3D COORDINATE TRANSFORMATIONS - MyGeodesy
3D COORDINATE TRANSFORMATIONSR E DEAKINDepartment of Land InformationRMIT UniversityGPO Box 2476V MELBOURNE VIC 3001AUSTRALIAPhone: 61 3 9925 2213Fax: 61 3 9663 2517e-mail: deakin@rmit.edu.au(published in Surveying and Land Information Systems, Vol.58, No.4, Dec.1998, pp.223-34)ABSTRACTand control systems used in large manufacturingprojects such as the construction of the ANZACfrigates for the Australian and New Zealand Navies(Bellman & Anderson 1995) and in photogrammetry they are used in the orientation(interior and exterior) of aerial photographs. Intwo-dimensional (2D) form, transformations areused in cadastral survey re-establishments (Bebb1981, Sprott 1983 and Bird 1984), matchingdigitized cadastral maps (Shmutter and Doytsher1991) and "sewing together" the edges of strips ofdigital images (Bellman, Deakin and Rollings1992).A three-dimensional (3D) conformal coordinatetransformation, combining axes rotations, scalechange and origin shifts is a practical mathematicalmodel of the relationships between different 3Dcoordinate systems. Applications in geodesy andphotogrammetryoftenusesimplifiedtransformation models under the assumption ofsmall or negligible rotations, but in other areas ofinterest rotations may be large. In such other cases,approximate values of rotations are required toperform initial transformations before the simplifiedmodels are employed. This paper uses an exampleapplicable to the construction industry todemonstrate methods of calculating approximaterotations and performing initial transformationsprior to computing transformation parameters. Arigorous development and proof of the 3Dconformal transformation is given as well as thenecessary assumptions for the simplified model. Inaddition, this paper also explains how least squaresmay be used in determining transformationparameters.In general, the effect of a transformation on a 2Dor 3D object will vary from a simple change oflocation and orientation (with no change in shapeor size) to a uniform change in scale (no change inshape) and finally to changes of shape and size ofdifferent degrees of nonlinearity (Mikhail 1976).The most common transformations in surveyingapplications, and the only type dealt with in thispaper, are conformal, i.e. transformations thatpreserve angles and thus the shape of objects.Theory and applications of other coordinatetransformations, such as affine, polynomial,projective etc. can be found in Mikhail (1976) andMoffitt and Mikhail (1980).INTRODUCTIONCoordinate transformations, conformal andotherwise, are widely used in surveying and relatedprofessions.For instance, in geodesy, 3Dtransformations are used to convert coordinatesrelated to the Australian Geodetic Datum to thenew Geocentric Datum of Australia (Featherstone1996), in engineering they form part of monitoring1
3D conformal coordinate transformations alsoknown as similarity transformations (Blais 1972and Bervoets 1992) and in the 2D case, sometimescalled Helmert transformations after the Germangeodesist F.R. Helmert (1843-1917) are oftengiven in the formBAGUEN TE X E N λ R κ φ ω Y TN TU Z U D(1)TheOBJECTHE, N and U (East, North, Up) and X, Y and Z are3D "design" and "survey" coordinates respectively,λ is a scale factor, R κ φ ω is a rotation matrix (theDesignoriginEZproduct of rotations ω, φ and κ about the X, Yand Z axes in turn) and TE , TN and TU areU′N′κtranslations between the origins of the twocoordinate systems. In the context of this paper,(1) – often called a seven-parametertransformation, three rotations three translationsand one scale factor – represents the mathematicalrelationship between a constructed object in theXYZ survey system and its transformed position inthe ENU design system. Its practical use in theconstruction industry, where components