Proposed single-zone map projection system for Turkey

In Turkey, base maps in the scale of 1:25,000 are produced in Gauss–Kruger (GK) mapping system with 6° zones. On the other hand, large-scale cadastral maps in the scale of 1:1000 and standard topo-graphic maps in the scale of 1:5000 and 1:10,000 are produced in GK mapping system with 3° zones (LSMMIPR, 2020).

For Turkey, while the central meridians of GK projection with 3°-wide zones (MUTM) are 27°, 30°, 33°, 36°, 39°, 42° and 45°, these central meridians are 27°, 33°, 39° and 45° for GK projection with 6°-wide zones (UTM). There are four zones for the UTM system and the numbers of these zones are 35, 36, 37 and 38. In order to avoid misunderstandings, zone numbers are added before the easting value in UTM coordinates (Figure 1).

Figure 1

In engineering projects that do not fit into a single zone, if there are overlapping parts with the neighbour zones in their 0.5° or 1° overlap area, the coordinate system is obtained by doing a zone transformation. If the application area overflows the overlap area, zone transformation would not be a suitable process. In this situation, it is not proper to work with a single coordinate system. A transformation from the UTM to different coordinate systems has to be done. The transformation and switching to other coordinate systems cause errors and loss of time because the users do not possess sufficient information. Therefore, the UTM is not sufficient for engineering projects that do not fit into a single zone. Hence, different representation methods should be used for these projects. However, the methods that implement different approaches other than the classical zone width approach have taken their place in the literature (Yildirim, 2004).

Given Turkey's geodetic location and considering the origin of a single coordinate system, projection zone width is increased (l ± 3.5°) UTM, Composite (CP), LCC2 and Double (DP). 585 test points have been selected in GRS80 ellipsoid and ITRF96 datum reference surface by creating a geographical grid network (0.5° × 0.5°) within the geographical borders of Turkey (B = [35.5°N–42.5°N]; L = [26°E–45°E]). In order to distribute the distortions that occur in all projections evenly, the starting coordinate has been taken as the centre of the geodetic boundaries (B₀ = 39°N; L₀ = 35.5°E). Thus, it has been aimed to examine the distortions that may occur in the border regions as they move away from the beginning. Common formulas have been used for the arc to chord (T – t) and distance (S – s) distortions in each projection. For these distortions, each test point has been chosen as a starting point and 45° azimuth angle and 1-km edges have been produced from this point. The coordinates of the second point of these edges have been calculated with the help of the direct solution of geodetic (Vincenty, 1975). After the calculation of projection distortions, arc to chord (T – t) map and distance (S – s) map correction values have been calculated with the help of projection coordinates. Nomenclature of variables used in projection algorithms has been given in the list below:

Inspired by the Mercator projection that had been developed in the 16th century, J. H. Lambert improved the Transversal Mercator (TM) projection for sphere in the 17th century. TM has been developed by C. F. Gauss in the early 19th century, but published by Schreiber and Kruger for ellipsoid. Therefore, TM is also used as GK. Later, ellipsoid formulas were re-examined by Hristow and the univariate and bivariate power series used today were obtained. The current usage type (Easting and Northing) of UTM was developed by the US Army in 1947. Many countries worldwide use the UTM system in mapping applications. Due to the increase in distortion in the east–west direction of the UTM system, 6° zone meridians have been used. In addition, UTM can be modified for large-scale maps and 3° zone meridians can be used regionally (Deakin et al., 2010; Snyder, 1987).

