地图投影计算机代数分析研究进展

doi:10.11947/j.AGCS.2017.20170396

摘要/Abstract

摘要： 地图投影是现代地图学的重要组成部分，涉及大量的椭圆函数幂级数展开、隐函数复合函数微分、椭圆积分、复变函数运算等一系列烦琐的数学分析过程，人工推导不但费时费力，而且容易出错，有时由于难以忍受的复杂性等各种原因，甚至根本无法实现。本文主要从椭球各纬度间正反解符号表达式、不同变形性质地图投影间的直接变换、高斯投影的复变函数表示、斜轴墨卡托投影数学分析、极区海图投影及变换等5个方面，论述了地图投影计算机代数分析取得的研究进展，讨论了该领域有待进一步解决的主要问题，对推动地图投影学的发展具有积极意义。

关键词: 地图投影, 投影变换, 高斯投影, 极区投影, 计算机代数

Abstract: Map projection is an important component of modern cartography, and involves many fussy mathematical analysis processes, such as the power series expansions of elliptical functions, differential of complex and implicit functions, elliptical integral and the operation of complex numbers. The derivation of these problems by hand not only consumes much time and energy but also makes mistake easily, and sometimes can not be realized at all because of the impossible complexity. The research achievements in mathematical analysis of map projection by computer algebra are systematically reviewed in five aspects, i.e., the symbolic expressions of forward and inverse solution of ellipsoidal latitudes, the direct transformations between map projections with different distortion properties, expressions of Gauss projection by complex function, mathematical analysis of oblique Mercator projection, polar chart projection with its transformation. Main problems that need to be further solved in this research field are analyzed. It will be helpful to promote the development of map projection.

Key words: map projection, projection transformation, Gauss projection, polar projection, computer algebra

中图分类号:

P282

边少锋, 李厚朴, 李忠美. 地图投影计算机代数分析研究进展[J]. 测绘学报, 2017, 46(10): 1557-1569.

BIAN Shaofeng, LI Houpu, LI Zhongmei. Research Progress in Mathematical Analysis of Map Projection by Computer Algebra[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(10): 1557-1569.

参考文献

[1] 王家耀, 成毅. 论地图学的属性和地图的价值[J]. 测绘学报, 2015, 44(3):237-241. DOI:10.11947/j.AGCS.2015.20140406. WANG Jiayao, CHENG Yi. Discussions on the Attributes of Cartography and the Value of Map[J]. Acta Geodaetica et Cartographica Sinica, 2015, 44(3):237-241. DOI:10.11947/j.AGCS.2015.20140406.
[2] 王家耀, 何宗宜, 蒲英霞, 等. 地图学[M]. 北京:测绘出版社, 2016. WANG Jiayao, HE Zongyi, PU Yingxia, et al. Cartography[M]. Beijing:Surveying and Mapping Press, 2016.
[3] 李志林, 刘启亮, 高培超. 地图信息论:从狭义到广义的发展回顾[J]. 测绘学报, 2016, 45(7):757-767. DOI:10.11947/j.AGCS.2016.20160235. LI Zhilin, LIU Qiliang, GAO Peichao. Entropy-Based Cartographic Communication Models:Evolution from Special to General Cartographic Information Theory[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(7):757-767. DOI:10.11947/j.AGCS.2016.20160235.
[4] 吴忠性, 杨启和. 数学制图学原理[M]. 北京:测绘出版社, 1989. WU Zhongxing, YANG Qihe. Principles of Mathematical Cartography[M]. Beijing:Surveying and Mapping Press, 1989.
[5] 杨启和. 地图投影变换原理与方法[M]. 北京:解放军出版社, 1989. YANG Qihe. Principles and Methods of Map Projection Transformation[M]. Beijing:PLA Press, 1989.
[6] YANG Qihe, SNYDER J P, TOBLER W R. Map Projection Transformation:Principles and Applications[M]. London:Taylor & Francis, 2000.
[7] 吕晓华, 李少梅. 地图投影原理与方法[M]. 北京:测绘出版社, 2016. LÜ Xiaohua, LI Shaomei. Principles and Methods of Map Projection[M]. Beijing:Surveying and Mapping Press, 2016.