ofstructures are manufactured "off-site" and broughttogether "on-site" can be explained in the followingway.CFφSurveyoriginωYE′XFigure 1. An object with coordinates in two 3Drectangular systemsThe seven parameters in (1) can be determined bysolving a system of equations derived from"common" points whose coordinate values areknown in both the survey and design systems.Each common point (or control point) generatesthree equations, thus a minimum of three suchpoints is required to solve for the parameters, but itis usual (and also good practice) to include extracontrol points in surveys of this type; the additionalpoints leading to a redundant system of equations(more equations than unknowns). Least squarescan then be employed to determine the bestestimates of the parameters which minimize thesum of the squares of the residuals (smallcorrections to the transformed coordinates) at thecontrol points. The residuals, three for each pointin the direction of the coordinate axes, can then beused as a measure of how well the constructedobject fits its design values.At the off-site location the object is measured insitu and coordinated in an arbitrary XYZ surveysystem. To confirm that it will fit in its designedon-site location its XYZ coordinates are scaled androtated until the coordinate axes are parallel withthe ENU design axes. The origins of the twocoordinate systems are then brought together byadding the translations. This has the effect ofsuperimposing the object over its design locationand its "fit" can be checked by comparingcoordinate differences.2
Translation parameters may be eliminated from thesolution by using "centroidal" coordinatesE , N and U and X , Y and Z having a commoniterative, requiring an initial transformation toapproximately align the two coordinate axes, asimple method of determining the parameters ofthe initial transformation is given and a workedexample of a problem is provided.origin at the centroid of the control points. Thetransformation then combines scale and rotationonly E X N λ R κ φ ω Y U Z 2D CONFORMAL TRANSFORMATIONS(2a)C.F. Gauss showed that the necessary andsufficient condition for a conformal transformationfrom the ellipsoid to the map plane is given by thecomplex expression (Lauf 1983)[Translations can also be eliminated by moving theorigins to a common control point, both techniqueshaving the desirable effect of reducing the size ofthe numbers involved in the computations.]y i x f (χ i ω)whereDue to the nature of the matrix R κ φ ω it is notfunctionf (χ i ω)isanalytic,containing isometric parameters χ (isometriclatitude) and ω (longitude). i is the imaginarynumber ( i 2 1 ) and the xy rectangularcoordinates have the y-axis "up the page" and thex-axis "across the page". [It should be noted herethat isometric means: of equal measure, and on thesurface of the ellipsoid (or sphere) latitude andlongitude are not equal measures of length. This isobvious if we consider a point near the pole wheresimilar distances along a meridian and a parallel oflatitude will correspond to vastly different angularvalues of latitude and longitude.Hence inconformal map projections, isometric latitude isdetermined to ensure that angular changescorrespond to linear changes.]possible to solve directly for the individualrotations ω, φ and κ . But if the coordinate axesare approximately parallel, a matrix RS ofindependent small rotations δω, δφ and δκ canbe developed, which together with an approximatevalue of the scale factor λ ′ with its small unknowncorrection δλ ( λ λ ′ δλ ) enables a leastsquares solution of δω , δφ , δκ and δλ using E X N ( λ ′ δλ ) RS Y U Z the(3)(2b)This solution technique requires iteration, eachiteration preceded by a transformation using (2b)with the process being terminated when δω, δφ ,A necessary condition for an analytic function isthat it must satisfy the Cauchy-Riemann equationsδκ and δλ reach some predetermined negligiblevalues. y x y x and χ ω ω χThis paper shows how the general principles ofconformal transformation, originally developed byC.F. Gauss (1777-1855), are used to derive atransformation between two plane rectangularcoordinate systems which is equivalent to 2Drotation, scaling and translation. This concept isextended to 3D systems and the derivation of (1)and (2) is set out together with a proof showingthat angles are preserved in the transformation.