Scale-adaptive Mercator projection plane geodesic precise plotting algorithm

doi:10.11947/j.AGCS.2024.20230549

Abstract

Abstract:

Addressing the issues of mismatched data volume and mapping precision in the current Mercator projection plane geodesic line plotting methods, as well as the excessive arch height errors in equator-crossing geodesic line plotting, this paper, based on the analysis of the geometric characteristics of geodesic line projection curve on the Mercator projection plane, drawing on the idea of binary search, establishes a model for precise calculation of the equatorial division points of geodesic lines, achieving the resolution of these points under any given scale for equator-crossing geodesic lines. By analyzing the changing relationship between the tangent azimuth angles of interpolation points and the arch apex, the study constructs a set of search rules for the apex, guided by precise geodetic azimuth angles, thus facilitating rapid calculation of arch height errors for any segment of the projection curve on the Mercator projection plane. Drawing analogy with the Douglas-Peucker algorithm concept, the study adopts permissible cartographic error as threshold for curve simplification, ultimately enabling rapid and precise plotting of geodesic lines at any given scale. Experimental results demonstrate that this algorithm significantly improves computational efficiency and reduces interpolation redundancy. In typical application scenarios, under the strict control of limiting arch height error to not exceed permissible cartographic error, the maximum reduction in the number of interpolation points obtained by this algorithm can reach approximately one-thousandth of existing algorithms, and the computation time can be reduced to approximately one-hundredth.

Key words: geodesic line plotting, Mercator projection, binary search, Douglas-Peucker algorithm, geodesic theme calculation

CLC Number:

P226

Tian XIE, Jian DONG, Lulu TANG, Mengkai MA, Mingyang ZHANG, Zikang SONG, Dong WANG. Scale-adaptive Mercator projection plane geodesic precise plotting algorithm[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(12): 2338-2348.

