Proposed single-zone map projection system for Turkey

Yildirim, Faruk; Kadi, Fatih

doi:10.2478/rgg-2021-0006

Figures & Tables

Distortion sizes for Mittermayer method: (a) scale factor, counter interval 0.0005; (b) meridian convergence, contour intervals 1°; (c) the arc to chord (T – t) distortion, contour intervals 0.2°; (d) the distance (S – s) distortion, contour intervals 75 cm.

Distortion sizes for LCC2: (a) scale factor, counter interval 0.0005; (b) meridian convergence, contour intervals 1°; (c) the arc to chord (T – t) distortion, contour intervals 0.2″; (d) the distance (S – s) distortion, contour intervals 40 cm.

Distortion sizes for CP: (a) scale factor, counter interval 0.0005; (b) meridian convergence, contour intervals 1°; (c) the arc to chord (T – t) distortion, contour intervals 0.2″; (d) the distance (S – s) distortion, contour intervals 40 cm.

Distortion sizes for double projection: (a) scale factor, counter interval 0.0005; (b) meridian convergence, contour intervals 1°; (c) the arc to chord (T – t) distortion, contour intervals 0.2″; (d) the distance (S – s) distortion, contour intervals 40 cm.

Correction values based on correction formulas for accuracy analysis in CP: (a) Ss difference for 1 km, counter interval 2 mm; (b) Ss difference for 2 km, counter interval 2 mm; (c) Tt difference for 1 km, contour intervals 0.001″; (d) Tt difference for 2 km, contour intervals 0.001″.

(B, L)	Ellipsoid geographical latitude and longitude;
(φ, λ)	Sphere geographical latitude and longitude;
(B₀, L₀)	Central meridian (latitude) and longitude of central meridian;
(x, y)	Projected coordinates or plane coordinates;
(N, E)	Northing, Easting map of scale or sheet coordinates;
q	Isometric latitude for ellipsoid: atanh(sin B) – e atanh(e sin B);
ω	Isometric latitude for sphere: atanh(sin φ);
q, l	Isometric coordinates;
l	L – L₀;
A	Ellipsoid azimuth;
S	Ellipsoid geodetic or distance on ellipsoid;
s	Plane or projection distance;
t	Plane or projection azimuth;
γ	Grid convergence of ellipsoid;
(T – t)	The arc to chord distortion of projection (A–t–γ);
(S – s)	Distance distortion of projection (S–s);
(T – t)_map	The arc to chord correction calculated from (x, y);
(S – s)_map	Distance correction calculated from (x, y);
a	Semi-major axis of ellipsoid;
b	Semi-minor axis of ellipsoid;
c	Polar radius of curvature;
e²	Eccentricity of ellipsoid squared;
e′²	Second eccentricity of ellipsoid squared;
e	2.71828182845904523. . . Euler's number;
G	Meridian arc length from equator to latitude, meridian distance;
M	Radius of curvature in the meridian:
	c/(1 + e′² cos² B)^3/2;
N	Radius of curvature in the prime vertical: c/(1 + e′² cos² B)^1/2;
R	Radius of Gauss sphere: (MN)^1/2 = c/(1 + e′² cos² B);
r	Radius of curvature of the parallel (N cos B);
η²	e′² cos² B;
t	tan B;
m	Point grid scale factor;
m₀	Grid scale factor assigned to central meridian (longitude) or parallels (latitude);
(B₁, B₂)	Standard parallels for Lambert conformal conic.

Coefficients for correction formulas

k	*t_k*	*s_k*
1	−1.230817046856762e-14	3.410737419053955e-04
2	−4.102723489522539e-15	−1.230817046856762e-14
3	−1.560602822242498e-21	−1.230817046856762e-14
4	−7.675969852812854e-22	−5.117313235208569e-22
5	7.803014111212491e-22	1.560602822242498e-21
6	−2.882524686831016e-28	7.799576438254665e-29
7	9.649191199496467e-29	2.876383454639922e-28
8	2.882524686831016e-28	−4.824471121337579e-29

ak coefficients in metres

	GK		LCC2		CP

k	Hayford (ED50 Datum)	GRS80 (ITRF Datum)	Hayford (ED50 Datum)	GRS80 (ITRF Datum)	Hayford (ED50 Datum)	GRS80 (ITRF Datum)
1	4963550.54140	4963327.33863	4961856.51702	4961633.36122	4961858.21105	4961635.05520
2	−1561831.78385	−1561761.55083	−1561480.06295	−1561409.82430	−1561480.41467	−1561410.17603
3	−174039.01315	−174022.53500	327595.12206	327580.38357	327093.48792	327078.78065
4	344383.18219	344355.40058	−51546.55823	−51544.23875	−51150.62849	−51148.33911
5	−97650.60029	−97639.86801	6488.61350	6488.32148	6384.47429	6384.19329
6	−37371.53049	−37368.38135	−680.64850	−680.61786	−717.33938	−717.30563
7	39232.35134	39,226.49432	61.19939	61.19663	100.37054	100.36193
8	−7671.34989	−7668.97005	−4.81481	−4.81460	−12.48135	−12.47875
9	−6850.12639	−6849.30900	0.33671	0.33670	−6.51375	−6.51295
10	5154.82567	5153.41787	−0.02119	−0.02119	5.13365	5.13225

