be foud i Mikhail (976) ad Moffitt ad Mikhail (980). I the theor that follows, trasformatios are epressed i the form of equatios likig oordiates i oe sstem with oordiates i aother sstem ad these equatios otai rotatio agles (usuall deoted b θ ), as well as sale s ad traslatios t ad t (or ad t ) where the subsripts relate to the oordiate t E N sstem aes labels,,; E,N; u,; et. The idea of rotatio is importat ad as we will see there are seeral differet tpes of rotatios, i.e., a objet a hae a atual rotatio where it is rotated about a poit; or a apparet rotatio where its oordiates hage beause the oordiate aes are rotated; ad these rotatios a be lokwise or atilokwise. I this paper we will ol be osiderig apparet rotatios aused b atilokwise rotatio of oordiate aes ad to larif these issues some rules ad diagrams are helpful. I geeral we osider poits i spae as beig oeted to the origi O of a 3D right-haded retagular oordiate sstem,,z. Suh a sstem a be isualised as the orer of a room where the itersetio of two walls ad the floor proide three referee lies O, O ad Oz, kow as the -, - ad z-aes that are (usuall) at

3 right agles to oe aother. The -z ad -z plaes are the walls ad the - plae is the floor. z The three mutuall perpediular aes, ad z are related b the right-had rule as follows: If the thumb, the forefiger ad the seod figer of the right had are plaed mutuall at right agles the the thumb poits i the z-diretio, the forefiger poits i the -diretio ad the seod figer poits i the -diretio. z The aes, ad z (i the li order z) are a right-haded sstem (or detral sstem) sie a rotatio from towards adaes a right-haded srew i the diretio of z. Similarl, a rotatio from towards z adaes a right-haded srew i the diretio of ad so o. The diagram o the left shows the right-hadsrew rule for the positie diretios of θ rotatios ad aes of a right-haded retagular oordiate sstem. These rotatios are osidered positie atilokwise whe lookig alog the ais towards the origi; the positie sese of rotatio beig determied b the right-had-grip rule where a imagiar right had grips the ais with the thumb poitig i the positie diretio of the ais ad the atural url of the figers idiatig the positie diretio of rotatio. 3

4 The right-haded oordiate sstem ad positie atilokwise rotatios (gie b the right-had-grip rule) are osistet with oetios used i mathematis ad phsis ad i mathematis, agles are measured positie atilokwise from the - ais; a oetio we also use i these otes. As sureors, we deal almost elusiel with agular quatities (bearig, azimuths, diretios, et) osidered as positie lokwise ad usuall measured from orth (or the -ais i the - sstem or the -ais i the u- sstem) ad this sureig oetio of positie lokwise rotatio from orth ould be desribed b a left-had-grip rule but we do ot usuall do this. CONFORMAL TRANSFORMATIONS IN TWO-DIMENSIONAL (D) SPACE I D oformal trasformatios all poits lie i a plae ad suh poits are osidered to hae ol, (or u,) oordiates, i.e., the lie i the - (or u-) plae with a z-alue = 0 (or w-alue = 0). I these otes it is assumed that D oformal trasformatios are trasformatios from oe retagular oordiate sstem (u,) whih we ould all the sure sstem to aother retagular sstem (,) that we ould all the title sstem. Both of these oordiate sstems ould be thought of as arbitrar ad it is immaterial where the origis of both sstems lie. I additio, uless stated otherwise, a rotatio is a agle osidered to be positie i a atilokwise diretio as determied b the right-had-grip rule ad rotatios of polgos (or objets) are apparet, sie we are osiderig rotatios of oordiate aes rather tha atual rotatios of polgos about a etre more about this later. Also, trasformatio equatios are oeietl epressed usig matri otatio ad a rotatio matri R (whose elemets are futios of the rotatio agle θ ) is a ompoet of a oformal trasformatio equatio. Rotatio matries are orthogoal, whih is a er useful propert, ad there is a eplaatio of this propert i the followig setios. The geeral oformal trasformatio formula are deeloped i a simple wa. First, b osiderig trasformatios iolig rotatio ol; the, iolig both sale ad rotatio. Ad fiall, the geeral ase, iolig sale, rotatio ad traslatios. We the show that this geeral oformal trasformatio (ombiig sale, rotatio, traslatio) is the same as that obtaied b usig the mathematial priiples of oformal mappig deeloped b C. F. Gauss. 4

5 Coformal Trasformatio iolig Rotatio ol u, oordiates (sure sstem) are trasformed to, (title sstem) oordiates b osiderig a rotatio of the u, oordiate aes through a positie atilokwise agle θ. The trasformatio equatios a be epressed i the followig wa = uos θ + si θ = usi θ + os θ () or i matri otatio os θ si θ u = si θ os θ () As a eample osider the polgo ABCD whose u, oordiates are rotated b a positie atilokwise agle θ = 30. Figure shows the iitial loatio of the polgo i the u, sure sstem ad Figure shows its trasformed (rotated) loatio i the, title sstem. B Poit u C A A B C D D u Figure Polgo ABCD with u, oordiates i metres 5