[8] 何宗宜. 计算机地图制图[M]. 北京:测绘出版社, 2008. HE Zongyi. Computer Aided Cartography[M]. Beijing:Surveying and Mapping Press, 2008.
[9] 闾国年, 吴平生, 周晓波. 地理信息科学导论[M]. 北京:中国科学技术出版社, 1999. LÜ Guonian, WU Pingsheng, ZHOU Xiaobo. Introduction to Geographical Information Science[M]. Beijing:China Science Technology Press, 1999.
[10] 龚健雅. 地理信息系统基础[M]. 北京:科学出版社, 2001. GONG Jianya. Foundation of Geographical Information System[M]. Beijing:Science Press, 2001.
[11] 郭仁忠. 空间分析[M]. 2版. 北京:高等教育出版社, 2001. GUO Renzhong. Spatial Analysis[M]. 2nd ed. Beijing:Higher Education Press, 2001.
[12] 庞小平. 遥感制图与应用[M]. 北京:测绘出版社, 2016. PANG Xiaoping. Remote Sensing Mapping and Its Application[M]. Beijing:Surveying and Mapping Press, 2016.
[13] 孔祥元, 郭际明, 刘宗泉. 大地测量学基础[M]. 武汉:武汉大学出版社, 2005. KONG Xiangyuan, GUO Jiming, LIU Zongquan. Foundation of Geodesy[M]. Wuhan:Wuhan University Press, 2005.
[14] 边少锋, 纪兵, 李厚朴. 卫星导航系统概论[M]. 2版. 北京:测绘出版社, 2016. BIAN Shaofeng, JI Bing, LI Houpu. Introduction to Satellite Navigation System[M]. 2nd ed. Beijing:Surveying and Mapping Press, 2016.
[15] ADAMS O S. Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-area Meridional Projection[M]. Washington D C:U.S. Government Printing Office, 1921.
[16] THOMAS P D. Conformal Projections in Geodesy and Cartography[M]. Washington D C:U.S. Government Printing Office, 1952.
[17] 方俊. 地图投影学-第二册[M]. 北京:科学出版社, 1958. FANG Jun. Map Projection[M]. Beijing:Science Press, 1958.
[18] 方炳炎. 地图投影学[M]. 北京:地图出版社, 1978. FANG Bingyan. Map Projection[M]. Beijing:Map Press, 1978.
[19] SNYDER J P. Map Projections:A Working Manual[M]. Washington D C:U.S. Government Printing Office, 1987.
[20] 华棠. 海图数学基础[M]. 北京:海潮出版社, 1985. HUA Tang. Mathematical Foundation of Chart[M]. Beijing:Haichao Press, 1985.
[21] 丁佳波. 关于等角投影解析变换的补充[J]. 测绘学报, 1985, 11(1):46-50. DING Jiabo. A Complement to the Analytical Transformation Between Conformal Projections[J]. Acta Geodaetica et Cartographica Sinica, 1985, 11(1):46-50.
[22] 孙群, 杨启和. 底点纬度解算以及等量纬度和面积函数反解问题的探讨[J]. 解放军测绘学院学报, 1985(2):64-75. SUN Qun, YANG Qihe. The Research on the Computation of the Foot-Point Latitude and the Inverse Solution of Isometric Latitude and Area Function[J]. Journal of PLA of Surveying and Mapping, 1985(2):64-75.
[23] BOWRING B R. The Transverse Mercator Projection-A Solution by Complex Numbers[J]. Survey Review, 1990, 30(237):325-342.
[24] 胡毓矩, 龚剑文, 黄伟. 地图投影[M]. 北京:测绘出版社, 1981. HU Yuju, GONG Jianwen, HUANG Wei. Map Projection[M]. Beijing:Surveying and Mapping Press, 1981.
[25] CHENG Yang. The Conformal Space Projection[J]. Cartography and Geographic Information Systems, 1996, 23(1):37-50.