(4)Using this theorem, a conformal transformationfrom the XY rectangular coordinate system(isometric parameters) to the EN (East, North)rectangular system (also isometric parameters) isgiven by the complex expressionN i E f (Y i X)(5)A function f ( Y i X ) which satisfies the Cauchy-The application of the least squares principle to thesolution of the transformation parameters isexplained and the necessary system of equations isdeveloped in matrix form. Since the solution isRiemann equations is a complex polynomial, hence(5) can be given as3
N iE n (ak i bk )( Y i X )kN′(6)αY cos αExpanding (6) to the first power (k 1) andequating real and imaginary parts givesPNαX cos αE′TN(7)E b0 a1X b1YX sin αY sin αk 0N a0 a1Y b1XYXwhich are essentially the same equations as inJordan/Eggert/Kneissal (1963, pp. 70-73) in thesection headed "Das Helmertsche Verfahren(Helmertsche Transformation)" although as notedby Bervoets (1992) in his bibliography, there is noreference to the original source. It is probable thatHelmert developed this conformal transformationin his masterpiece on geodesy, Die mathematischenund physikalischen Theorem der höheren Geodäsie,(The mathematics and physical theorems of highergeodesy) on which he worked from 1877 andpublished in two parts: vol. 1, Die mathematischenTheorem (1880) and vol. 2, Die physikalischenTheorem (1884) [DSB 1972].ETEFigure 2. 2D Conformal Transformation(Rotation and Translation)Thus it is seen that the 2D conformaltransformation is equivalent to a translation androtation of rectangular axes with a scale factorbetween the EN and XY coordinates.THE 3D ROTATION MATRIXThe 2D transformation can be extended to threedimensions by firstly considering a sequence ofrotations of ω, φ and κ about the X, Y and Z-axesIn (7), a0 and b0 are translations between thecoordinate axes and the coefficients a1 and b1 canbe considered as functions of scale λ and rotationα between the coordinate axesa1 λ cos αin turn.Z′′ ( Z′′′ )Z′Z(8)b1 λ sin ακSubstituting (8) into (7), re-arranging and usingmatrix notation gives the familiar 2D conformaltransformation E cos α sin α X TE N λ sin α cos α Y T N Y′′′ω(9)Y′ ( Y′′ )X ( X′ )where the coefficient matrix on the right-hand-sideis the rotation matrix Rα cos α sin α Rα sin α cos α The 2D conformal transformationrepresented by the familiar diagramφYX′′(10)canX′′′Figure 3. 3D rotations ω, φ and κbeRotations are considered as positive anti-clockwisewhen looking along the axis towards the origin; thepositive sense of rotations being determined by theright-hand-grip rule where an imaginary right handgrips the axis with the thumb pointing in thepositive direction of the axis and the natural curl ofthe fingers indicating positive direction of rotation.4