6 B Poit A A C B C D D Figure Rotated polgo ABCD with, oordiates i metres Comparig Figures ad it appears that the size ad shape of the polgo ABCD has ot haged but its orietatio with respet to the oordiate aes has. This a be erified b osiderig the dimesios (bearigs ad distaes) of the polgo ABCD deried from the two oordiate sets. Lie Bearig Distae AB BC CD DA Polgo dimesios i the u, sstem Lie Bearig Distae AB BC CD DA Polgo dimesios i the, sstem This eample demostrates that a rotatio of the oordiate aes auses a apparet rotatio, i a opposite diretio, of a polgo defied withi the oordiate sstem. The size ad shape of the polgo does ot hage. 6

7 Equatio () ad its matri equialet () a be obtaied b osiderig Figure 3. si θ os θ u si θ u u os θ P θ u Figure 3, oordiates of P as futios of u, oordiates ad rotatio θ Rotatio matries Equatio () a be epressed as os θ si θ u u = si θ os θ = R (3) os θ si θ where R = is kow as a rotatio matri. Rotatio matries are si θ os θ orthogoal, i.e., the sum of squares of the elemets of a row or olum is equal to uit ad a orthogoal matri has the uique propert that its ierse is equal to T its traspose, i.e., R = R. This useful propert allows us to write the trasformatio from, oordiates to u, oordiates as follows. R R R R T u = R u = u = I ad rearragig gies u os θ si θ T = R = si θ os θ (4) 7

8 We ould write (4) as u os θ si θ = si θ os θ = R whih i words meas: the, oordiates are trasformed (rotated) to u, oordiates. Equatio (3) o the other had meas: the u, oordiates are trasformed (rotated) to, oordiates ad it is iterestig to ote that R ad are i fat the same rotatio matri eept i the former, θ is positie atilokwise ad i the latter θ is positie lokwise. Note that si( θ) = si θ ad os( θ) = os θ. R Orthogoal Matries Orthogoal matries are etremel useful sie their ierse is equal to their traspose. Rotatio matries R are orthogoal, hee R be foud i Alla (997) ad is repeated here. T = R. A proof of this a Cosider the effet of a rotatio o the oordiates of a poit P, epressed as X = R X is the trasformed (or rotated) oordiates ad R is the rotatio matri. Multiplig both sides of the equatio b the ierse of R gies R X = R R but from matri algebra R R = I ad I = so or R X = = R X The legth (atuall squared legth) of the lie from the origi to the origial positio of poit P is gie b (rotated) positio is gie b T it is iariat uder rotatio. Hee ad the legth from the origi to the ew T XX. This legth does ot hage due to rotatio, i.e., T but X = = T XX R so T = ( R) T R = T T RR 8

9 For this result to be possible but T RR R R = I = I Therefore R = R T Thus the ierse of a rotatio matri is equal to its traspose. Rotatio of Aes ersus Rotatio of Objet I these otes it is assumed that a rotatio agle is a positie atilokwise agle as determied b the right-had-grip rule ad that "apparet" rotatios of objets (polgos) are aused b a rotatio of the oordiate aes. This is ot the ol wa that a objet a be rotated. Cosider Figure 4 where P with oordiates, moes to o d φ θ d P' P P' with oordiates ',' b a positie atilokwise rotatio φ. The oordiates of P' are = dos( θ + φ) = d( os θosφ si θsi φ) (5) = dsi( θ + φ) = d( si θosφ + os θsi φ) Figure 4 The oordiates of P are = dos θ ad = d si θ whih a be substituted ito (5) to gie = os φ si φ = os φ + si φ or i matri form os φ si φ = si φ osφ = R (6) Where R is a rotatio matri ad the rotatio agle φ is a "right-haded" rotatio. Ispetio of equatios (3) ad (6) shows that R is ot the same form as R, i fat T it is idetial i form to R. The rotatio matri R auses a apparet rotatio of the objet b rotatio of the oordiate aes whilst the rotatio matri R rotates the objet itself. Both R ad R are "right-had" rotatio matries (oe is the traspose of the other) ad there is ofte ofusio amogst users of trasformatio software i defiig the tpe of 9

10 rotatio ad the positie diretio of rotatio. You must be er areful i defiig rotatio, i.e., ou must state what is beig rotated, either aes or objet ad what is the positie diretio of rotatio. I these otes it is alwas assumed that the oordiate aes are beig rotated ad the rotatios are alwas positie atilokwise as defied b the right-had-grip rule. Coformal Trasformatio iolig Rotatio θ ad a Sale hage s u, oordiates (sure sstem) are trasformed to, oordiates (title sstem) b osiderig a rotatio of the u, oordiate aes through a positie atilokwise agle θ ad a salig of the u, oordiates b a fator s. The trasformatio equatios a be epressed i the followig wa = ( sos θ) u + ( ssi θ) = ( ssi θ) u + ( sos θ) (7) or i matri otatio os θ si θ u = s si θ os θ (8) Ofte, the oeffiiets of u ad i (7) are writte as a = sos θ ad b = ssi θ giig a b u = b a ad the sale fator s ad the rotatio agle θ are gie b s = a + b θ ta b a = (9) (0) 0