[26] GRAFAREND E W, SYFFUS R. The Solution of the Korn-Lichtenstein Equations of Conformal Mapping:the Direct Generation of Ellipsoidal Gauβ-Krüger Conformal Coordinates or the Transverse Mercator Projection[J]. Journal of Geodesy, 1998, 72(5):282-293.
[27] GRAFAREND E W, YOU R J, SYFFUS R. Map Projection[M]. 2nd ed. Berlin:Springer-Verlag, 2014.
[28] KESSLER F C. A Visual Basic Algorithm for the Winkel Tripel Projection[J]. Cartography and Geographic Information Science, 2000, 27(2):177-183.
[29] OZTAN O C, IPBUKER C, ULUGTEKIN N. A Numerical Approach to Pseudo-Projections on Example Franz Mayr Projection[J]. Journal of General Command of Mapping, 2001, 125:37-50.
[30] 吕晓华, 刘宏林. 地图投影数值变换方法综合评述[J]. 测绘学院学报, 2002, 19(2):150-153. LÜ Xiaohua, LIU Honglin. A Comprehensive Appraisal of Numerical Transformation Method for Map Projection[J]. Journal of Institute of Surveying and Mapping, 2002, 19(2):150-153.
[31] 任留成. 空间投影理论及其在遥感技术中的应用[M]. 北京:科学出版社, 2003. REN Liucheng. Theory of Space Map Projection with Its Applications in Remote Sensing[M]. Beijing:Science Press, 2003.
[32] IPBUKER C, BILDIRICI I O. Computer Program for the Inverse Transformation of the Winkel Projection[J]. Journal of Surveying Engineering, 2005, 131(4):125-129.
[33] 孙达, 蒲英霞. 地图投影[M]. 南京:南京大学出版社, 2005. SUN Da, PU Yingxia. Map Projection[M]. Nanjing:Nanjing University Press, 2005.
[34] 钟业勋. 数理地图学-地图学及其数学原理[M]. 北京:测绘出版社, 2007. ZHONG Yexun. Mathematical Cartography-Cartography and Its Mathematical Elements[M]. Beijing:Surveying and Mapping Press, 2007.
[35] KARNEY C F F. Transverse Mercator with an Accuracy of A Few Nanometers[J]. Journal of Geodesy, 2011, 85(8):475-485.
[36] OSBORNE P. The Mercator Projection[M]. Edinburgh:Edinburgh University Press, 2013.
[37] 边少锋, 许江宁. 计算机代数系统与大地测量数学分析[M]. 北京:国防工业出版社, 2004. BIAN Shaofeng, XU Jiangning. Computer Algebra System and Mathematical Analysis in Geodesy[M]. Beijing:National Defense Industry Press, 2004.
[38] 杨启和. 测量和地图学中应用的六种纬度及其变换关系式[J]. 测绘科技通讯, 1995, 18(3):14-19. YANG Qihe. Six Kinds of Latitudes with Their Transformation Relations in Surveying and Mapping[J]. Science and Technology Newsletter, 1995, 18(3):14-19.
[39] 熊介. 椭球大地测量学[M]. 北京:解放军出版社, 1988. XIONG Jie. Ellipsoidal Geodesy[M]. Beijing:PLA Press, 1988.
[40] 李厚朴, 边少锋, 钟斌. 地理坐标系计算机代数精密分析理论[M]. 北京:国防工业出版社, 2015. LI Houpu, BIAN Shaofeng, ZHONG Bin. Precise Analysis Theory of Geographic Coordinate System by Computer Algebra[J]. Beijing:National Defense Industry Press, 2015.
[41] 过家春, 李厚朴, 庄云玲, 等. 依不同纬度变量的子午线弧长正反解公式的级数展开[J]. 测绘学报, 2016, 45(5):560-565. DOI:10.11947/j.AGCS.2016.20140575. GUO Jiachun, LI Houpu, ZHUANG Yunling, et al. Series Expansion for Direct and Inverse Solutions of Meridian in Terms of Different Latitude Variables[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(5):560-565. DOI:10.11947/j.AGCS.2016.20140575.