The three rotations in order are:(i)Rκ φ ωRotation of ω about the X-axis. This rotatesthe Y and Z axis to Y ′ and Z ′ with the Xand X ′ axes coincident.where, for instance, c κ sφsω cos κ sin φ sin ω .Coordinates in the new system will be givenby the matrix equation00 X 1 X ′ Y ′ 0 cos ω sin ω Y 0 sin ω cos ω Z Z ′ (Rω )(ii)Rotation matrices, e.g. Rα , R κ , R φ , Rω andR κ φ ω are orthogonal, i.e. the sum of squares of theelements of any row or column is equal to unity.They have the unique property that their inverse isequal to their transpose, i.e. R 1 R T which willbe used in later developments.(11)Rotation of φ about the new Y ′ axis. Thisrotates the X ′ and Z ′ to X ′′ an d Z ′′ withTHE 3D CONFORMAL TRANSFORMATIONthe Y ′ and Y ′′ axes coincident.The 3D conformal coordinate transformation is anextension of the 2D case; (14) and (15) combinedwith a scale factor and translations to give (aspreviously stated)Coordinates in the new system will be givenby the matrix equation cos φ 0 sin φ X ′ X ′′ 0 Y′ Y ′′ 0 1 sin φ 0 cos φ Z ′ Z ′′ (R φ )(iii) TE X E N λ R κ φ ω Y TN TU Z U (12)(1)To prove that this transformation is indeedconformal, i.e. angles between points in the XYZsurvey system are preserved when transformed intothe ENU design system, consider the following.Rotation of κ about the new Z ′′ axis. Thisrotates the X ′′ and Y ′′ to X ′′′ and Y ′′′ withthe Z ′′ and Z ′′′ axes coincident.(i)Coordinates in the new system will be givenby the matrix equation cos κ sin κ 0 X ′′ X ′′′ Y ′′′ sin κ cos κ 0 Y ′′ 0 Z ′′′ 01 Z ′′ R( κ) c φc κ c ω sκ sω sφc κ sω sκ c ω sφc κ c φsκ c ω c κ sω sφsκ sωc κ c ω sφsκ (15) sφ sω c φcωc φ (13)Let three points a, b and c in the surveysystem be transformed into A, B and C inthe design system; points in both systemslocated by vectors a, b, c and A, B, C.Using (1) with R as the rotation matrix and tas the vector of translations, we may writethe transformation of a to A asThe coefficient matrices R κ , R φ , Rω above areA λRa t3D rotation matrices which can be multipliedtogether (in that order) to give another rotationmatrix R κ φ ω (Mikhail and Moffitt 1980).and similarly for the other vectors. X ′′′ X X Y ′′′ R κ R φ Rω Y R κ φ ω Y (14) Z ′′′ Z Z with5(16)
CUZc uθNθ′U AE(b)vLet θ be the angle at a in the survey systembetween the vectors u b a andv c a ; and θ′ the complementary angleat A in the design system between vectorsU B A and V C A .V Rvcos θ′ ( Ru) Rv u T R T Rv u T vT(21)Comparing (17) and (21) shows that θ θ′ andconstitutes a proof that angles are preserved by thetransformation (1). [Baetslé (1966) has a similarproof but in a slightly different form.]components of the two unit vectors, notinguthat a unit vector is defined as u uConformal mapping has a long mathematicalhistory. Vlcek (1966) in a discussion paper on thetopic notes that the famous French mathematicianLiouville (1847) determined all the conformaltransformations in an analytical way and in atheorem bearing his name showed that the onlyways of conformal representation of space on itselfare:where u is the magnitude.The dot product is equivalent to a matrixmultiplication, hence(17)In the design system we may use (16) towrite1.U B A λ R b t (λ R a t) λ R u2.by translation and rotation, accompaniedby constant magnification,by inversion with respect to the sphere,which, at the very least, demonstrates thatconformal 3D transformations have been with usfor a long time.and the unit vector URuλRu λRuRu(20)and (20) we may writewhere u 1 , u 2 , u 3 and v1 , v 2 , v3 are theU (19)In the design system the angle θ′ between Uand V is given by the dot productcos θ′ U V U T V and using (19)(v)In the survey system, the angle between thevectors is found from the vector dot productof unit vectors u and vcos θ u v u T vRuu R Ruuuand using similar reasoningcos θ u v u 1v1 u 2 v 2 u 3 v3(iv)Tcan be substituted into (18) to giveFigure 4. Conformal transformationof two vectors.(iii)2YU (ii)Ru ( Ru) Ru u T R T Ru u T uEquating (a) and (b) gives Ru u which aX2u uT uand remembering that for orthogonalrotation matrices R 1 R T , henceR T R I where I is the identity matrixVB b (a) (18)Now using matrix algebra we may write tworesults (a) and (b)6