The formulas used in UTM could only be used for the maximum l = ±3.5° from the initial longitude chosen in the 19th century due to the lack of computational techniques. However, with the development of mathematical computer software for numerical computing (MATLAB, Maxima, Mathematica, Maple, etc.), new solutions (Bermejo-Solera and Otero, 2009; Bowring, 1990, 1993a,b; Deakin et al., 2010; Engsager and Poder, 2007; Grafarend, 1995; Guo et al., 2020; Ingwersen, 1996; Karney, 2011; Klotz, 1993; Lee, 1962, 1963; Mittermayer, 1993; Thompson, 1975; Turiño, 2004, 2008) have been developed for l > ±3.5° by increasing the number of coefficients in UTM. The coefficients used in the calculations are presented ready to the user. Mittermayer's method has been preferred in this study as it provides ease of calculation. The correlations in this method are given below: (1) xmit=G90[q¯+A(q¯,l¯)] {x_{mit}} = {G_{90}}\left[ {\overline q + A(\overline q,\overline l)} \right] (2) ymit=G90[l¯+B(q¯,l¯)] {y_{mit}} = {G_{90}}\left[ {\overline l + B(\overline q,\overline l)} \right] (3) A(q¯,l¯)+iB(q¯,l¯)=∑k=0n¯=9bk(q¯+il¯)k A\left({\overline q,\overline l} \right) + iB\left({\overline q,\overline l} \right) = \sum\limits_{k = 0}^{\overline n = 9} {b_k}{(\overline q + i\overline l)^k} (4) n¯=n−12, n=19 \overline n = {{n - 1} \over 2},\quad n = 19 (5) q¯=q¯(q,l)=2πarctan(sinhqcosl) \overline q = \overline q \left({q,l} \right) = {2 \over \pi}\arctan \left({{{\sinh q} \over {\cos l}}} \right) (6) l¯=l¯(q,l)=2πarctanh(sinlcoshq) \overline l = \overline l \left({q,l} \right) = {2 \over \pi}\arctan {\rm{h}}\left({{{\sin l} \over {\cosh q}}} \right) where G₉₀ symbolises the meridian arc length between 0° and 90° latitudes. Meridian convergence and differential scale factor used in determining distortions are calculated with the help of the following equations: (7) γ=γ(q,l)=arctan(1+C)tanl tanhq−DD tanl tanhq+(1+C) \gamma = \gamma \left({q,l} \right) = \arctan {{\left({1 + C} \right)\tan l{\kern 1pt} \tanh q - D} \over {D{\kern 1pt} \tan l{\kern 1pt} \tanh q + (1 + C)}} (8) m=m(q,l)=EG90NcosB(1+C)2+D2 m = m\left({q,l} \right) = {{E{G_{90}}} \over {N\cos B}}\sqrt {{{(1 + C)}^2} + {D^2}} (9) E=2πcosl coshqcosh2q−sin2l E = {2 \over \pi}{{\cos l{\kern 1pt} \cosh q} \over {\mathop {\cosh}\nolimits^2 q - \mathop {\sin}\nolimits^2 l}} (10) C(q¯,l¯)+iD(q¯,l¯)=∑k=0n¯=9(2bk+1)(q¯+il¯)2k C\left({\overline q,\overline l} \right) + iD\left({\overline q,\overline l} \right) = \sum\limits_{k = 0}^{\overline n = 9} (2{b_k} + 1){(\overline q + i\overline l)^{2k}}

The b_k coefficients in the Equation (3) and (10), are given in Table 1, in the range of 0–9. The geodetic boundaries of convergence of distortion and meridional in Turkey obtained using this method are shown in Figure 2.

Figure 2

J. H. Lambert developed a single and double standard parallel conical projection for both sphere and ellipsoid in the 17th century. Double standard parallel (B₁; B₂) LCC2 projection is preferred to work with a single coordinate system in the entire country in order to reduce the distortions that occur from the selected B₀ towards north and south in the x-axis direction. Projection coordinate data are calculated with the help of the following equations: (11) xlcc2=R0¯−R¯cosl′ {x_{lcc2}} = \overline {{R_0}} - \overline R \cos l{'} (12) ylcc2=R¯sinl′ {y_{lcc2}} = \overline R \sin l{'} (13) l′=lδ l{'} = l\delta (14) l′=δ(L−L0) l{'} = \delta (L - {L_0}) (15) R¯=R0¯e_[−δ(q−q0)] \overline R = \overline {{R_0}} {\underline e ^{[ - \delta (q - {q_0})]}} (16) R0¯=N1cosB1δe_[δ(q1−q0)]=N2cosB2δe_[δ(q2−q0)] \overline {{R_0}} = {{{N_1}\cos {B_1}} \over \delta}{\underline e ^{[\delta ({q_1} - {q_0})]}} = {{{N_2}\cos {B_2}} \over \delta}{\underline e ^{[\delta ({q_2} - {q_0})]}} (17) δ=ln(N1cosB1)−ln(N2cosB2)q2−q1 \delta = {{\ln \left({{N_1}\cos {B_1}} \right) - \ln \left({{N_2}\cos {B_2}} \right)} \over {{q_2} - {q_1}}}

The meridian convergence and differential scale factor used in determining the distortions are calculated with the help of the following equations (Bugayevskiy and Snyder, 1995; Grossmann, 1976; Hooijberg, 2012; Thomas, 1952; Yang et al., 1999): (18) γ=l′=lδ \gamma = l{'} = l\delta (19) m=N1cosB1NcosBe_[−δ(q−q1)]=N2cosB2NcosBe_[−δ(q−q2)] m = {{{N_1}\cos {B_1}} \over {N\cos B}}{\underline e ^{[ - \delta (q - {q_1})]}} = {{{N_2}\cos {B_2}} \over {N\cos B}}{\underline e ^{[ - \delta (q - {q_2})]}}