Figures/Tables 15

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Tab. 1

Tab. 2

Fig. 9

Tab. 3

Tab. 4

Tab. 5

Tab. 6

References 24

[1]	李松林, 陈成, 边少锋, 等. 常用海图投影平面上大椭圆航线的表象与曲率分析[J]. 测绘学报, 2019, 48(10): 1331-1338. DOI:. doi: 10.11947/J.AGCS.2019.20180348
	LI Songlin, CHEN Cheng, BIAN Shaofeng, et al. Representation and curvature analysis of great ellipse on common chart projection plane[J]. Acta Geodaetica et Cartographica Sinica, 2019, 48(10): 1331-1338. DOI:. doi: 10.11947/J.AGCS.2019.20180348
[2]	CHANG Sisi, JI Bing, HU Qiongfang, et al. Two algorithms of geodesic line length calculation considering elevation in eLoran systems[J]. IET Radar, Sonar & Navigation, 2023, 17(10): 1469-1478.
[3]	PATRIKALAKIS N M, BARDIS L. Offsets of curves on rational B-spline surfaces[J]. Engineering with Computers, 1989, 5(1): 39-46.
[4]	WEINTRIT A, KOPACZ P. A novel approach to loxodrome (rhumb line), orthodrome (great circle) and geodesic line in ECDIS and navigation in general[M]//Methods and algorithms in navigation. Boca Raton: CRC Press, 2011: 123-132.
[5]	CHENG Peng, MIAO Chunyan, LIU Yongjin, et al. Solving the initial value problem of discrete geodesics[J]. Computer-Aided Design, 2016, 70: 144-152.
[6]	RAVI KUMAR G V V, SRINIVASAN P, DEVARAJA H, et al. Geodesic curve computations on surfaces[J]. Computer Aided Geometric Design, 2003, 20(2): 119-133.
[7]	彭认灿, 许坚, 沈文周. 椭球面上实施海域精确划界的几个关键问题[J]. 测绘学院学报, 2001, 18(3): 210-212.
	PENG Rencan, XU Jian, SHEN Wenzhou. The solutions to some key problems of accurately delimitating sea area boundary on the ellipsoid[J]. Journal of Institute of Survey and Mapping, 2001, 18(3): 210-212.
[8]	张建辉, 张学峰, 肖利, 等. 等距离海洋划界模型研究[J]. 海洋湖沼通报, 2022, 44(4): 1-9.
	ZHANG Jianhui, ZHANG Xuefeng, XIAO Li, et al. Research on equidistant maritime delimitation model[J]. Transactions of Oceanology and Limnology, 2022, 44(4): 1-9.
[9]	梁德清, 许坚, 彭认灿. 大地线克莱劳方程在大地线绘制中的应用[J]. 海洋测绘, 2001, 21(2): 20-23.
	LIANG Deqing, XU Jian, PENG Rencan. Geodetic line Clausius equation in the plotting of geodetic lines[J]. Hydrographic Surveying and Charting, 2001, 21(2): 20-23.
[10]	BASELGA S, OLSEN M J. Approximations, errors, and misconceptions in the use of map projections[J/OL]. Mathematical Problems in Engineering, 2021: 1-12[2023-08-01]. https://doi.org/10.1155/2021/1094602.
[11]	THOMAS C M, FEATHERSTONE W E. Validation of Vincenty's formulas for the geodesic using a new fourth-order extension of Kivioja's formula[J]. Journal of Surveying Engineering, 2005, 131(1): 20-26.
[12]	VINCENTY T. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations[J]. Survey Review, 1975, 23(176): 88-93.
[13]	TSENG W K, TSAI K C, LIOU C, et al. Algorithms for the generalized inverse solution and direct solution: using an algebra computer-based system to obtain meridian arc length[J]. Journal of Marine Science and Technology, 2023, 31(2): 185-192.
[14]	何菊, 胡鹏, 胡海. 一种适用任意距离的大地主题解算方法的研究和试验[J]. 测绘科学, 2006, 31(2): 29-31.
	HE Ju, HU Peng, HU Hai. Investigation and experiment of algorithm fitting arbitrary distance[J]. Science of Surveying and Mapping, 2006, 31(2): 29-31.
[15]	丁士俊, 杨艳梅, 史俊波, 等. 大地主题解算几种不同算法在计算中应注意的问题[J]. 黑龙江工程学院学报(自然科学版), 2013, 27(3): 1-5.
	DING Shijun, YANG Yanmei, SHI Junbo, et al. Some issues concerning several different algorithms against geodetic problems in the calculation[J]. Journal of Heilongjiang Institute of Technology, 2013, 27(3): 1-5.
[16]	纪兵, 边少锋. 大地主题问题的非迭代新解[J]. 测绘学报, 2007, 36(3): 269-273.
	JI Bing, BIAN Shaofeng. The new non-iterative solution to the geodetic problem[J]. Acta Geodaetica et Cartographica Sinica, 2007, 36(3): 269-273.
[17]	国家市场监督管理总局, 国家标准化管理委员会. 中国航海图编绘规范：GB 12320—2022[S]. 北京: 中国标准出版社, 2022.
	State Administration for Market Regulation, Standardization Administration of the People's Republic of China. Compilation specifications for Chinese nautical charts: GB 12320—2022[S]. Beijing: Standards Press of China, 2022.
[18]	唐红涛, 王微, 杨永崇, 等. 椭球面上绘制大地线的算法[J]. 测绘科学, 2015, 40(4): 7-10.
	TANG Hongtao, WANG Wei, YANG Yongchong, et al. An algorithm of expressing geodesic on the ellipsoid surface[J]. Science of Surveying and Mapping, 2015, 40(4): 7-10.
[19]	HOTZ I, HAGEN H. Visualizing geodesics[C]//Proceedings of 2000 Visualization. Los Alamitos: IEEE Computer Society, 2000: 311-318.
[20]	李彬彬, 彭认灿, 董箭, 等. 绝对精度阈值约束的大地线内插算法[J]. 海洋测绘, 2019, 39(4): 31-35.
	LI Binbin, PENG Rencan, DONG Jian, et al. Geodetic interpolation algorithm with absolute precision threshold constraints[J]. Hydrographic Surveying and Charting, 2019, 39(4): 31-35.
[21]	董箭, 李彬彬, 彭认灿, 等. 顾及拱高误差的墨卡托大地线快速展绘算法[J]. 测绘科学, 2020, 45(9): 43-51.
	DONG Jian, LI Binbin, PENG Rencan, et al. Rapid plotting algorithm for the Mercator geodetic line with regard to the arch height error[J]. Science of Surveying and Mapping, 2020, 45(9): 43-51.
[22]	华棠, 丁佳波, 边少锋, 等. 地图海图投影学[M]. 西安: 西安地图出版社, 2018.
	HUA Tang, DING Jiabo, BIAN Shaofeng, et al. Map and chart projection science[M]. Xi'an: Xi'an Map Publishing House, 2018.
[23]	屈婉玲. 算法设计与分析[M]. 北京: 清华大学出版社, 2011.
	QU Wanling. Design and analysis of algorithms[M]. Beijing: Tsinghua University Press, 2011.
[24]	DOUGLAS D H, PEUCKER T. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature[J]. The Canadian Cartographer, 1973, 10(2): 112-122.