11 As a eample osider the polgo ABCD whose u, oordiates (sure sstem) are rotated b a positie atilokwise agle θ = 30 ad saled b a fator s = 0.6. Figure shows the iitial loatio of the polgo i the u, sstem ad Figure 5 shows its trasformed (rotated ad saled) loatio i the, sstem (title sstem). Poit A A D B C B C D Figure 5 Rotated ad saled polgo ABCD with, oordiates i metres Comparig Figures ad 5 it appears that the shape of the polgo ABCD has ot haged but its size ad orietatio with respet to the oordiate aes has. This a be erified b osiderig the dimesios (bearigs ad distaes) ad area of the polgo ABCD deried from the two oordiate sets. Lie Bearig Distae AB BC CD DA Area=, m Polgo dimesios i the u, sstem Lie Bearig Distae AB BC CD DA Area=8,39.3 m Polgo dimesios i the, sstem Ispetio of the two sets of dimesios reeals that bearigs hae bee rotated b a agle θ = 30 ad distaes saled b a fator s = 0.6. Note that the shape of the polgo is uhaged but the area of the trasformed figure has bee redued b a fator of s.

12 Coformal Trasformatio with Rotatio θ, Sale hage s ad Traslatios t, t u, oordiates (sure sstem) are first trasformed to oordiates b osiderig a rotatio of the u, oordiate aes through a positie atilokwise agle θ ad a salig of the u, oordiates b a fator s. The oordiates are the trasformed ito, oordiates (title sstem) b the additio of traslatios ad t., The trasformatio equatios a be epressed i the followig wa, t = ( sos θ) u + ( ssi θ) + t = ( ssi θ) u + ( sos θ) + t () or i matri otatio or os θ si θ u t = s si θ os θ + t u t = s R + t () Similarl to before writig a = sos θ ad b = ssi θ gies a b u t = + t b a (3) This trasformatio is referred to b seeral ames (i) Four-parameter trasformatio, the four parameters beig abt,,, t, (ii) D Liear Coformal trasformatio, (iii) Similarit trasformatio ad (i) Helmert's trasformatio, after the Germa geodesist F.R. Helmert (843-97). Note that "liear" is sometimes used i the desriptio of a oformal trasformatio to differetiate it from a polomial oformal trasformatio. Polomial oformal trasformatios are rarel used so the distitio will ot be used hereafter.

13 The D (liear) oformal trasformatio equatios ma be deried b osiderig Figure 6. The, oordiates are obtaied b rotatig ad salig the u, oordiates; ad the the, oordiates obtaied b addig the traslatios t to the, oordiates. This two-step proess is gie b the equatios: os θ si θ u s = si θ os θ t = + t t ad ' os θ si θ u si θ u P u os θ t θ ' u t Figure 6. Shemati diagram of rotated ad traslated aes Note that i Figure 6, t ad t are both positie quatities, but i geeral, the ma be positie or egatie. 3

14 As a eample of a D Coformal trasformatio, osider the polgo ABCD whose u, oordiates are rotated b a positie atilokwise agle θ = 30, saled b a fator s = 0.6 ad traslated b t = m ad t = m Figure shows the iitial loatio of the polgo i the u, sure sstem ad Figure 7 shows its trasformed (rotated, saled ad traslated) loatio i the, title sstem. A B C Poit A B D C D Figure 7 Rotated, saled ad traslated polgo ABCD with, oordiates i metres Comparig Figures ad 7 it appears that the shape of the polgo ABCD has ot haged but its area ad orietatio with respet to the oordiate aes has. This a be erified b osiderig the dimesios (bearigs ad distaes) ad area of the polgo ABCD deried from the two oordiate sets. Lie Bearig Distae AB BC CD DA Area=, m Polgo dimesios i the u, sstem Lie Bearig Distae AB BC CD DA Area=8,39.3 m Polgo dimesios i the, sstem Ispetio of the two sets of dimesios reeals that bearigs ad distaes of the polgo i the u, sstem hae bee has bee rotated b a agle b a fator s = 0.6. θ = 30 ad saled Note that the shape of the polgo is uhaged but the area of the trasformed figure has bee redued b a fator of preious trasformatio demostrates that shape ad orietatio of a polgo. 4 s. Compariso with the traslatio has o effet o the area,