THE 3D ROTATION MATRIX FOR SMALL ANGLESCENTROIDAL COORDINATESFor small angles δω , δφ , δκ the rotation matrixIn a solution for the transformation parameters, thethree translations TE , TN and TU can beR κ φ ω may be simplified by the approximationseliminated by adopting a system of "centroidal"coordinates, i.e. a system of coordinates whoseorigin is at the centroid of the n control points.cos δω 1sin δφ δφ (radians)sin δκ sin δω 0Denoting the coordinates of the centroid (in bothsystems) with a subscript gand (15) becomes the anti-symmetric (or skewsymmetric) matrix (Harvey 1986).δκ δφ 1 δω RS δκ 1 δφ δω 1 X1 X 2 X nnY1 Y2 YnYg nZ1 Z 2 Z nZg nXg (22)It should be noted that RS is no longer orthogonalbut its inverse will, nevertheless, be given by itstranspose ( RS 1 RST ), since it is the approximateand similarly for E g , N g and U g .(24)Centroidalcoordinates (denoted with an over-bar) areform of the orthogonal matrix R κT φ ω (Hotine 1969,X i X i Xgp. 263).Yi Yi YgIn the least squares development to follow it isuseful to split RS (the rotation matrix for smallangles) into two parts(25)Zi Zi Zgwith similar expressions for E , N and U .δκ δφ 1 0 0 0 RS I δR 0 1 0 δκ 0δω (23) 0 0 1 δφ δω 0 Thus, using centroidal coordinates [see (2), wherethe origins of both systems are common] thetransformation is reduced to a combination of scaleand rotation only and the size of the numbersinvolved in the computations is reduced. Aftersolving for the scale factor λ and the elements ofthe rotation matrix R κ φ ω , the translations can beAssuming small angles is a convenient andpractical technique of simplifying R κ φ ω (whosenon-independent elements are functions of therotation angles ω , φ and κ ) to RS (whosefound by substituting the coordinates of thecentroid into (1) and re-arrangingindependent elements are δω , δφ and δκ ), thusenabling the solution of δω, δφ and δκ via a Eg X g TE TN N g λ R κ φ ω Yg U g Zg TU system of linear equations (see the followingsection on least squares solution). Solutions, basedon this assumption, will not be theoretically correctbut will be "practically correct" if an iterativeprocess is adopted where each n th iteration ispreceded by a transformation using previouslyderived values, and the process terminated whenδω n , δφ n and δκ n converge to negligible values.(26)LEAST SQUARES SOLUTION OF TRANSFORMATIONPARAMETERSA practical test on the assumption of small anglesin the initial transformation will be revealed bynoting whether the solution converges.To develop the least squares solution of theparameters, consider (2b) which pre-supposes thatthe rotation angles are small which may be due to(i) an initial transformation which approximatelyaligns the coordinate axes, or (ii) a knowledge that7
the rotations are small. Bearing this in mind(noting that a method of determining theparameters of an initial transformation is discussedlater) we may use (23) to write for each of the ncontrol points an equation of the form E N U δω δφ x , δκ δλ X vE ( λ ′ δλ ) ( I δR) Y v N (27) Z vU Equation (29a) is a "standard" least squares form(Mikhail 1976)v Bx fwhere vE , v N and vU are residuals at the control( 1(30a)B T Wfwhere W is the weight matrix associated with theobservations, which in this case are the triplets ofcentroidal coordinates X i , Yi , Z i and E i , N i , U i .Mikhail (1976 pp. 64-66) defines weight matricesW, cofactor matrices Q and variance-covariancematrices Σ as follows(28) X λ′ Y Z An equation of the form of (28) can be written foreach control point and represented symbolically inpartitioned matrix form as v1 B1 f1 v B f 2 2 x 2 v n Bn fn )x BT WBarranged as E N U (29b)with the solution for the parameters aspoints. Now, since the rotation angles and δλ aresmall, their products will be negligible( δλ δR 0 ), thus (27) can be expanded and re- δω 0 Z Y X vE δφ 0 X Y vN λ ′ Z δκ Y X vU Z0 δλ Ei λ′ Xi fi N i λ ′ Yi U i λ ′ Z i W Q 1(30b)Σ σ 02 Q(30c)where the elements of Q are estimates of thevariances and covariances of the