transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface

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1 . CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections), o a plane (azimuthal pojections). These sufaces ae imagined as enveloping o touching the datum suface and by some means, usually geometic, the meidians, paallels and featues ae pojected onto these sufaces. In the case of the cone, a plane containing the axis of the cone cuts the cone on a line joining the base and the apex. If the cone is cut along this line (a geneato of the cone) it can be laid flat (developed). If the axis of the cone coincides with the axis of the Eath, the pojection is said to be nomal aspect, if the axis lies in the plane of the equato the pojection is known as tansvese and in any othe oientation it is known as oblique. [It is usual that the descipto "nomal" is implied in the name of a pojection, but fo diffeent oientations, the wods "tansvese" o "oblique" ae added to the name.] This simplified appoach is not adequate fo developing a geneal theoy of pojections (which as we can see is quite mathematical) but is useful fo descibing chaacteistics of cetain pojections. In the case of conical pojections, some chaacteistics ae a common featue: (i) Meidians ae equally spaced staight lines adiating fom a cental point O. (ii) Paallels, in geneal, ae unequally spaced concentic cicles having a cente at O. (iii) Meidians of longitude and paallels of latitude fom an othogonal netwok of lines. Pojection suface cone tansfomation Y θ P Datum suface Eath V-cuve (meidian) λ X U-cuve (paallel) Figue. Conical pojection. u,v cuves on the datum suface pojected as U,V cuves on the pojection suface Figue. shows a schematic diagam of a conical pojection demonstating the basic chaacteistics common to all conical pojections (nomal aspect). Conical pojections have a geneal cicula shape when the whole of the Eath is displayed; the U-cuves (paallels of latitude) ae concentic cicles and the V-cuves (meidians of longitude) ae equally spaced adial lines. In Figue., the cente fo the cicula U-cuves and the adial V-cuves is the 3, R.E. Deakin Map Pojection Theoy (3) 4

2 pole, but this is not always the case. In some conical pojections, the pole is shown as a line. The oigin of the X,Y Catesian coodinates is shown at the intesection of a cental V-cuve (a cental meidian λ ) and a selected U-cuve (a paallel of latitude ). A point P is shown on the pojection with pola coodinates, θ whee is a adial distance fom the cente of the pojection, the oigin of the pola coodinate system and θ is an angle measued positive anticlockwise, negative clockwise fom the cental meidian. All pojection equations fo conical pojections ae given in tems of pola coodinates... The Gaussian Fundamental Quantities of Conical Pojections Fo conical pojections, the pojection suface is a plane and the U,V cuvilinea coodinate system is an othogonal system of U and V-cuves that ae concentic cicula acs of adius and staight adial lines at angles θ fom a cental V- cuve. The functional elationships connecting the U,V coodinate system with the X,Y,Z Catesian coodinate system wee given peviously by equations (.) and ae estated hee in moe explicit fom 3 (, ) (, θ ) (, ) (, θ ) (, ) X = F U V = F Y = F U V = F Z = F U V = (.) Refeing to Figue., the X,Y Catesian coodinates ae elated to the, θ pola coodinates by the equations X = sinθ Y = cosθ (.) whee is the adius of the U-cuve (a paallel of latitude) passing though the X,Y coodinate oigin. The Gaussian Fundamental Quantities of the pojection suface ae given by equations (5.3) and ae estated hee ecognising that U-cuves ae cicula acs of adius and V-cuves ae adial lines at angles θ fom a cental meidian and Z = E F G X Y X Y = + = + U U X X Y Y X X Y Y = + = + U V U V X Y X Y = + = + V V (.3) X X The patial deivatives, etc can be obtained fom (.) 3, R.E. Deakin Map Pojection Theoy (3) 43

3 X Y = sinθ = cosθ X Y = cosθ = sinθ (.4) Substituting equations (.4) into (.3) gives X Y E = + = sin θ + cos θ = X X Y Y F = + = sinθcosθ sinθ cosθ = X Y G = + = cos θ + sin θ = (.5) Note that F =, which eflects the fact that the -cuves and θ -cuves (U and V-cuves that ae concentic cicles and adial lines) intesect at ight angles. Now, consideing the datum suface to be a sphee of adius R and the u,v cuves as paallels and meidians, λ, the Gaussian Fundamental Quantities E,F,G elating the functional elationships X = f Y = f ae (, λ) (, λ) (.6) E F G X Y = + X X Y Y = + λ λ X Y = + λ λ (.7) Expessions fo E,F,G can be obtained fom the Tansfomation Matix (6.7) in the following fom E E θ θ θ θ F = + F λ λ λ λ G G λ λ λ λ (.8) Noting that has eplaced U, θ has eplaced V, has eplaced u and λ has eplaced v. 3, R.E. Deakin Map Pojection Theoy (3) 44

4 Now fom equations (.5) we have E =, F = and Tansfomation Matix (.8) gives G =, and substituting into the E F G = + = + λ λ = + λ λ (.9) These expessions can be simplified if the following conditions fo nomal aspect Conical pojections ae enfoced (these conditions can be "undestood" by inspection of Figue.) (i) f ( ) = i.e., the adius of a paallel of latitude (a U-cuve) on the pojection is a function of the latitude only and (ii) θ = n ( λ λ ) i.e., the pola angle θ (the angle between a V-cuve and the cental meidian) is a linea function of λ only. n is a scala quantity known as the cone constant. These two conditions mean that =, =, = n λ λ (.) Substituting these diffeential elationships into (.9) gives the Gaussian Fundamental Quantities E,F,G fo nomal aspect Conical pojections E =, F =, G = λ (.) Using these diffeential elationships and paticula scale conditions we can deive CONFORMAL, EQUAL AREA and EQUIDISTANT Conical pojections. The scale conditions ae: Fo CONFORMAL Conical pojections: E G m = = e g J EQUAL AREA Conical pojections: j = E EQUIDISTANT Conical pojections: e = 3, R.E. Deakin Map Pojection Theoy (3) 45