15 D Coformal Trasformatio deried usig oformal mappig theorems C.F. Gauss ( ) showed that the eessar ad suffiiet oditio for a oformal trasformatio from the ellipsoid to the map plae is gie b the omple epressio (Lauf 983) where the futio f ( + i = f χ + iω ) ( ) (4) χ + iω is aalti, otaiig isometri parameters χ (isometri latitude) ad ω (logitude) ad i this equatio the -ais is east-west ad the -ais is orth-south. i is the imagiar umber ( i = ). It should be oted here that isometri meas of equal measure, ad o the surfae of the ellipsoid (or sphere) latitude ad logitude are ot equal measures of legth. This is obious if we osider a poit ear the pole where similar distaes alog a meridia ad a parallel of latitude will orrespod to astl differet agular alues of latitude ad logitude. Hee i oformal map projetios, isometri latitude is determied to esure that agular hages orrespod to liear hages. A eessar oditio for a aalti futio is that it must satisf the Cauh- Riema equatios = ad = χ ω ω χ (5) Usig this theorem, a oformal trasformatio from oe plae retagular oordiate sstem u, (isometri parameters) to aother plae retagular sstem, (also isometri parameters) is gie b the omple epressio A futio f ( iu) + i = f + iu ( + that satisfies the Cauh-Riema equatios, is a omple polomial, hee (6) a be gie as ( k k)( ) ) (6) k (7) 0 + i = a + ib + iu Equatio (7) a be epaded to the first power (k = ) giig + i = ( a + ib )( + iu) + ( a + ib )( + iu) = a + b i + a + a ui + b i + b ui 0 0 5

16 Equatig real ad imagiar parts (rememberig that i = ) gies = b + a u + b 0 = a bu + a 0 (8) or i matri otatio with traslatios ad b betwee the oordiate aes a0 0 a b u b = + a 0 b a 0 (9) These equatios are of similar form to equatios (3) i the setio headed "Coformal Trasformatios with Rotatio, Sale ad Traslatios" ad properl desribe a D Coformal trasformatio. Note that the elemets of the leadig diagoal of the oeffiiet matri (a rotatio matri multiplied b a sale fator) are idetial ad the off-diagoal elemets the same magitude but opposite sig. Equatios (8) are essetiall the same equatios as i Jorda/Eggert/Keissal (963, pp ) i the setio headed "Das Helmertshe Verfahre (Helmertshe Trasformatio)" (Helmert's Trasformatio) although as oted b Beroets (99) i his bibliograph, there is o referee to the origial soure. It is probable that F.R. Helmert deeloped this oformal trasformatio i his masterpiee o geodes, Die mathematishe ud phsikalishe Theorem der höhere Geodäsie, (The mathematis ad phsial theorems of higher geodes) o whih he worked from 877 ad published i two parts: ol., Die mathematishe Theorem (880) ad ol., Die phsikalishe Theorem (884) [DSB 97]. This probabl aouts for the ommo usage of the term Helmert trasformatio whe desribig a D Coformal trasformatio. The partial deriaties of (8) are = a, = b, = b ad = u u whih satisf the Cauh-Riema equatios = ad = u u so erifig that the trasformatio is oformal. a 6

17 SOLVING FOR CONFORMAL TRANSFORMATION PARAMETERS Coordiate trasformatios, as used i pratie, are models desribig the assumed mathematial relatioships betwee poits i two retagular oordiate sstems; i these otes, the u, (sure) ad the, (title) sstems. To determie the parameters of a trasformatio, oordiates of poits ommo to both sstems must be kow. These poits are kow as otrol poits or ommo poits. The umber of ommo poits required for the solutio of trasformatio parameters depeds o the umber of parameters i the trasformatio. I D trasformatios, eah ommo poit gies rise to two equatios, thus ommo poits will gie equatios. Therefore, if the four parameters of a D Coformal trasformatio are to be determied, the a miimum of two ommo poits are required to sole for the parameters. It is good measuremet pratie to determie oordiate trasformatio parameters b usig more tha the miimum umber of ommo poits. This itrodues redudat equatios ito the solutio for the parameters ad the theor of least squares is emploed to alulate the best estimates. Parameters alulated i this maer are usuall more reliable ad the least squares proess allows preisio estimatio of the parameters as well as a assessmet (ia residuals) of how well the trasformatio model fits the ommo poits. B usig least squares, seeral tpes of trasformatios a be "tested" o the ommo poits to assess their suitabilit. The solutio for the trasformatio parameters ioles the followig steps (i) Selet the ommo poits esurig that there are suffiiet to allow a redudat set of equatios. (ii) Selet the appropriate weight matri W for the model. (iii) Sole for the parameters (otaied i the etor ) ad residuals (otaied i the etor ). (i) Assess the suitabilit of the model b aalsis of the parameters ad residuals. 7