算例	端点坐标	比例尺	大地线长度/m	最大拱高/mm	最小拱高/mm	平均拱高/mm
算例1	41°23'59.56″N，70°18'52.37″E	1∶200 000	8 000 000.09	2.16	1.54×10^-9	1.79×10^-4
算例1	20°30'15.20″S，30°18'25.14″E	1∶200 000	8 000 000.09	2.16	1.54×10^-9	1.79×10^-4
算例2	08°22'28.12″N，57°19'25.13″E	1∶10 000	1 999 935.32	1.06×10^-1	1.63×10^-8	5.15×10^-5
算例2	04°33'05.38″S，69°55'23.13″E	1∶10 000	1 999 935.32	1.06×10^-1	1.63×10^-8	5.15×10^-5
算例3	01°07'08.25″N，115°51'48.11″E	1∶2000	700 000.05	4.12×10^-1	7.78×10^-9	3.34×10^-4
算例3	02°00'28.19″S，110°23'41.57″E	1∶2000	700 000.05	4.12×10^-1	7.78×10^-9	3.34×10^-4

算例	算法1			算法2			算法3
算例	最大拱高	最小拱高	平均拱高	最大拱高	最小拱高	平均拱高	最大拱高	最小拱高	平均拱高
算例1	2.49×10^-2	3.64×10^-6	4.75×10^-5	9.73×10^-2	2.61×10^-2	4.32×10^-2	6.32×10^-5	1.08×10^-9	1.77×10^-5
算例2	2.20×10^-2	6.52×10^-6	9.66×10^-5	9.25×10^-2	2.60×10^-2	4.53×10^-2	1.77×10^-4	6.15×10^-9	7.35×10^-5
算例3	2.89×10^-2	2.56×10^-5	3.49×10^-4	9.13×10^-2	2.39×10^-2	4.85×10^-2	2.56×10^-4	1.68×10^-7	1.08×10^-4

方位角/（°）	算法1				算法2
	100 km		1000 km		100 km		1000 km
	内插点数	时间/ms	内插点数	时间/ms	内插点数	时间/ms	内插点数	时间/ms
30	34 654	1 584.65	568 238	25 915.61	33	10.69	257	48.57
60	31 730	1 436.19	447 537	20 472.97	33	3.29	257	39.59
90	820	38.43	81 316	3 805.85	33	2.10	257	27.50
120	28 856	1 336.32	164 119	7 619.46	33	3.29	257	31.90
150	30 551	1 420.54	153 837	7 175.61	33	3.16	257	28.45
180	2	0.45	2	0.17	2	0.92	2	1.23
210	30 551	1 449.62	153 837	7 331.62	33	2.98	257	27.25
240	28 856	1 364.12	164 119	7 788.16	33	3.18	257	27.99
270	820	39.05	81 316	3 889.28	33	2.03	257	23.12
300	31 730	1 513.91	447 537	21 534.94	33	3.71	257	29.27
330	34 654	1 661.31	568 238	27 204.77	33	3.16	257	27.94
360	2	0.16	2	0.15	2	0.34	2	0.03
平均值	21 102	987.06	235 842	11 061.55	28	3.24	215	26.07

方位角/（°）	算法1				算法2
	100 km		1000 km		100 km		1000 km
	内插点数	时间/ms	内插点数	时间/ms	内插点数	时间/ms	内插点数	时间/ms
30	3467	177.34	56 825	2 608.17	9	8.64	67	19.23
60	3175	144.99	44 761	2 055.19	9	0.95	129	13.20
90	78	3.72	8133	380.23	9	0.45	129	11.55
120	2887	134.51	16 413	764.49	9	0.98	123	13.11
150	3056	142.11	15 385	717.27	9	0.94	65	9.43
180	2	0.13	2	0.16	2	0.70	2	1.08
210	3056	145.26	15 385	731.75	9	0.90	65	9.07
240	2887	137.61	16 413	780.43	9	0.95	123	9.32
270	78	3.79	8133	389.36	9	0.44	129	6.16
300	3175	152.35	44 761	2 155.62	9	1.03	129	9.46
330	3467	166.49	56 825	2 726.66	9	0.95	67	9.23
360	2	0.14	2	0.16	2	0.32	2	0.03
平均值	2111	100.70	23 587	1 109.12	8	1.44	86	9.24

方位角/（°）	算法1				算法2
	100 km		1000 km		100 km		1000 km
	内插点数	时间/ms	内插点数	时间/ms	内插点数	时间/ms	内插点数	时间/ms
30	348	27.24	5684	267.56	3	8.43	33	12.17
60	319	22.99	4478	205.52	5	0.36	33	4.75
90	11	0.55	811	37.98	5	0.22	33	3.71
120	290	13.49	1642	76.41	5	0.37	33	4.73
150	307	14.31	1540	71.57	3	0.24	17	3.26
180	2	0.12	2	0.39	2	0.77	2	1.09
210	307	14.59	1540	73.02	3	0.23	17	3.12
240	290	13.80	1642	78.08	5	0.36	33	4.10
270	11	0.58	811	38.82	5	0.22	33	2.78
300	319	15.32	4478	215.12	5	0.37	33	4.18
330	348	16.77	5684	271.92	3	0.25	33	3.40
360	2	0.12	2	0.12	2	0.09	2	0.03
平均值	213	11.66	2360	111.38	4	0.99	25	3.94