The selection of standard parallels varies according to the shape and size of the study area to be projected. Therefore, the developed equations are as follows: (20) B1=Bs+Bn−Bsk, B2=Bn+Bn−Bsk {B_1} = {B_s} + {{{B_n} - {B_s}} \over k},\quad \quad {B_2} = {B_n} + {{{B_n} - {B_s}} \over k} where B_s and B_n symbolise the southern and northern latitude boundaries of the study area respectively. Also, k is a coefficient that changes according to the shape of the region. If the region is rectangular, k = 5; if it is a circle, k = 4; and if it is close to a rhombus, k = 3 (Bugayevskiy and Snyder, 1995). Standard parallels B₁ = 37°30′N; B₂ = 40°30′N and k = 5 are taken for Turkey. Calculated distortions and meridian convergence are shown in Figure 3.

Figure 3

The CP transcribes the map's geometry to scale, to the map's height-to-width ratio and to the central latitude of the shown area by replacing projections and adjusting their parameters. The CP indicates the entire globe including poles; it portrays continents or larger countries with less distortion and it can morph to the web Mercator projection for maps displaying small areas. By mixing different projections in order to reduce the deformation in scaled maps, applications have been developed for small-scale map applications (Jenny, 2012; Jenny and Šavrič, 2018; Šavrič and Jenny, 2014) as well as large- and medium-scale maps.

CPs (Gojamanov and Ismayilov, 2019; Huryeu and Padshyvalau, 2007, 2008; Padshyvalau et al., 2005; Pędzich, 2005) including Lagrange polyconic projection (Bugayevskiy and Snyder, 1995; Yang et al., 1999) and Chebyshev–Grave criteria (Milnor, 1969; Nestorov, 1997) are defined. CP is a type of projection that minimises distortions that increase in all directions as it moves away from the selected coordinate origin and aims to provide users with formula and calculation convenience. Basic formulas for CP are given below: (21) xcp=k1x1+k2x2 {x_{{\rm{cp}}}} = {k_1}{x_1} + {k_2}{x_2} (22) ycp=k1y1+k2y2 {y_{{\rm{cp}}}} = {k_1}{y_1} + {k_2}{y_2} where (x₁,y₁) and (x₂,y₂) values are the GK and LCC2 projection co-ordinates used in this study, respectively. Considering the physical size of Turkey's geodetic boundaries, it is seen that there are distortions in both north–south and east–west directions. These two projection methods have been chosen to reduce these distortions in both directions. k₁ and k₂ are mixed projection coefficients and calculated using the following equation: (23) dk=n[x1y1]−[y1][x1]n[x12]−[x12] {d_k} = {{n\left[ {{x_1}{y_1}} \right] - \left[ {{y_1}} \right]\left[ {{x_1}} \right]} \over {n\left[ {x_1^2} \right] - \left[ {x_1^2} \right]}} (24) k1=−|dk21+dk2| or k1=|1+dk2dk2| {k_1} = - \left| {{{d_k^2} \over {1 + d_k^2}}} \right|\quad {\rm{or}}\quad {k_1} = \left| {{{1 + d_k^2} \over {d_k^2}}} \right| (25) k2=1−k1 {k_2} = 1 - {k_1}

The value of n in the Equation (23) symbolises the number of points (585 points) calculated in practice. Coordinates of the second projection can also be used in the calculation of d_k. In this study, the d_k coefficients have been tested in both projections and LCC2 has been preferred because it reduces distortions. In order to prefer CP, the projection equations (x₁,y₁) and (x₂,y₂) in Equation (6) should be calculated. For this, isometric coordinate parameter pairs can be written in terms of complex forms in conform projections (Gojamanov and Ismayilov, 2019; Huryeu and Padshyvalau, 2007, 2008; Padshyvalau et al., 2005; Pędzich, 2005): (26) xj+iyj=F(q+il), j=1,2 {x_j} + i{y_j} = F\left({q + il} \right),\quad j = 1,2 (27) Δxj+iyj=∑k=110ak(Δq+il)k \Delta {x_j} + i{y_j} = \sum\limits_{k = 1}^{10} {a_k}{(\Delta q + il)^k} (28) Δxj=xj−x0, x0=G0 \Delta {x_j} = {x_j} - {x_0},\quad {x_0} = {G_0} (29) Δq=q−q0, l=L−L0 \Delta q = q - {q_0},\quad \;\;\;{\kern 1pt} l = L - {L_0}

