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

作　　者：谢天 董箭[1] 唐露露 马梦锴 张明阳 宋子康 王栋 XIE Tian;DONG Jian;TANG Lulu;MA Mengkai;ZHANG Mingyang;SONG Zikang;WANG Dong(Department of Military Oceanography and Surveying,Dalian Naval Academy,Dalian 116018,China;Troops 31457,Shenyang 110079,China;Chart Information Center,Tianjin 300450,China)

机构地区：[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

出　　处：《测绘学报》2024年第12期2338-2348,共11页Acta Geodaetica et Cartographica Sinica

基　　金：国家自然科学基金(42071439)。

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。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.

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 

分 类 号：P226[天文地球—大地测量学与测量工程]