observations andσ 02 is the reference variance whose posterioriestimate is (Mikhail 1976, p. 288)(29a)σ 02(TTTv T Wv f Wf x B Wf n un u)(30d)n and u are the number of observations andunknown parameters respectively.where each component v i ( i 1, 2, , n ) of thevector of residuals v and Bi of the matrix ofcoefficients B isThe partitioned form of W in (30a) is 0 ZiYi X i vE i 0 X i Yi v i v N i , Bi λ ′ Z i Yi X i vU i 0 Z i W1 0 0 W2W 0 0and the vector of parameters x and thecomponents fi of the vector of numeric terms fare0 0 Wn (30e)where each diagonal element Wi contains weightsassociated with the coordinate triplets of each i thcontrol point, and the elements of Wi must beobtained from a pre-analysis of the surveytechniques used to determine their values. Theoff-diagonal elements of W are null matrices,8
indicating that the observations (coordinatetriplets) are considered to be independent of eachother.(ii)In this paper, and the example following, it isassumed that the coordinates (derived from surveymeasurements) are independent of each other andall having equal precision. This allows W to bereplaced by the identity matrix I in (30a) to give()x BT B 1cos θ a b a1b1 a2 b 2 a3 b3components of the two unit vectors.A unit vector p , perpendicular to the planecontaining a and b, can be obtained fromthe vector cross productCALCULATION OF APPROXIMATE ROTATIONSp The solution above requires that rotations be small,i.e. RS approximates R κ φ ω . To ensure this awherepreliminary (or initial) transformation is madeusing centroidal coordinates and approximationsfor the rotation matrix R A and scale factor λ A X Y Z IN ITIAL X λ A RA Y Z O RIGIN ALb3 a1 b1 a3sin θp3 b1 a2 b 2 a1sin θq a p q1 i q 2 j q 3 kq 1 a 2 p 3 a3 p 2q 2 a3 p1 a1 p 3q 3 a1 p 2 a2 p1Thus q , a and p are the unit vectors ofanother orthogonal centroidal coordinatesystem ξ( Xi) , η( Et a) and ζ( Zeta) .ζA ηv1vv, v 2 2 , v3 3rrrv12 v 22 v32is(35)wherevectors in the directions of the X, Y and Zaxes respectively) the unit vectorv v1 i v 2 j v3 khasv vv r p2 unit vector q perpendicular to both a andpcomponents are v1 , v 2 , v3 ( i , j , k being unitwhereb 2 a3 b 3 a 2sin θ(34)A second cross product (a p) gives a thirdShifting both the survey and design systemsto the centroid and choosing two controlpoints A and B gives the vector pairs a andb in the survey system, and A and B in thedesign system whose components are thecentroidal coordinates of the respectivepoints. Each vector can be reduced to a unitvectorusingthefollowing:ifv v1 i v 2 j v3 k is a vector whosecomponents v1 b a p1 i p 2 j p 3 ksin θp1 (32)An approximate vale of λ A can be obtained byratios of distances in the survey and design systems,but for most practical applications will be very closeto unity. The approximate rotation matrix R A canbe derived in the following manner.(i)(33)where a1 , a2 , a3 and b1 , b 2 , b3 are the(31)BT fIn the survey system, the angle θ betweenthe vectors is found from the dot productξ p athe qCentroidmagnitude.9
Figure 5. ξ ηζ coordinate system(iii)where the left-hand-side of (39) can beconsidered as an initial transformation of thesurvey coordinates.Thus with anapproximate rotation matrix R A R 2T R1The ξ ηζ and X Y Z centroidal systems,are rotated with respect to each other, theξ -axis making angles α1 , β1 , γ 1 with theand the approximate scale factor λ A wehave (32) as given above whereX , Y and Z -axes respectively. Similarly,the η and ζ -axes make angles α 2 , β 2 , γ 2and α 3 , β3 , γ 3 . The elements of q are theRA direction cosines cosα1 , cosβ1 and cos γ 1R 2T(Wolf 1974) andξ X cos α1 Y cos β1 Z cos λ1the approximate rotations ω A , φ A and κ A can becalculated fromξ X q1 Y q 2 Z q 3(36b)tan κ A R 21R 11(41)cos φ A R 11cos κ ′(42)cos ω A R 33 cos κ ′R 11(43)Similarly for the elements of a and p wemay writeη X a1 Y a 2 Z a 3ζ X p1 Y p2 Z p3(36c)Combining (36b) and (36c) in the form of arotation matrix R1 gives X q1 q 2 q3 X ξ a a a R 1 Y 1 2 3 Y η Z p1 p2 p3 Z ζ An example of the computation of approximaterotations is contained in the Appendix.