The following equations are used to avoid calculation overhead in complex functions: (30) xj=x0+∑k=110akPk; yj=∑k=110akQk, j=1,2 {x_j} = {x_0} + \sum\limits_{k = 1}^{10} {a_k}{P_k};\quad \quad \;{\kern 1pt} {y_j} = \sum\limits_{k = 1}^{10} {a_k}{Q_k},\quad j = 1,2 (31) Pj=Pj−1P1−Qj−1Q1, P1=q−q0, P0=1 {P_j} = {P_{j - 1}}{P_1} - {Q_{j - 1}}{Q_1},\quad {P_1} = q - {q_0},\quad {\kern 1pt} {P_0} = 1 (32) Qj=Pj−1Q1−Qj−1P1, Q1=L−L0, Q0=0 {Q_j} = {P_{j - 1}}{Q_1} - {Q_{j - 1}}{P_1},\quad {Q_1} = L - {L_0},\quad {Q_0} = 0

Since the GK mapping accepts the equator as the beginning of the x-axis, the G₀ value must be calculated. On the other hand, in the LCC2 description, since the x-axis is accepted as the beginning of the B₀ latitude, it is not necessary to calculate the G₀ value (G₀ = 0). The a_k coefficients are obtained by using the Taylor series at latitude B₀ selected by taking l = 0 in accordance with the conformal description conditions of the analytical function derivative (Grossmann, 1976): (33) a1=dxdq=(dGdq)0=dF(q)dq=F′(q)=NcosB }GK {a_1} = {{dx} \over {dq}} = {\left({{{dG} \over {dq}}} \right)_0} = {{dF(q)} \over {dq}} = F{'}\left(q \right) = N\cos B\quad \quad \;\;\,\} {\rm{GK}} (34) a1=dxdq=(dGdq)0=dF(q)dq=F′(q)=R0¯e_δ(q−q0) }LCC2 {a_1} = {{dx} \over {dq}} = {\left({{{dG} \over {dq}}} \right)_0} = {{dF(q)} \over {dq}} = F{'}\left(q \right) = \overline {{R_0}} {\underline e ^{\delta (q - {q_0})}}\quad \} {\rm{LCC}}2 or (35) an=1n!=(dnxdqn)=(−1)n+1n!δnR0¯e_−δ(q−q0), q=q0 }LCC2 {a_n} = {1 \over {n!}} = \left({{{{d^n}x} \over {d{q^n}}}} \right) = {{{{(- 1)}^{n + 1}}} \over {n!}}{\delta ^n}\overline {{R_0}} {\underline e ^{- \delta (q - {q_0})}},\;q = {q_0}\;\} {\rm{LCC}}2 (36) a2=12(d2Gdq2)0=12(ddBdGdqdBdq)0, (dBdq)0​=N0cosB0M0 {a_2} = {1 \over 2}{\left({{{{d^2}G} \over {d{q^2}}}} \right)_0} = {1 \over 2}{\left({{d \over {dB}}{{dG} \over {dq}}{{dB} \over {dq}}} \right)_0},\;{\left({{{dB} \over {dq}}} \right)_0} = {{{N_0}\cos {B_0}} \over {{M_0}}} (37) a3=16(d3Gdq3)0=16(ddBd2Gdq2dBdq)0 {a_3} = {1 \over 6}{\left({{{{d^3}G} \over {d{q^3}}}} \right)_0} = {1 \over 6}{\left({{d \over {dB}}{{{d^2}G} \over {d{q^2}}}{{dB} \over {dq}}} \right)_0}

The values of δ and R0¯ \overline {{R_0}} in the equations given above for LCC2 are calculated using the Equations (11–17). The effect of the a_k coefficients on the (x, y) coordinates falls below 1 mm after the 10th iteration, considering the geographical borders of Turkey. These coefficients are difficult to derive and calculate manually. MATLAB software has been used for this process. These coefficients have been calculated using a number of functions (syms, diff and complex) available in the MATLAB library. The a_k coefficients for CP are calculated by using Equations (26–29) or Equations (30–32) instead of Equations (21) and (22) (Gojamanov and Ismayilov, 2019; Huryeu and Padshyvalau, 2007, 2008; Padshyvalau et al., 2005; Pędzich, 2005): (38) ajCP=k1ajGK+k2ajLCC2 a_j^{{\rm{CP}}} = {k_1}a_j^{{\rm{GK}}} + {k_2}a_j^{{\rm{LCC}}2}

Thus, the ease of calculation for CP is presented to the users with the coefficients in Table 2.

