依比例尺自适应调整的墨卡托投影面大地线精确展绘算法

doi:10.11947/j.AGCS.2024.20230549

测绘学报 ›› 2024, Vol. 53 ›› Issue (12): 2338-2348.doi: 10.11947/j.AGCS.2024.20230549

依比例尺自适应调整的墨卡托投影面大地线精确展绘算法

谢天¹(), 董箭¹(), 唐露露¹, 马梦锴¹, 张明阳², 宋子康³, 王栋¹

^1.海军大连舰艇学院军事海洋与测绘系，辽宁　大连　116018
^2.31457部队，辽宁　沈阳　110079
^3.海图信息中心，天津　300450

收稿日期:2023-12-13 出版日期:2025-01-06 发布日期:2025-11-06
通讯作者: 董箭 E-mail:xietian0229@163.com;navydj@163.com
作者简介:第一谢天（2000—），男，硕士生，研究方向为地理信息方法与海洋划界。E-mail：xietian0229@163.com
基金资助:
国家自然科学基金(42071439)

Scale-adaptive Mercator projection plane geodesic precise plotting algorithm

Tian XIE¹(), Jian DONG¹(), Lulu TANG¹, Mengkai MA¹, Mingyang ZHANG², Zikang SONG³, Dong WANG¹

^1.Department of Military Oceanography and Surveying, Dalian Naval Academy, Dalian 116018, China
^2.Troops 31457, Shenyang 110079, China
^3.Chart Information Center, Tianjin 300450, China

Received:2023-12-13 Online:2025-01-06 Published:2025-11-06
Contact: Jian DONG E-mail:xietian0229@163.com;navydj@163.com
About author:XIE Tian (2000—), male, postgraduate, majors in geographic information methods and maritime delimitation. E-mail: xietian0229@163.com
Supported by:
The National Natural Science Foundation of China(42071439)

摘要/Abstract

摘要：

针对现有墨卡托投影面大地线展绘方法计算数据量与制图精度不匹配、纵跨赤道大地线展绘拱高误差超限等问题，本文在分析墨卡托投影面大地线表象曲线几何特性的基础上，借鉴二分法搜索思想，建立了大地线赤道分割点对分逼近精准解算模型，实现了任意给定比例尺条件下纵跨赤道大地线赤道分割点求解；通过分析内插点与拱顶点切线方位角的变化规律，构建了大地方位角精确引导的拱顶点搜索规则，实现了墨卡托投影面任意弧段大地线表象曲线拱高误差快速解算；类比道格拉斯普克算法思想，将制图允许误差作为曲线抽稀阈值，最终实现了任意给定比例尺条件下大地线快速精确展绘。试验结果表明，本文算法显著提高了计算效率，降低了内插冗余。在典型应用场景中，本文算法在严格控制拱高误差不超过制图允许误差条件下，计算所得内插点数最大程度可减少为现有算法的约千分之一，计算所用时间缩短至约百分之一。

关键词: 大地线展绘, 墨卡托投影, 二分法搜索, 道格拉斯普克算法, 大地主题解算

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

中图分类号:

P226

谢天, 董箭, 唐露露, 马梦锴, 张明阳, 宋子康, 王栋. 依比例尺自适应调整的墨卡托投影面大地线精确展绘算法[J]. 测绘学报, 2024, 53(12): 2338-2348.

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.

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参考文献 24

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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

依比例尺自适应调整的墨卡托投影面大地线精确展绘算法

Scale-adaptive Mercator projection plane geodesic precise plotting algorithm

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方位角/（°）	100 km		1000 km
方位角/（°）	内插点数	时间/ms	内插点数	时间/ms
30	101	490	1001	490
60	101	470	1001	490
90	101	650	1001	500
120	101	680	1001	580
150	101	480	1001	490
180	101	490	1001	510
210	101	520	1001	490
240	101	490	1001	790
270	101	530	1001	490
300	101	500	1001	480
330	101	570	1001	520
360	101	500	1001	510
平均值	101	531	1001	528