18 Mathematial model for solutio of D Coformal Trasformatio Parameters The D Coformal trasformatio, or the mathematial model, osistig of rotatio, salig ad traslatio is set out aboe [see equatio (3)] ad the trasformatio for the k =,, 3,, ommo poits is gie i the form of obseratio equatios (0) k k a b uk t + k = k b a + (0) k t where ad are small ukow orretios or residuals simpl added to the k k equatios to aout for the assumed iosiste i the model. We ould thik of these residuals as osistig of two parts; oe part assoiated with the u, (sure) sstem ad the other assoiated with the trasformed, (title) sstem; the subsripts ad attahed to the residuals simpl reflet the fat that the hae bee added to the "trasformed" side of the model. Re-arragig (0) so that all the "ukows" are o to the left of the equals sig ad the obseratios are to the right gies au b t = k k k a + bu t = k k k k k () For ommo poits ad u = 4 ukow parameters, the partitioed matri represetatio of the equatios () is u 0 a u 0 b u t 3 t u 0 + = u u3 0 u 0 u 0 () 8

19 These equatios are represeted b the matri equatio + B = f (3) where B f is a (,) olum etor of residuals is a (,u) matri of oeffiiets is a (u,) etor of ukow parameters is a (,) olum etor of umeri terms (oordiates) The ormal equatios for the least squares solutio of parameters ad residuals are gie i matri form as or T ( ) BWB= T BWf (4) N = t (5) where T N = B WB t= T B Wf is the (u,u) smmetri oeffiiet matri of the ormal equatios is the (u,) etor of umeri terms of the ormal equatios w w 0 0 w 0 W = w 0 w w is the (,) diagoal weight matri where the weights o the upper-left diagoal are repeated o the lower-right diagoal. Weights are usuall iteger alues ad high weights are assoiated with "strog" poits ad low weights assoiated with "weak" poits. w k 9

20 The geeral form of the ormal equatios N = t are wk ( uk + k) 0 wkuk wkk wk ( ukk + kk) a wk ( uk + k) wk w k wkuk k ( kk ukk) b = t wk 0 w k k t smmetri w k w k k (6) Cetroidal oordiates Computatioal saigs a be made b reduig oordiates to a weighted etroid. For the ommo poits, the oordiates of the weighted etroid sstem are, i the, w + w + w + + w = = w + w + w + + w w + w + w + + w = = w + w + w + + w w k k w k k w w k k (7) Note here that oordiates of the weighted etroid, are just the weighted arithmeti meas of the oordiates of the ommo poits. Also, ote that if all poits hae the same weight the the oordiates of the etroid = = k, k, Now, the etroidal oordiates of the ommo poits i the, sstem are the = = = = = = = = are (8) 0

21 Similar relatioships a be writte for etroidal oordiates i the u, sstem. A useful propert of the etroidal oordiates of the ommo poits is that their sums equal zero, i.e., w = 0, w = 0, wu = 0, w = 0 (9) k k k k k k k k Thus, replaig, ad u, oordiates with their etroidal outerparts, ad u, redues the obseratio equatios (0) to a etroidal form k a b k uk + = k k b a k It should be oted here that traslatios t ad t are both zero whe etroidal oordiates are used idiatig that the etroids, ad u, are the same poit. (30) For ommo poits ad u = ukow parameters, the partitioed matri represetatio of the obseratio equatios resultig from the etroidal model (30) is u u a u b 3 u + = (3) u u 3 3 u 3 3 u These equatios are represeted b the matri equatio (3) ad the ormal equatios hae the followig simple form otaiig ol three differet umbers wk ( uk + k ) 0 a wk ( ukk + kk) b = 0 wk ( uk k ) + wk ( kk ukk) (3)

22 The solutios for the parameters a ad b are a = b = w ( u + ) k k k k k ( + ) w u k k k w ( u ) k k k k k ( + ) w u k k k (33) (34) The traslatios t ad t are obtaied b re-arragig (3) ad replaig, ad u, with the oordiates of the etroid, ad u, giig t = t b a a b u t = au b or (36) t = + bu a (35) After alulatio of the parameters, usig (3). T = a b t t the residuals are alulated The least squares solutio for the trasformatio parameters looks formidable, but it reall is er simple. The parameters for a D Coformal trasformatio a be omputed usig a poket alulator ad this solutio depeds o formig ol three umbers from a sstem of etroidal oordiates. Alteratiel, a simple omputer program spreadsheet (suh as Eel) ould be used. I the followig pages a eample of a D Coformal trasformatio as a aid to adastral re-establishmet will be disussed.