The meridian convergence and differential scale factor used in determination of the distortions are calculated as follows. (39) m=K12+K22NcosB, ​γ=arctanK1K2 m = {{\sqrt {K_1^2 + K_2^2}} \over {N\cos B}},\quad \quad \quad \gamma = \arctan {{{K_1}} \over {{K_2}}} (40) K1=−∑j=110jajCPQ(j−1), K2=∑j=110jajCPP(j−1) {K_1} = - \sum\limits_{j = 1}^{10} ja_j^{{\rm{CP}}}{Q_{(j - 1)}},\quad {K_2} = \sum\limits_{j = 1}^{10} ja_j^{{\rm{CP}}}{P_{(j - 1)}}

The distortions and meridian convergence calculated are shown in Figure 4.

Figure 4

The conformal projection of ellipsoid to sphere is fixed and there are many algorithms for conformal projection from sphere to plane. DP consists of two stages: firstly, from the isometric coordinates in the ellipsoid (q, l) to the isometric coordinates in the sphere (ω,Δλ), and then from the geodetic coordinates in the sphere (φ,λ) to the Gaussian sphere (Grossmann, 1976). (41) ω+iΔλ=K1(q+il)+K2 \omega + i\Delta \lambda = {K_1}\left({q + il} \right) + {K_2} (42) K1=1+e′2cos4B0 {K_1} = \sqrt {1 + {e^{{'}2}}\mathop {\cos}\nolimits^4 {B_0}} (43) K2=arctanh(1K1sinB0)−K1q0 {K_2} = \arctan {\rm{h}}\left({{1 \over {{K_1}}}\sin {B_0}} \right) - {K_1}{q_0} (44) ω=K1q+K2 \omega = {K_1}q + {K_2} (45) Δλ=λ−λ0=K1l, λ0=L0 \Delta \lambda = \lambda - {\lambda _0} = {K_1}l,\quad {\lambda _0} = {L_0} (46) φ=arcsin(tanh(ω)) \varphi = \arcsin (\tanh (\omega))

The double standard parallel projection equations from sphere to plane are as follows: (47) xdp=R0¯−R¯cos(δ(λ−λ0)) {x_{{\rm{dp}}}} = \overline {{R_0}} - \overline R \cos (\delta (\lambda - {\lambda _0})) (48) ydp=R0¯−R¯sin(δ(λ−λ0)) {y_{{\rm{dp}}}} = \overline {{R_0}} - \overline R \sin (\delta (\lambda - {\lambda _0})) (49) R¯=R0¯e_[−δ(ω−ω0)] \overline R = \overline {{R_0}} {\underline e ^{[ - \delta (\omega - {\omega _0})]}} (50) R0¯=R1cosφ1δe_[δ(ω1−ω0)]=R2cosφ2δe_[δ(ω2−ω0)] \overline {{R_0}} = {{{R_1}\cos {\varphi _1}} \over \delta}{\underline e ^{[\delta ({\omega _1} - {\omega _0})]}} = {{{R_2}\cos {\varphi _2}} \over \delta}{\underline e ^{[\delta ({\omega _2} - {\omega _0})]}} (51) δ=ln(R1cosφ1)−ln(R2cosφ2)ω2−ω1 \delta = {{\ln ({R_1}\cos {\varphi _1}) - \ln ({R_2}\cos {\varphi _2})} \over {{\omega _2} - {\omega _1}}} (52) R1=c/(1+e′2cos2B1); R2=c/(1+e′2cos2B2) {R_1} = c/(1 + {e{'}^2}\mathop {\cos}\nolimits^2 {B_1});\quad {R_2} = c/(1 + {e{'}^2}\mathop {\cos}\nolimits^2 {B_2})

The meridian convergence and differential scale factor used in determination of distortions are calculated as follows (Bugayevskiy and Snyder, 1995; Grossmann, 1976): (53) γ=δ(λ−λ0) \gamma = \delta (\lambda - {\lambda _0}) (54) m=R1cosφ1Rcosφe_[−δ(ω−ω1)]=R2cosφ2Rcosφe_[−δ(ω−ω2)] m = {{{R_1}\cos {\varphi _1}} \over {R\cos \varphi}}{\underline e ^{[ - \delta (\omega - {\omega _1})]}} = {{{R_2}\cos {\varphi _2}} \over {R\cos \varphi}}{\underline e ^{[ - \delta (\omega - {\omega _2})]}}

The calculated distortions and meridian convergence are shown in Figure 5.