[42] 杨启和, 杨晓梅. 测量和地图学中应用的三种纬度函数及其反解变换的线性插值方法[J]. 测绘学报, 1997, 26(1):92-94, 封3. YANG Qihe, YANG Xiaomei. Three Kinds of Latitudes in Surveying and Mapping and the Linear Interpolation Method for Inverse Solution[J]. Acta Geodaetica et Cartographica Sinica, 1997, 26(1):92-94, 封3.
[43] 边少锋, 纪兵. 等距离纬度等量纬度和等面积纬度展开式[J]. 测绘学报, 2007, 36(2):218-223. DOI:10.3321/j.issn:1001-1595.2007.02.018. BIAN Shaofeng, JI Bing. The Expansions of Rectifying Latitude, Conformal Latitude and Authalic Latitude[J]. Acta Geodaetica et Cartographica Sinica, 2007, 36(2):218-223. DOI:10.3321/j.issn:1001-1595.2007.02.018.
[44] 李厚朴, 边少锋. 辅助纬度反解公式的Hermite插值法新解[J]. 武汉大学学报(信息科学版), 2008, 33(6):623-626. LI Houpu, BIAN Shaofeng. Derivation of Inverse Expansions for Auxiliary Latitudes by Hermite Interpolation Method[J]. Geomatics and Information Science of Wuhan University, 2008, 33(6):623-626.
[45] 王瑞, 李厚朴. 辅助纬度反解公式的Lagrange级数法推演[J]. 海洋测绘, 2008, 28(3):18-23. WANG Rui, LI Houpu. The Derivation of the Inverse Expansions for Auxiliary Latitudes by Lagrange Series Method[J]. Hydrographic Surveying and Charting, 2008, 28(3):18-23.
[46] 李忠美, 边少锋, 孔海英. 符号迭代法解算椭球大地测量学反问题[J]. 海洋测绘, 2013, 33(1):27-29, 33. LI Zhongmei, BIAN Shaofeng, KONG Haiying. Symbolic Iterative Method for Solving Inverse Problems in Ellipsoidal Geodesy[J]. Hydrographic Surveying and Charting, 2013, 33(1):27-29, 33.
[47] 李忠美, 李厚朴, 边少锋. 常用纬度差异极值符号表达式[J]. 测绘学报, 2014, 43(2):214-220. DOI:10.13485/j.cnki.11-2089.2014.0031. LI Zhongmei, LI Houpu, BIAN Shaofeng. Symbolic Expressions of Difference Extrema Between Regular Latitudes[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(2):214-220. DOI:10.13485/j.cnki.11-2089.2014.0031.
[48] 李厚朴, 边少锋, 刘敏. 地图投影中三种纬度间变换直接展开式[J]. 武汉大学学报(信息科学版), 2013, 38(2):217-220. LI Houpu, BIAN Shaofeng, LIU Min. Direct Expansions of Transformations Between Three Kinds of Latitudes Used in Map Projection[J]. Geomatics and Information Science of Wuhan University, 2013, 38(2):217-220.
[49] BIAN Saofeng, CHEN Yongbing. Solving An Inverse Problem of a Meridian Arc in Terms of Computer Algebra System[J]. Journal of Surveying Engineering, 2006, 132(1):7-10.
[50] 李厚朴, 刘敏, 孔海英, 等. 子午线弧长和等面积纬度函数变换的直接展开式[J]. 海洋测绘, 2011, 31(1):17-19. LI Houpu, LIU Min, KONG Haiying, et al. The Direct Expansions for Transformations Between Meridian Arc and Authalic Latitude Function[J]. Hydrographic Surveying and Charting, 2011, 31(1):17-19.
[51] 李厚朴, 边少锋, 王克平. 等距离纬度和等量纬度变换的直接展开式[C]//第二十三届海洋测绘综合性学术研讨会. 呼伦贝尔:[s.n.], 2013:189-195. LI Houpu, BIAN Shaofeng, WANG Keping. The Direct Expansions for Transformations Rectifying and Isometric Latitudes[C]//Hydrographic Surveying and Charting Symposium of 24th General Meeting. Hulunbeier:[s.n.], 2013:149-153.