(37)CONCLUSIONReplacing survey system vectors a and bwith design system vectors A and B in steps(ii) and (iii) yields unit vectors Q , A and P ,This paper has presented the necessary equationsand computation technique for the practical use of3D conformal transformations in surveymeasurement. These transformations, combinedwith least squares, can be usefully employed onlarge construction projects where components,often manufactured "off-site", must be broughttogether "on-site" to fit within a design coordinatesystem. The least squares transformation methodcan be used to compare the off-site component,measured in situ in XYZ survey coordinates, withits proposed location in the on-site ENU designcoordinates; comparison being via the coordinateresiduals at common points. This pre-analysis"tool" can be used to prevent costly (and oftenembarrassing) misalignment of components in alarge engineering structure.and a second rotation matrix R 2 Q1 Q 2 Q3 E E ξ η A1 A 2 A3 N R 2 N P P P U U ζ 23 1(v)(38)Re-arranging (38) and substituting into (37),using the orthogonal property of rotation()matrices R 2 1 R 2T , gives E N U (40)R A has the same form as R κ φ ω (15) and values of(36a)or(iv) R 11 R 12 R 13 R1 R 21 R 22 R 23 R 31 R 32 R 33 R 2T X R1 Y Z (39)10
An example computation of the transformationparameters and residuals of a simple plane figureABC with measured XYZ coordinates and ENUdesign coordinates is provided in the Appendix.Harvey, B.R. 1986, 'Transformation of 3D coordinates', The Australian Surveyor, Vol. 33, No.2, June 1986, pp. 105-125.REFERENCESHelmert, F.R. 1880, Die mathematischen undphysikalischen Theorem der höheren Geodäsie,Vol. 1, Die mathematischen Theorem, Leipzig.Baetslé, P.L. 1966, 'Conformal transformations inthree dimensions', Photogrammetric Engineering,Vol. 32, No. 5, September 1966, pp. 816-824.Helmert, F.R. 1884, Die mathematischen undphysikalischen Theorem der höheren Geodäsie,Vol. 2, Die physikalischen Theorem, Leipzig.Bebb, G. 1981,'The applications oftransformationstocadastral surveying',Information-Innovation-Integration: Proceedings ofthe 23rd Australian Survey Congress, Sydney,March 28 – April 3, 1981, The Institution ofSurveyors Australia, pp. 105-117.Hotine, M. 1969, Mathematical Geodesy, ESSAMonograph 2, United States Department ofCommerce, Washington, D.C.Jordan/Eggert/Kneissl, 1963,Handbuch derVermessungskunde (Band II), MetzlerscheVerlagsbuchhandlung, Stuttgart.Bellman, C. and Anderson, L. 1995, 'Close rangephotogrammetry for dimensional control inshipbuilding', SAMS'95: Proceedings of the 3rdSymposium on Surveillance and MonitoringSurveys, Melbourne, Australia, November 1-2,1995, ed, M.R. Shortis and C.L. Ogleby,Department of Geomatics, University ofMelbourne, pp. 1-8.Lauf, G.B. 1983, Geodesy and Map Projections,TAFE Publications, Collingwood.Liouville, J. 1847, 'Note au sujet de l'articleprecedent', Journal de Math. pures et appl., Vol.12.Bellman, C. Deakin, R. and Rollings, N. 1992,'Colour photomosaics from digitized aerialphotographs', Looking North: Proceedings of the34th Australian Surveyors Congress, Cairns,Queensland, May 23-29, 1992, The Institute ofSurveyors Australia, pp. 481-495.Mikhail, E.M. 1976,Observations and LeastSquares, IEP A Dun-Donelley, New York.Moffitt, F.H. and Mikhail, E.M. 1980,Photogrammetry, 3rd ed, Harper & Row, NewYorkBervoets, S.G. 1992, 'Shifting and rotating afigure', Survey Review, Vol. 31, No. 246,October 1992, pp. 454-464.Shmutter, B and Doytsher, Y. 1991, 'A newmethod for matching digitized maps', TechnicalPapers 1991 ACSM-ASPRS Annual Convention,Baltimore, USA, Vol. 