5 In addition, since the datum suface is a sphee of adius R, the Gaussian Fundamental Quantities of the datum suface ae e= R f = g = R cos (.).. Confomal Conical Pojections Fo a Confomal Conical pojection the scale condition to be enfoced is E G m = = (.3) e g Altenatively, using the notation fo meidian and paallel scale factos we may wite the scale condition as h = k (.4) whee E h = = (.5) e R and G n k = = = g Rcos λ Rcos (.6) Note that in equation (.6) the cone constant n = (.7) λ is used, see equations (.) and the conditions fo nomal aspect Conical pojections. Enfocing the scale condition h = k gives the diffeential elationship n = R Rcos and eaanging gives d = d (.8) cos Note: The minus sign is intoduced to eflect the fact that the adius inceases as the latitude deceases. Integating (.8) gives 3, R.E. Deakin Map Pojection Theoy (3) 46

6 whee lnc = ( C C of logaithms, log ) d = d cos π ln + C = nln tan + + C 4 π ln = nln tan + + lnc 4 is the natual logaithm of the constants of integation. Using the laws p a M ploga M Taking antilogaithms of both sides gives = and log MN = log M + log N we may wite a a a π ln = ln tan + lnc 4 + = ln C tan + 4 = C tan + 4 (.9) and fom condition (ii) above ( ) θ = n λ λ (.) Equations (.9) and (.) ae the geneal equations fo Confomal Conic pojections, but the constants n and C must be detemined. To detemine the constants n (the cone constant) and C, geometic constaints elating to standad paallels ae employed, emembeing that a standad paallel is defined as a paallel of latitude along which the scale facto is constant and equal to unity. 3, R.E. Deakin Map Pojection Theoy (3) 47

7 Conside the case of a single standad paallel = tangent length N = meidian distance tansfomation S Eath cone standad paallel λ Figue. Schematic diagam of a Conical pojection with a single standad paallel having a adius. Setting the scale facto k in equation (.6) equal to unity along a standad paallel gives k n = R cos = That can be eaanged to give an equation fo the cone constant n as n R cos = (.) Inspection of equation (.) shows that if the adius of the standad paallel is fixed then the cone constant n can be detemined. Refeing to Figue., the two choices fo fixing ae: (a) (b) Make the adius of the standad paallel equal to the tangent length of the cone. = Rcot (.) Make the adius of the standad paallel equal to the meidian distance on the Eath fom the pole to the tangent point of the cone. = R π (.3) 3, R.E. Deakin Map Pojection Theoy (3) 48

8 ... Confomal Conic Pojection with a single standad paallel (adius = tangent length) Pojection equations: = C tan + 4 θ = n λ λ ( ) (.4) Rcos Rcos Cone constant: n = = = sin (.5) Rcot To detemine the constant C, conside the adius = C + = tan 4 tan R C = tan tan + 4 n R (.6) Figue.3 shows a Confomal Conic pojection of the nothen hemisphee. The pojection has a single standad paallel whose adius is equal to the tangent length of the cone. The gaticule inteval is 5, cental meidian λ = 3 and the X,Y coodinate oigin is at = 45. A point P is shown whose coodinates ae P = and λ P = 9. The pojection paametes, scale of the pojection and the X,Y coodinates of P can be computed in the following way. λ and = C tan + 4 θ = n λ λ ( ) R = 637 m R = = 637. m tan n = tan + = 4 n = sin =.777 C m 3, R.E. Deakin Map Pojection Theoy (3) 49

9 = 45 o Y X equato P λ 9 o = λ = 3 o Figue.3 Confomal Conic pojection, single standad paallel = 45 (adius of standad paallel = tangent length of cone). Gaticule inteval 5, cental meidian λ = 3, X,Y coodinate oigin at λ and = 45 The scale of the pojection can be obtained fom the geneal elationship ( ) ( ) distance on pojection map scale = = distance on Eath Eath Fom measuements on Figue.3, the length = 4.5 mm and fom the pevious calculations = 637 mm on the Eath. Note that this is eally the tangent length of a cone touching the Eath at latitude = 45. The scale of the pojection is then ( ) ( ) map 4.5 scale = = = o :55 million Eath , R.E. Deakin Map Pojection Theoy (3) 5

10 The Catesian equations fo Conical pojections (of the nothen hemisphee) ae The adius X = sinθ Y = cosθ of the paallel is given by (.4) and since the Catesian oigin is at the intesection of the cental meidian λ and the standad paallel then =. Fo the point P at = and λ = 9 and with C and n fo the pojection = C tan m 4 = θ = n ( λ λ ) = X = m = 5. mm Y = m = 5.6 mm 3, R.E. Deakin Map Pojection Theoy (3) 5

ME 210 Applied Mathematics for Mechanical Engineers

ME 210 Applied Mathematics for Mechanical Engineers Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the