Numerical values for CP (for 1 km) (k1 = 0_001, k2 = 1 − k1)

B [°]	L [°]	X_CP[m]	Y_CP[m]	M	m0=1m {m_0} = {1 \over m}	γ [°]	(T – t) [″]	(T-t)_map	Tt[m]	(S – s) [m]	(S-s)_map	Ss [m]
35.5	26.0	−343547.200	−861653.097	1.001502	0.998500	−5.97880628	0.6806	0.6805	0.0000	−1.499	−1.489	−0.010
36.0	26.5	−292879.879	−810972.322	1.001018	0.998983	−5.66419331	0.5850	0.5852	−0.0002	−1.015	−1.005	−0.010
36.5	27.0	−241925.869	−760876.119	1.000607	0.999394	−5.34957328	0.4890	0.4893	−0.0003	−0.604	−0.595	−0.009
37.0	27.5	−190685.986	−711365.636	1.000269	0.999731	−5.03494624	0.3924	0.3927	−0.0003	−0.267	−0.259	−0.008
37.5	28.0	−139161.028	−662442.072	1.000005	0.999995	−4.72031223	0.2952	0.2955	−0.0003	−0.004	0.003	−0.007
38.0	28.5	−87351.773	−614106.679	0.999816	1.000184	−4.40567130	0.1974	0.1976	−0.0002	0.185	0.191	−0.006
38.5	29.0	−35258.976	−566360.762	0.999702	1.000298	−4.09102350	0.0989	0.0990	−0.0001	0.299	0.303	−0.005
39.0	29.5	17116.629	−519205.677	0.999662	1.000338	−3.77636888	−0.0002	−0.0002	0.0000	0.338	0.341	−0.003
39.5	30.0	69774.333	−472642.839	0.999699	1.000301	−3.46170748	−0.1000	−0.1002	0.0002	0.301	0.303	−0.002
40.0	30.5	122713.451	−426673.712	0.999812	1.000188	−3.14703936	−0.2006	−0.2009	0.0003	0.187	0.188	−0.001
40.5	31.0	175933.327	−381299.820	1.000002	0.999998	−2.83236456	−0.3019	−0.3024	0.0005	−0.003	−0.004	0.001
41.0	31.5	229433.329	−336522.741	1.000269	0.999731	−2.51768313	−0.4040	−0.4047	0.0007	−0.271	−0.273	0.002
42.0	32.0	338751.594	−290207.607	1.001040	0.998961	−2.20301786	−0.6108	−0.6117	0.0009	−1.043	−1.047	0.004
42.5	32.5	392851.626	−246933.182	1.001545	0.998457	−1.88831991	−0.7155	−0.7166	0.0011	−1.548	−1.553	0.005
35.5	33.0	−385425.899	−227133.656	1.001494	0.998508	−1.57336268	0.6816	0.6812	0.0004	−1.490	−1.484	−0.006
36.0	33.5	−331015.720	−180494.967	1.001010	0.998991	−1.25870410	0.5861	0.5855	0.0005	−1.007	−1.002	−0.005
36.5	34.0	−276364.523	−134461.147	1.000600	0.999401	−0.94403851	0.4900	0.4894	0.0007	−0.597	−0.593	−0.004
37.0	34.5	−221473.213	−89033.279	1.000263	0.999737	−0.62936594	0.3934	0.3926	0.0008	−0.261	−0.258	−0.003
38.0	35.0	−110853.214	−43907.730	0.999811	1.000189	−0.31468987	0.1983	0.1974	0.0009	0.189	0.191	−0.002
39.0	35.5	0.000	0.000	0.999659	1.000341	0.00000000	0.0007	−0.0004	0.0011	0.341	0.341	0.000
39.5	36.0	55609.954	42993.428	0.999696	1.000304	0.31470008	−0.0992	−0.1004	0.0012	0.303	0.303	0.001
40.0	36.5	111461.668	85375.887	0.999810	1.000190	0.62940691	−0.1997	−0.2010	0.0013	0.189	0.188	0.002
40.5	37.0	167554.386	127145.900	1.000000	1.000000	0.94412043	−0.3010	−0.3024	0.0014	−0.002	−0.004	0.002
41.0	37.5	223887.372	168301.932	1.000268	0.999732	1.25884059	−0.4031	−0.4047	0.0015	−0.270	−0.273	0.003
41.5	38.0	280459.918	208842.389	1.000615	0.999386	1.57356734	−0.5061	−0.5077	0.0016	−0.617	−0.621	0.004
42.0	38.5	337271.342	248765.618	1.001040	0.998961	1.88830062	−0.6099	−0.6116	0.0018	−1.043	−1.047	0.004
42.5	39.0	394320.989	288069.904	1.001545	0.998457	2.20304037	−0.7145	−0.7164	0.0019	−1.549	−1.554	0.005
35.5	39.5	−380561.376	363342.720	1.001495	0.998507	2.51738164	0.6827	0.6812	0.0015	−1.491	−1.484	−0.007
36.0	40.5	−320609.223	451047.199	1.001012	0.998989	3.14676397	0.5872	0.5856	0.0016	−1.009	−1.002	−0.007
36.5	41.0	−262583.287	492747.097	1.000603	0.999398	3.46148011	0.4912	0.4895	0.0016	−0.600	−0.593	−0.007
37.0	41.5	−204364.419	533824.364	1.000266	0.999734	3.77620340	0.3946	0.3929	0.0017	−0.264	−0.258	−0.006
37.5	42.0	−145953.593	574277.932	1.000004	0.999996	4.09093380	0.2974	0.2956	0.0018	−0.003	0.004	−0.006
38.0	42.5	−87351.773	614106.679	0.999816	1.000184	4.40567130	0.1996	0.1977	0.0019	0.185	0.191	−0.006
38.5	43.0	−28559.910	653309.428	0.999703	1.000297	4.72041584	0.1011	0.0991	0.0020	0.298	0.304	−0.006
39.0	43.5	30421.060	691884.946	0.999665	1.000335	5.03516738	0.0020	−0.0002	0.0023	0.335	0.341	−0.006
39.5	44.0	89590.217	729831.941	0.999703	1.000297	5.34992587	−0.0978	−0.1003	0.0025	0.297	0.303	−0.006
40.0	44.5	148946.655	767149.065	0.999817	1.000183	5.66469127	−0.1983	−0.2012	0.0029	0.182	0.189	−0.007
42.5	45.0	429557.127	780679.388	1.001552	0.998451	5.97971209	−0.7135	−0.7181	0.0046	−1.555	−1.551	−0.004