Figure 5

Considering the distortions in Figures 2–5, it is seen that the distortion in the Mittermayer method increases as it moves away from the starting meridian in the east–west direction compared to other methods. This means distortion of the size of the map in the whole country in large- and medium-sized maps. When the deformations in the other three methods are examined, it is seen that the edge distortion of CP gives better results between standard parallels compared to LCC2 and DP. In this study, for the first time in Turkey, CPU, GK and LCC2 depictions are used together. The combination of GK and LCC2 equations for the CP depiction equations can be seen as an additional computational burden on users. However, considering the geographical borders of Turkey, the coefficients to be used in the solution of Equations (21, 22) and (26–29) are calculated and presented to the users in Tables 2 and 3. Thus, both ease of calculation is provided to the users and the edge distortion between the latitudes (37°30′ −40°30′) falls below 20 cm. The coefficients given in Tables 3 and 4 for CP are valid only for B₀ = 39°. For the use of other countries, it is necessary to calculate a new average latitude value of the study area. Calculations for some points for CP are given in Table 5.

It is necessary to know the geodetic coordinates for transformation from the map coordinates of the newly defined system to the old system and for transition to geocentric coordinates in the Global Navigation Sattellite System (GNSS) application. This is done by inverse transformation from either GK or LCC2 projection coordinates (Equations (41–44)): (55) Δq+il=∑k=110bk(Δx+iy)k \Delta q + il = \sum\limits_{k = 1}^{10} {b_k}{(\Delta x + iy)^k} or (56) q=q0+∑k=110bkPk′, L=L0+∑k=110bkQk′ q = {q_0} + \sum\limits_{k = 1}^{10} {b_k}P_k^{'},\quad \quad \;\;L = {L_0} + \sum\limits_{k = 1}^{10} {b_k}Q_k^{'} (57) Pj′=Pj−1′P1′−Qj−1′Q1′, P1′=Δx, P0′=1 P_j^{'} = P_{j - 1}^{'}P_1^{'} - Q_{j - 1}^{'}Q_1^{'},\quad P_1^{'} = \Delta x,\quad P_0^{'} = 1 (58) Qj′=Pj−1′Q1′−Qj−1′P1′, Q1′=y, Q0′=0 Q_j^{'} = P_{j - 1}^{'}Q_1^{'} - Q_{j - 1}^{'}P_1^{'},\quad Q_1^{'} = y,\quad Q_0^{'} = 0

For the value of B₀ = 39°, the b_k coefficients are calculated using the formulas given below: (59) b1=(dqdG)0=1NcosB }GK {b_1} = {\left({{{dq} \over {dG}}} \right)_0} = {1 \over {N\cos B}}\quad \quad \quad \quad \quad \;\;{\kern 1pt} \} {\rm{GK}} (60) b1=(dqdG)0=1N1cosB1e_δ(q−q0) }LCC2 {b_1} = {\left({{{dq} \over {dG}}} \right)_0} = {1 \over {{N_1}\cos {B_1}{{\underline e}^{\delta (q - {q_0})}}}}\quad \} {\rm{LCC}}2 or (61) bn=1n!=1δnR0¯e_−δ(q−q0), q=q0 }LCC2 {b_n} = {1 \over {n!}} = {1 \over {{\delta ^n}\overline {{R_0}} {{\underline e}^{- \delta (q - {q_0})}}}},\quad q = {q_0}\quad \} {\rm{LCC}}2 (62) b2=12(d2qdG2)0=12(ddBdqdGdBdG)0, (dBdG)0=1M0 {b_2} = {1 \over 2}{\left({{{{d^2}q} \over {d{G^2}}}} \right)_0} = {1 \over 2}{\left({{d \over {dB}}{{dq} \over {dG}}{{dB} \over {dG}}} \right)_0},\quad {\left({{{dB} \over {dG}}} \right)_0} = {1 \over {{M_0}}} (63) b3=16(d3qdG3)0=16(ddBd2qdG2dBdG)0 {b_3} = {1 \over 6}{\left({{{{d^3}q} \over {d{G^3}}}} \right)_0} = {1 \over 6}{\left({{d \over {dB}}{{{d^2}q} \over {d{G^2}}}{{dB} \over {dG}}} \right)_0}