1, Surveying, pp. 241-246.Bird, D, 1984, 'Letters to the Editors: Least squaresreinstatement', The Australian Surveyor, Vol. 32,No. 1, March 1984, pp. 65-66.Blais, J.A.R. 1972, 'Three-dimensionalsimilarity', The Canadian Surveyor, Vol. 26, No.1, March 1972, pp. 71-76.DSB, 1972, Dictionary of Scientific Biography, Vol.VI, C.C. Coulston Editor in Chief, CharlesScribner's Sons, New York.Featherstone, W.E. 1996, 'An updated explanationof the Geocentric Datum of Australia and itseffects upon future mapping', The AustralianSurveyor, Vol. 41, No. 2, June 1996, pp. 121130.11
Sprott, J. S. 1983, 'Least squares reinstatement',The Australian Surveyor, Vol. 31, No. 8,December 1983, pp. 543-556.matrixRκ φ ωwithvaluesκ 10 ,φ 94 , ω 310 then (iii) rounding thetransformed centroidal coordinates to thenearest 0.1m and "adding back"E g , N g and U g .Vlcek, J. 1966, 'Discussion paper: Simultaneousthree-dimensional transformation',Photogrammetric Engineering, Vol. 32, No. 2,March 1966, pp. 178-180.The solution for the transformation parameters isaccomplished in the following steps.Wolf, P. R. 1974, Elements of Photogrammetry,McGraw–Hill, New York.Step 1:APPENDIXShift the origin of the survey system tothe centroid by subtracting X g , Yg andZ g from the coordinates of A B and CCalculation of transformation parameters betweensurvey and design locations of the figure ABC.then calculate the unit vectors a and b .XYZSurvey system X, Y, ZBA1000.0 X1000.0 Y1000.0 ZA-240.00020.000120.0001620.0 X740.0 Y340.0 ble 1. Survey system centroidal coordinates 1240.0 X gCentroid 980.0 Y g880.0 Z gCa 240 i 20 j 120 ka 0.891953 i 0.074329 j 0.445976 kb 380 i 240 j 540 k1100.0 X1200.0 Y1300.0 Zb 0.540874 i 0.341605 j 0.768610 kStep 2:Use the vector dot product (33) tocalculate angle θ between a and bDesign system E, N, UBA1911.9 E1435.2 N554.1 U a b
projects such as the construction of the ANZAC frigates for the Australian and New Zealand Navies (Bellman & Anderson 1995) and in photo-grammetry they are used in the orientation (interior and exterior) of aerial photographs. In two-dimensional (2D) form, transformat
Geometric transformations Geometric transformations will map points in one space to points in another: (x',y',z') f(x,y,z). These transformations can be very simple, such as scaling each coordinate, or complex, such as non-linear twists and bends. We'll focus on transformations that ca
Geometric Transformations Changes in size, shape are accomplished with geometric transformation. It alter the coordinate descriptions of object. The basic transformations are Translation, Roatation, Scaling. Other transformations are Reflection and shear.Basic transformations u
Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .
Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .
8: IIR Filter Transformations Continuous Time Filters Bilinear Mapping Continuous Time Filters Mapping Poles and Zeros Spectral Transformations Constantinides Transformations Impulse Invariance Summary MATLAB routines DSP and Digital Filters (2019-6707) IIR Transformations: 8 - 2 / 10
Four coordinate systems in navigation system are considered: global coordinate system, tractor coordinate system, IMU coordinate system and GPS-receiver coordinate system, as superscripts or subscripts YKJ, t, i and GPS. Tractor coordinate system origin is located at the ground level of the tractor, under the center point of the
EXAMPLE 1 coordinate proof GOAL 1 Place geometric figures in a coordinate plane. Write a coordinate proof. Sometimes a coordinate proof is the most efficient way to prove a statement. Why you should learn it GOAL 2 GOAL 1 What you should learn 4.7 Placing Figures in a Coordinate Plane Draw a right triangle with legs of 3 units and 4 units on a .
vRelease Version July 2019 CUDA Runtime API API Reference Manual