23 CONFORMAL TRANSFORMATION EXAMPLE Figure 8 shows a Pla of Subdiisio (LP48556) with distaes i liks ( hai = 00 liks = 66 feet) ad bearigs related to True North. The pla shows two Referee Marks (RM's), oe ear the south-west orer of Lot ad the other ear the south-east orer of the 00 lik wide aess to Lot. The subdiisio was reated ad marked o the groud i the 90's. TRUE N Distaes i liks 0 RM LP ROAD RM 0 00 Figure 8 Pla of Subdiisio LP48556 Figure 9 shows a Abstrat of Fieldotes of a reet sure oduted for the purposes of boudar re-establishmet prior to purhase of Lot, LP At the time of sure ol oe of the RM's alog the road was foud ad old pegs, thought to be origial, were foud at the south-west ad orth-east orers of Lot. Most of the feig was fairl reet, probabl replaig origial feig. The post at the orth-east orer of Lot, whih is ew, is er lose to the old peg whih ma hae bee disturbed whe the ew post was put i. The other old peg at the south-west orer of Lot appeared to be origial ad udisturbed. 3

24 00 N p & w p & w post ad wire.605 OP Fd A ABSTRACT OF FIELDNOTES DATUM A-B Distaes i metres post post ad wire post ad wire & wire post ad wire post ad wire OP Fd. p & w ROAD B p & w RM foud Figure 9 Abstrat of Fieldotes of sure of Lot, LP48556 The datum of the sure was the post A (south-west orer of Lot ) ad the RM B foud ear the south-east orer of the road aess to Lot. A traerse lie offset.00 m (0 liks) from the post at A ad passig through the RM was adopted for the bearig datum of For this eample we will perform a adastral re-establishmet usig a D Coformal trasformatio (sale, rotatio ad traslatios) with weights based o the RM ad the two old pegs of LP I light of the iformatio aboe, the RM will be gie a weight of 0, the old peg at the south-west orer of Lot will be gie a weight of 5 ad the other old peg (orth-east orer of Lot ) will be gie a weight of. 4

25 The parameters of the trasformatio (sale, rotatio ad traslatios) will be determied ad a ispetio of residuals will gie some idiatio as to the "orretess" of the re-establishmet. For the purposes of omputig the trasformatio parameters, two arbitrar oordiate sstems will be used. Oe sstem of oordiates, i metres, alled TITLE will hae alues of E ad N for the RM ear the south-east orer of the road aess to Lot. For the purpose of omputig the TITLE oordiates the origial dimesios i liks will be oerted to metres where hai = 00 liks = 66 feet, ad foot = metres (eatl) giig liks = metres. The origial dimesios of 000 liks, 3000 liks ad 578 liks will be oerted to metres (3 deimal plaes) ad the other dimesios "omputed to lose" ad oted to 4 deimal plaes. The road aess frotage will be deried b omputatio after oertig the 00 lik width to 0.7 metres. This omputatio proess should esure that oordiates are mathematiall orret to 3 deimal plaes. The other sstem of oordiates, also i metres, ad alled SURVEY will hae alues of E ad N for the RM foud. The traerse dimesios are mathematiall orret (to a millimetre) ad should ield SURVEY oordiates of traerse poits ad oupatio orret to 3 deimal plaes. 5

26 TRUE N Distaes i liks Distaes i metres RM E N E N 0.7 TITLE 0 00 ROAD RM E N Figure 0 TITLE oordiates (metres) TITLE CENTROIDAL TITLE POINT Desriptio Weight E N E N RM Old Peg a Old Peg etroid Coordiates of the etroid omputed usig equatio (7) ad etroidal oordiates alulated usig equatio (8). 6

27 00 ABSTRACT OF FIELDNOTES post DATUM A-B Distaes i metres N p & w p & w post ad wire OP Fd. 37 A O.P post ad wire post ad wire post & wire post post ad wire post ad wire OP Fd. p & w O.P. post SURVEY post ROAD B p & w RM foud E N RM post Figure SURVEY oordiates (metres) SURVEY CENTROIDAL SURVEY POINT Desriptio Weight U V U V RM Old Peg a Old Peg etroid Coordiates of the etroid omputed usig equatio (7) ad etroidal oordiates alulated usig equatio (8). 7

28 Usig the etroidal TITLE ( EN, ) ad etroidal SURVEY (, ) UV oordiates we a form the weighted etroidal oordiate produts wk ( Uk + Vk ) wk ( UkEk + VkNk) ad wk ( VkEk UkNk), that are used i equatios (33) ad (34) to ompute the parameters a ad b. These produts are POINT wk ( Uk + Vk ) wk ( UkEk + VkNk) wk ( VkEk UkNk) a sums The trasformatio parameters a ad b are ( + V N ) wk UkEk k k a = = = w U ( + V ) k k k ( U N ) wk VkEk k k b = = = w U ( + V ) k k k The traslatios t ad t are obtaied from equatios (36) as E N te = E au bv = t = N + bu av = N where the oordiates of the etroid i both sstems are gie i the tables below Figures 9 ad 0. Haig obtaied a, b, t ad t, residuals at the ommo poits a be obtaied from either equatio (0) or (30). E N These results, together with the trasformed oordiates are show o the Eel worksheet o the followig page. 8