All calculated b_k coefficients are given in Table 3. Calculation of the values of δ and R0¯ \overline {{R_0}} used in the above equations are given in Equations (11–17). The following equations are used for transition from isometric latitude to geodetic latitude given in Equations (55–58) (Kaya, 1994): (64) B=χ+1/2sin2χ[e2+e4(cos2χ−16sin2χ)++e6(cos4χ−56sin2χcos2χ+130sin4χ)++​e8​​(​​cos6​χ​−​2sin2​χcos4​χ​+​49120sin4​χcos4​χ​+​31252sin2​χ​)​​] \matrix{B \hfill & {= \chi + 1/2\sin 2\chi [{e^2} + {e^4}\left({\mathop {\cos}\nolimits^2 \chi - {1 \over {6\mathop {\sin}\nolimits^2 \chi}}} \right) +} \hfill \cr {} \hfill & {+ {e^6}\left({\mathop {\cos}\nolimits^4 \chi - {5 \over {6\mathop {\sin}\nolimits^2 \chi \mathop {\cos}\nolimits^2 \chi}} + {1 \over {30\mathop {\sin}\nolimits^4 \chi}}} \right) +} \hfill \cr {} \hfill & {+ {e^8}\left({\mathop {\cos}\nolimits^6 \chi - 2\mathop {\sin}\nolimits^2 \chi \mathop {\cos}\nolimits^4 \chi + {{49} \over {120\mathop {\sin}\nolimits^4 \chi \mathop {\cos}\nolimits^4 \chi}} + {{31} \over {252\mathop {\sin}\nolimits^2 \chi}}} \right)]} \hfill \cr} (65) χ=arcsin(tanhq) \chi = \arcsin (\tanh q)

Also, b_k coefficients can be obtained by using Equation (66): (66) bjCP=k1bjGK+k2bjLCC2 b_j^{{\rm{CP}}} = {k_1}b_j^{{\rm{GK}}} + {k_2}b_j^{{\rm{LCC}}2}

The reverse conversion process is performed using Equations (55–58) and (66). This process is more practical for users. The coefficients calculated from Equations (59–63) and (66) are shown in Table 3.

In engineering projects where distance and direction are measured, it is necessary to reduce lengths and direction. Since map coordinates are used in the projects, giving the reduction formulas over these coordinates provides ease of calculation for the users. The reduction formulas commonly used for conformal descriptions (Draheim, 1953) can also be used for CP: (67) (T−t)map=t1x1Δy+t2ΔxΔy+t3x1y1Δx+t4x12Δy+ +t5y12Δy+t6x12y1Δx+t7y13Δx+t8x1y12Δy \matrix{{{{(T - t)}_{{\rm{map}}}}} \hfill & {= {t_1}{x_1}\Delta y + {t_2}\Delta x\Delta y + {t_3}{x_1}{y_1}\Delta x + {t_4}x_1^2\Delta y +} \hfill \cr {} \hfill & {\quad + {t_5}y_1^2\Delta y + {t_6}x_1^2{y_1}\Delta x + {t_7}y_1^3\Delta x + {t_8}{x_1}y_1^2\Delta y} \hfill \cr} where: (68) t1=3t2;t2=−1+η026N02;t3=−t0(1+η02)2N03;t4=−t0(1−3η02−4η04)4N03;t5=−t32;t6=40+108η02+63η04−3t0236N04;t7=−10η02−7η04+3t0212N04;t8=−t6 \matrix{{\matrix{{{t_1} = 3{t_2};} \hfill & {{t_2} = - {{1 + \eta _0^2} \over {6N_0^2}};} \hfill \cr}} \hfill & {{t_3} = - {{{t_0}\left({1 + \eta _0^2} \right)} \over {2N_0^3}};} \hfill \cr {{t_4} = - {{{t_0}\left({1 - 3\eta _0^2 - 4\eta _0^4} \right)} \over {4N_0^3}};} \hfill & {{t_5} = - {{{t_3}} \over 2};} \hfill \cr {{t_6} = {{40 + 108\eta _0^2 + 63\eta _0^4 - 3t_0^2} \over {36N_0^4}};} \hfill & {{t_7} = {{- 10\eta _0^2 - 7\eta _0^4 + 3t_0^2} \over {12N_0^4}};} \hfill \cr {{t_8} = - {t_6}} \hfill & {} \hfill \cr} (69) (S−s)map=s(s1+s2x12+s3x1Δx+s4x13+ +s5x1y12+s6x14+s7x12y12+s8y14) \matrix{{{{(S - s)}_{{\rm{map}}}}} \hfill & {= s({s_1} + {s_2}x_1^2 + {s_3}{x_1}\Delta x + {s_4}x_1^3 +} \hfill \cr {} \hfill & {\quad + {s_5}{x_1}y_1^2 + {s_6}x_1^4 + {s_7}x_1^2y_1^2 + {s_8}y_1^4)} \hfill \cr} where: (70) s1=N0cosB0b1CP−1;s2=s3=t1;s4=−t0(1−3η02−4η04)6N03;s5=−t3;s6=5+20η02+10η04−3t0224N04;s7=−13η02−8η04+9t024N04;s8=10η02+10η04−3t0224N04 \matrix{{{s_1} = {{{N_0}\cos {B_0}} \over {b_1^{{\rm{CP}}}}} - 1;} \hfill & {{s_2} = {s_3} = {t_1};} \hfill \cr {{s_4} = - {{{t_0}\left({1 - 3\eta _0^2 - 4\eta _0^4} \right)} \over {6N_0^3}};} \hfill & {{s_5} = - {t_3};} \hfill \cr {{s_6} = {{5 + 20\eta _0^2 + 10\eta _0^4 - 3t_0^2} \over {24N_0^4}};} \hfill & {{s_7} = {{- 13\eta _0^2 - 8\eta _0^4 + 9t_0^2} \over {4N_0^4}};} \hfill \cr {{s_8} = {{10\eta _0^2 + 10\eta _0^4 - 3t_0^2} \over {24N_0^4}}} \hfill & {} \hfill \cr}