bk coefficients

	GK		LCC2		CP

k	Hayford (ED50 Datum)	GRS80 (ITRF Datum)	Hayford (ED50 Datum)	GRS80 (ITRF Datum)	Hayford (ED50 Datum)	GRS80 (ITRF Datum)
1	2.01468684898033e-07	2.01477745023166e-07	2.01537468197565e-07	2.01546532602613e-07	2.01537399414265e-07	2.01546463815034e-07
2	1.27719412250272e-14	1.27730899691869e-14	1.27821479148768e-14	1.27832976273027e-14	1.27821377081870e-14	1.27832874196446e-14
3	1.90606580617672e-21	1.90630871057434e-21	1.08091269101801e-21	1.08105852690209e-21	1.08173784413317e-21	1.08188377708576e-21
4	2.33217785338216e-28	2.33259302576196e-28	1.02832386626856e-28	1.02850885526054e-28	1.02962772025567e-28	1.02981293943104e-28
5	3.40288110341843e-35	3.40362666718644e-35	1.04351317940802e-35	1.04374783445450e-35	1.04587254733203e-35	1.04610771328723e-35
6	4.92399104670312e-42	4.92529126527513e-42	1.10304880523434e-42	1.10334646159951e-42	1.10686974747581e-42	1.10716840640319e-42
7	7.46687045549524e-49	7.46916636213681e-49	1.19929486733841e-49	1.19967244054757e-49	1.20556244292657e-49	1.20594193446916e-49
8	1.14705437919241e-55	1.14745779438984e-55	1.33110423181849e-56	1.33158317899727e-56	1.34124367137860e-56	1.34172617376217e-56
9	1.79545605447638e-62	1.79616622523578e-62	1.50085099606334e-63	1.50145853535200e-63	1.51730470561204e-63	1.51791873906901e-63
10	2.84208801069137e-69	2.84333725613141e-69	1.71339747802719e-70	1.71416813538263e-70	1.74010496065608e-70	1.74088733980856e-70

Proposed single-zone map projection system for Turkey

Figures & Tables

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Coefficients for correction formulas

ak coefficients in metres

Numerical values for CP (for 1 km) (k1 = 0_001, k2 = 1 − k1)

bk coefficients

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