29 LEAST SQUARES SOLUTION OF PARAMETERS OF D LINEAR CONFORMAL TRANSFORMATION Coformal Trasformatio Eerise D Liear Coformal Trasformatio (with weights) E = +a*u + b*v + t(e) E,N are oordiates i Sstem. N = -b*u + a*v + t(n) U,V are oordiates i Sstem. t(e) ad t(n) are East ad North traslatios. Rotatios are osidered positie ati-lokwise. U,V oordiates (Sstem ) are trasformed (saled, rotated ad traslated) ito E,N oordiates (Sstem ). SYSTEM (TITLE) Cetroidal Coords SYSTEM (SURVEY) Cetroidal Coords Residuals Weighted Cetroidal Coordiate Produts POINT E N E() N() U V U() V() Weight E N W*(U()^+V()^) W*(U()*E()+V()*N()) W*(V()*E()-U()*N()) a Cetroid Cetroid Sums LEAST SQUARES SOLUTION a = t(e) = Sale = b = t(n) = Rotatio = degrees (positie ati-lokwise) TRANSFORMED COORDINATES E = +a*u + b*v + t(e) N = -b*u + a*v + t(n) SURVEY TITLE Poit U V E N RM (Cotrol Poit) OP (Cotrol Poit) 7a OP (Cotrol Poit) post post post post 7b post post RM South West orer 9

30 CONFORMAL TRANSFORMATION WITH SCALE FACTOR OF UNITY For ertai purposes it ma be desirable to determie the trasformatio parameters from ommo poits with the oditio that the sale fator be equal to uit, i.e., s =. This a be ahieed b the followig: Let's sa that we are able to obtai a ad b from equatios (33) ad (34) a least squares solutio with weights. The sale s ad rotatio agle θ are obtaied from equatios (0) whih are re-stated here agai as s = a + b (37) θ b a = ta (38) If we diide both sides of equatio (37) b s we hae s a + b = = s s = a + b s a b = + s s If we defie a b a = ad b = (39) s s the trasformatio, gie b equatio (3), beomes a b u t = b a + t ad this trasformatio has a sale fator of uit, sie (40) ( a ) ( b ) + = Also, we ote that b b = sie the sale s will ael i the diisio so that the a a rotatio agle θ omputed from equatio (38) is the same whether we use the parameters a,b or ew parameters a, b from equatio (39). 30

31 It should be oted that the "ew" trasformatio, with sale fator of uit, gie b equatio (40), has traslatios t, t of equatio (3). The traslatios t ad ad these will be differet from the traslatios t t a b u = t b a are obtaied b re-arragig equatio (40) ad replaig, ad u, with the oordiates of the etroid u, giig t, t, t = a u b or (4) t = + b u a ad (4) After alulatio of the parameters, usig (3). T = a b t t the residuals are alulated We defie a, b, t, t as the parameters of a oformal trasformatio with a sale fator of uit. Usig the omputed data from the eample: a = , b = giig s = ad from equatios (39) a a = = = s b b = = = s The traslatios t ad t E N are obtaied from equatios (4) as t E = E a U b V = t = N + b U a V = N where the oordiates of the etroid i both sstems are gie i the tables below Figures 9 ad 0. Haig obtaied a', b', t ad t E N, residuals at the ommo poits a be obtaied from either equatio (0) or (30) b replaig a, b, t ad t with a', b', t ad t. E N These results, together with the trasformed oordiates (where the sale fator is uit) are show o the Eel worksheet o the followig page. 3

32 LEAST SQUARES SOLUTION OF PARAMETERS OF D LINEAR CONFORMAL TRANSFORMATION Coformal Trasformatio Eerise D Liear Coformal Trasformatio (with weights) E = +a*u + b*v + t(e) E,N are oordiates i Sstem. N = -b*u + a*v + t(n) U,V are oordiates i Sstem. t(e) ad t(n) are East ad North traslatios. Rotatios are osidered positie ati-lokwise. U,V oordiates (Sstem ) are trasformed (saled, rotated ad traslated) ito E,N oordiates (Sstem ). SYSTEM (TITLE) Cetroidal Coords SYSTEM (SURVEY) Cetroidal Coords Residuals Weighted Cetroidal Coordiate Produts POINT E N E() N() U V U() V() Weight E N W*(U()^+V()^) W*(U()*E()+V()*N()) W*(V()*E()-U()*N()) a Cetroid Cetroid Sums LEAST SQUARES SOLUTION a = t(e) = Sale = b = t(n) = Rotatio = degrees (positie ati-lokwise) TRANSFORMED COORDINATES E = +a*u + b*v + t(e) N = -b*u + a*v + t(n) SURVEY TITLE Poit U V E N RM (Cotrol Poit) OP (Cotrol Poit) 7a OP (Cotrol Poit) post post post post 7b post post RM South West orer LEAST SQUARES SOLUTION (SCALE FACTOR OF UNITY) a' = t'(e) = Sale = b' = t'(n) = Rotatio = degrees (positie ati-lokwise) TRANSFORMED COORDINATES E = +a'*u + b'*v + t'(e) N = -b'*u + a'*v + t'(n) SURVEY TITLE Poit U V E N RM (Cotrol Poit) OP (Cotrol Poit) 7a OP (Cotrol Poit) post post post post 7b post post RM South West orer 3