[52] 李厚朴, 边少锋, 陈良友. 等面积纬度函数和等量纬度变换的直接解算公式[J]. 武汉大学学报(信息科学版), 2011, 36(7):843-846. LI Houpu, BIAN Shaofeng, CHEN Liangyou. The Direct Calculating Formulae for Transformations between Authalic Latitude Function and Isometric Latitude[J]. Geomatics and Information Science of Wuhan University, 2011, 36(7):843-846.
[53] 李厚朴, 边少锋. 不同变形性质正轴圆柱投影和正轴圆锥投影间的直接变换[J]. 测绘学报, 2012, 41(4):536-542. LI Houpu, BIAN Shaofeng. The Direct Transformations Between Normal Cylindrical and Conic Projections with Different Distortion Properties[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(4):536-542.
[54] BIAN Shaofeng, LI Houpu. Mathematical Analysis in Cartography By Means of Computer Algebra System[C]//Carlos Bateira. Cartography:A Tool for Spatial Analysis. Croatia:InTech, 2012.
[55] ENGSAGER K E, PODER K. A Highly Accurate World Wide Algorithm for the Transverse Mercator Mapping (Almost)[C]//Proceedings of XXⅢ International Cartographic Conference (ICC2007). Moscow:[s.n.], 2007.
[56] BERMEJO-SOLERA M, OTERO J. Global Optimization of the Gauss Conformal Mappings of An Ellipsoid to A Sphere[J]. Journal of Geodesy, 2010, 84(8):481-489.
[57] KARNEY C F F. Transverse Mercator with an Accuracy of A Few Nanometers[J]. Journal of Geodesy, 2011, 85(8):475-485.
[58] KAWASE K. A General Formula for Calculating Meridian Arc Length and Its Application to Coordinate Conversion in the Gauss-Krüger Projection[J]. Bulletin of the Geospatial Information Authority of Japan, 2011, 59:1-13.
[59] 程阳. 复变函数与等角投影[J]. 测绘学报, 1985, 14(1):51-60. CHENG Yang. Complex Function and Conformal Projection[J]. Acta Geodaetica et Cartographica Sinica, 1985, 14(1):51-60.
[60] BOWRING B R. The Transverse Mercator Projection——A Solution by Complex Numbers[J]. Survey Review, 1990, 30(237):325-342.
[61] KLOTZ J. Eine Analytische L?sung der Gauss-Krüger-Abbildung[J]. Zeitschrift für Versicherungswesen, 1993, 118(3):106-116.
[62] SCHUHR P. Transformationen Zwischen Ellipsoidischen Geographischen Konformen Gauss-Krüger-bzw. UTM-Koordinaten[J]. Forum, 1995(5):259-264.
[63] 边少锋, 柴洪洲, 金际航. 大地坐标系与大地基准[M]. 北京:国防工业出版社, 2005. BIAN Shaofeng, CHAI Hongzhou, JIN Jihang. Geodetic Coordinate System and Datum[M]. Beijing:National Defence Industry Press, 2005.
[64] 李厚朴, 边少锋. 高斯投影的复变函数表示[J]. 测绘学报, 2008, 37(1):5-9. DOI:10.3321/j.issn:1001-1595.2008.01.002. LI Houpu, BIAN Shaofeng. The Expressions of Gauss Projection By Complex Numbers[J]. Acta Geodaetica et Cartographica Ainica, 2008, 37(1):5-9. DOI:10.3321/j.issn:1001-1595.2008.01.002.
[65] 刘大海. 高斯投影复变换的数值计算方法[J]. 测绘科学技术学报, 2012, 29(1):9-11. LIU Dahai. Gauss Projection Numerical Method for Complex Transformation[J]. Journal of Geomatics Science and Technology, 2012, 29(1):9-11.
[66] 李厚朴, 王瑞, 边少锋. 复变函数表示的高斯投影非迭代公式[J]. 海洋测绘, 2009, 29(6):17-20. LI Houpu, WANG Rui, BIAN Shaofeng. The Non-Iterative Formulae of Gauss Projection By Complex Numbers[J]. Hydrographic Surveying and Charting, 2009, 29(6):17-20.