The Δx(x₂ – x₁) and Δy(y₂ – y₁) reductions are the difference between the map coordinate values of the edge to be calculated. In order to provide ease of calculation to the users, t_k and s_k coefficients in Equations (66–68) are calculated in B₀ = 39° latitude and ITRF96 datum are given in Table 4. These coefficient values that can be used for up to 2 km in length can be measured in the land of Turkey's geodetic boundaries.

These differences Tt = (T – t) – (T – t)_map and Ss = (S – s) – (S – s)_map are calculated on the sides of 1 and 2 km and are given in Figure 6. It is observed that the edge distortion (S – s) in Equations (66–68) is achieved with an accuracy of 1 cm for 1 km and less than 2 cm for 2 km. In addition, the map distortion (T – t) in Equations (66–68) provides an accuracy of less than 0.003″ for 1 km and 0.006″ for 2 km. In this context, Equations (66–68) are sufficient for the location accuracy of the angle and distance measurements made in the detail acquisition of projects using large-scale maps and in the application to the terrain.

Figure 6

Since large-scale maps (1:1000–1:10,000) cover small areas, m₀ is taken as 1 for the whole country (Figure 4a). Thus, in engineering projects that require detailed measurement from the field and application to the field, the land and the map remain the same in 1:1 scale. This provides the users with the ease of completing the application without any additional transformation process in the application of the data produced from large-scale maps to the terrain. However, this is not the case with medium-scale maps (1:25,000–1:250,000). In these maps, due to the area they occupy, distortion increases in the near regions of the depiction centre and as they move away from the depiction centre. For the CP representation, these maps also increase as distortions increase from the chosen initial latitude to the north and south, and particularly at the system edges. It may be possible to keep this growth within a certain boundary by preserving the conditions of representation, and especially to reduce it within the country borders. So, m could be slightly reduced for the whole country. For this, m₀ must be equal to 1 at x_max/2. Thus, in the initial latitude, m₀ will be slightly smaller than 1, but in the border latitudes (m − 1), the difference will be halved. x_max/2 corresponds to the latitude B = 40.50° and m₀ = 1 for CP. The m₀value can be chosen as 0.9995, the average value between B = 40.5° and B_max = 42° in CP (Figure 7).

Figure 7

In order to not work with negative coordinates in scaled maps, at the beginning of the coordinate of the description, negative x values south of latitude B₀ = 39° and negative y values west of longitude L₀ = 35.5° should be positive. For this, a fixed value (false) must be specified to add to all projection coordinates (x_cp, y_cp). Negative values of CP coordinates are x_max = −388 km and y_max = −861 km in Turkey (Table 5). Therefore, false northing = 500,000 m and false easting = 1,000,000 m values are determined for positive coordinates. Thus, scale maps for CP (scale map) or sheet (sheet) coordinates can be calculated in Turkey: (71) Northing(N)=m0xCP+500,000 {\rm{Northing}}\left(N \right) = {m_0}{x_{{\rm{CP}}}} + 500,000 (72) Easting(E)=m0yCP+1,000,000 {\rm{Easting}}\left(E \right) = {m_0}{y_{{\rm{CP}}}} + 1,000,000

In these equations, m₀ value is taken as 1 for large-scale maps and 0.9995 for medium- and small-scale maps.

The meanings of the parameters symbolised in Table 5 are explained in Section 2.2. This table has been produced by taking into account the latitude and longitude limit values of Turkey. Distortion values have been calculated in CP as a result of calculations made using MATLAB software.