33 WEIGHTING SCHEMES Whe solig for the trasformatio parameters, obseratio equatios are formed there are equatios, where is the umber of ommo poits or otrol poits ad the least squares priiple leads to a set of ormal equatios [see equatios (4) ad (5)] that iole a (diagoal) weight matri W where the elemets of the leadig diagoal w, w, w 3, et. are kow as weights ad are usuall itegers (see page 9). High weights (large itegers) are assoiated with "strog" poits ad low weights (small itegers) assoiated with "weak" poits. This assoiatio ma be best eplaied b referee to the eample (see Figures 8 ad 9) rememberig that weights are ol assiged to otrol poits. The three otrol poits are the Referee Mark (RM foud) ear the S.E. orer of Lot, the old peg (OP) b the post at the N.E. orer of Lot ad the OP at the S.W. orer of Lot. Most sureors would probabl regard RM's ad pegs (if the hae ot bee disturbed) as er strog idiators of title orers (ia titleoetios i the ase of RM's). Pegs might be slightl less well regarded as the ould hae bee disturbed, ad fee posts or fee itersetios, would rak below that of pegs ad RM's as importat idiators of title orers. This would be a fairl ormal hierarh that a sureor would gai from eperiee. Assigig weights is merel puttig umbers ito the trasformatio proess that reflet that hierarh. I the eerise, the RM has bee assiged a weight of 0, the OP at the S.W. orer of Lot has bee assiged a weight of 5 ad the other OP at the N.E. orer of Lot has bee assiged a weight of. Perhaps here, the itetio is to gie less weight to the OP b the fee post, sie there is a possibilit that the peg ould hae bee disturbed b the feig otrator perhaps. These are arbitrar umbers ad are refletios of the sureor's field eperiee. Cotrol poits of high weight will hae smaller residuals tha otrol poits of low weight. You a adjust the magitude of the weights to gie a partiular poit (or poits) lower residuals tha other poits. It is iterestig to ote that i the Eel spreadsheet used to ompute the trasformatio parameters, assigig a weight of zero effetiel remoes that poit as a otrol poit. This meas that iitiall, all the oupatio (RM's, OP's, posts, et.) a be otrol poits i a iitial trasformatio ad the remoed from the proess b assigig a weight of zero to poits that hae large residuals; idiatig 33

34 that the oupatio is ot at, or ear, a title orer. This adds fleibilit to the reestablishmet aalsis. REFERENCES Alla, Arthur L., 997, Maths for Map Makers, Whittles Publishig, UK. Bebb, G., 98, 'The appliatios of trasformatios to adastral sureig', Iformatio- Ioatio-Itegratio: Proeedigs of the 3rd Australia Sure Cogress, Sde, Marh 8 April 3, 98, The Istitutio of Sureors Australia, pp Bellma, C. Deaki, R. ad Rolligs, N., 99, 'Colour photomosais from digitized aerial photographs', Lookig North: Proeedigs of the 34th Australia Sureors Cogress, Cairs, Queeslad, Ma 3-9, 99, The Istitutio of Sureors Australia, pp Beroets, S.G., 99, 'Shiftig ad rotatig a figure', Sure Reiew, Vol. 3, No. 46, Otober 99, pp Bird, D., 984, 'Letters to the Editors: Least squares reistatemet', The Australia Sureor, Vol. 3, No., Marh 984, pp Deaki, R. E., 998, '3D oordiate trasformatios', Sureig ad Lad Iformatio Sstems, Vol. 58, No. 4, De. 998, pp. 3-34) DSB, 97, Ditioar of Sietifi Biograph, Vol. VI, C.C. Coulsto Editor i Chief, Charles Sriber's Sos, New York. Helmert, F.R., 880, Die mathematishe ud phsikalishe Theorem der höhere Geodäsie, Vol., Die mathematishe Theorem, Leipzig. Helmert, F.R., 884, Die mathematishe ud phsikalishe Theorem der höhere Geodäsie, Vol., Die phsikalishe Theorem, Leipzig. Jorda/Eggert/Keissl, 963, Hadbuh der Vermessugskude (Bad II), Metzlershe Verlagsbuhhadlug, Stuttgart. Lauf, G.B., 983, Geodes ad Map Projetios, TAFE Publiatios, Colligwood. Mikhail, E. M., 976, Obseratios ad Least Squares, IEP A Du-Doelle, New York. Moffitt, F. H. ad Mikhail, E. M., 980, Photogrammetr, 3rd ed, Harper & Row, New York. Shmutter, B ad Dotsher, Y., 99, 'A ew method for mathig digitized maps', Tehial Papers 99 ACSM-ASPRS Aual Coetio, Baltimore, USA, Vol., Sureig, pp Sprott, J. S., 983, 'Least squares reistatemet', The Australia Sureor, Vol. 3, No. 8, Deember 983, pp