[67] 李忠美, 于金星, 李厚朴, 等. 高斯投影与横墨卡托投影等价性证明[J]. 海洋测绘, 2013, 33(3):17-20. LI Zhongmei, YU Jinxing, LI Houpu, et al. The Verification of Equivalence Between Gauss and Transverse Mercator Projections[J]. Hydrographic Surveying and Charting, 2013, 33(3):17-20.
[68] 刘强, 边少锋, 李忠美. 球面高斯投影及其变形的闭合公式[J]. 海军工程大学学报, 2015, 27(1):45-49, 58. LIU Qiang, BIAN Shaofeng, LI Zhongmei. Closed Formulae of Gauss Projection with Its Deformation on Sphere[J]. Journal of Naval University of Engineering, 2015, 27(1):45-49, 58.
[69] 边少锋, 刘强, 李忠美. 不分带的高斯投影实数公式[J]. 测绘通报, 2016(6):6-9. DOI:10.13474/j.cnki.11-2246.2016.0178. BIAN Shaofeng, LIU Qiang, LI Zhongmei. Expressions of Non-Zonal Gauss Projection in Real Form[J]. Bulletin of Surveying and Mapping, 2016(6):6-9. DOI:10.13474/j.cnki.11-2246.2016.0178.
[70] 金立新, 许常文, 魏桂华. 高斯投影复变函数表示的实数解[J]. 海洋测绘, 2017, 37(2):27-31. JIN Lixin, XU Changwen, WEI Guihua. Real Solution of Gauss Projection Based on Complex Function[J]. Hydrographic Surveying and Charting, 2017, 37(2):27-31.
[71] 刘强, 边少锋, 李忠美. 利用复变函数实现高斯换带的方法[J]. 海军工程大学学报, 2016, 28(1):15-19. LIU Qiang, BIAN Shaofeng, LI Zhongmei. Transformation Between Gauss Projections with Different Central Meridians Based on Complex Function[J]. Journal of Naval University of Engineering, 2016, 28(1):15-19.
[72] 李厚朴, 边少锋, 李海波. 常用等角投影及其解析变换的复变函数表示[J]. 测绘科学技术学报, 2012, 29(2):109-112, 117. LI Houpu, BIAN Shaofeng, LI Haibo. Expressions of Commonly Used Conformal Projections and Their Analytical Transformations by Complex Numbers[J]. Journal of Geomatics Science and Technology, 2012, 29(2):109-112, 117.
[73] 李厚朴, 边少锋. 高斯投影与墨卡托投影解析变换的复变函数表达式[J]. 武汉大学学报(信息科学版), 2009, 34(3):277-279. LI Houpu, BIAN Shaofeng. Expressions for Analytical Transformation Between Gauss and Mercator Projections By Complex Numbers[J]. Geomatics and Information Science of Wuhan University, 2009, 34(3):279-273.
[74] 李胜全, 李厚朴, 边少锋. 拉格朗日投影与常用等角投影间解析变换的复变函数表示[J]. 武汉大学学报(信息科学版), 2012, 37(11):1382-1385. LI Shengquan, LI Houpu, BIAN Shaofeng. Expressions of Analytical Transformations Between Lagrange and the Commonly Used Conformal Projections by Complex Numbers[J]. Geomatics and Information Science of Wuhan University, 2012, 37(11):1382-1385.
[75] HOTINE M. The Orthomorphic Projection of the Spheroid[J]. Empire Survey Review, 1946, 8(62):300-311.
[76] SNYDER J P. Calculating Map Projections for the Ellipsoid[J]. The American Cartographer, 1979, 6(1):67-76.
[77] ENGELS J, GRAFAREND E. The Oblique Mercator Projection of the Ellipsoid of Revolution IE²_a,_b[J]. Journal of Geodesy, 1995, 70(1-2):38-50.
[78] 李全海. 斜圆柱投影及其对GPS空间坐标的转换模型[J]. 同济大学学报, 2002, 30(1):106-110. LI Quanhai. Model of Oblique Cylindrical Projection for Transforming GPS Space Coordinates to Plane Coordinates[J]. Journal of Tongji University, 2002, 30(1):106-110.
[79] 沙月进, 高洪兴, 胡伍生. 斜轴圆柱投影方法及其在交通工程中的应用研究[J]. 公路交通科技, 2004, 21(11):20-22. SHA Yuejin, GAO Hongxing, HU Wusheng. Study on Oblique Cylindrical Projection Method and Its Application in Traffic Engineering[J]. Journal of Highway and Transportation Research and Development, 2004, 21(11):20-22.
[80] 吕慧玲. 斜轴墨卡托投影方法在郑西客专中的应用研究[J]. 测绘信息与工程, 2009, 34(1):26-28. Lü Huiling. Application of Oblique Mercator Projection Method to Zhengxi Special Passenger Transport Line[J]. Journal of Geomatics, 2009, 34(1):26-28.
[81] 陆鹏程, 林冬伟. 斜轴墨卡托投影模型及其应用分析[J]. 铁道勘察, 2010, 36(4):26-29. LU Pengcheng, LIN Dongwei. Model for Oblique Mercator Projections as Well as Analysis on Its Application[J]. Railway Investigation and Surveying, 2010, 36(4):26-29.
[82] 刘家臣. 高速铁路精密工程控制测量精度研究[J]. 测绘与空间地理信息, 2011, 34(6):215-218. LIU Jiachen. Research on Accuracy of Precise Engineering Control Survey for High-Speed Railway[J]. Geomatics & Spatial Information Technology, 2011, 34(6):215-218.
[83] 王文庆, 帅明明, 龚俊. 斜轴墨卡托投影在东西走向高速铁路中的应用[J]. 地理空间信息, 2013, 11(4):152-153. WANG Wenqing, SHUAI Mingming, GONG Jun. Application of Oblique Mercator Projection in the East-West High-Speed Railway[J]. Geospatial Information, 2013, 11(4):152-153.
[84] 金立新, 付宏平. 法截面子午线椭球高斯投影理论[M]. 西安:西安地图出版社, 2012. JIN Lixin, FU Hongping. Gaussian Projection Theory Based on Ellipsoid with Normal Section as the Central Meridian[M]. Xi'an:Xi'an Map Publishing House, 2012.
[85] 金立新, 付宏平. 法截面子午线椭球空间几何理论[M]. 西安:西安地图出版社, 2013. JIN Lixin, FU Hongping. Spatial Geometry Theory Based on Ellipsoid with Normal Section as the Central Meridian[M]. Xi'an:Xi'an Map Publishing House, 2013.
[86] 金立新, 付宏平. 法截面子午线椭球工程应用研究[M]. 西安:西安地图出版社, 2016. JIN Lixin, FU Hongping. Engineering Application Based on Ellipsoid with Normal Section as the Central Meridian[M]. Xi'an:Xi'an Map Publishing House, 2016.
[87] 边少锋, 刘强, 李忠美, 等. 斜轴变形椭球高斯投影方法[J]. 测绘学报, 2015, 44(10):1071-1077. DOI:10.11947/j.AGCS.2015.20140290. BIAN Shaofeng, LIU Qiang, LI Zhongmei, et al. An Alteration of Gauss Projection Based on Oblique Deformed Ellipsoid[J]. Acta Geodaetica et Cartographica Sinica, 2015, 44(10):1071-1077. DOI:10.11947/j.AGCS.2015.20140290.
[88] FOX W A W. Transverse Navigation:An Alternative to the Grid System[J]. The Journal of Navigation, 1949, 2(1):25-35.
[89] KENDALL E C. Gyro/Grid Navigation[J]. The Journal of Navigation, 1956, 9(4):429-435.
[90] DYER G C. Polar Navigation-A New Transverse Mercator Technique[J]. The Journal of Navigation, 1971, 24(4):484-495.
[91] NAUMANN J. Grid Navigation with Polar Stereographic Charts[J]. European Journal of Navigation, 2011, 9(1):4-8.
[92] SKOPELITI A, TSOULOS L. Choosing A Suitable Projection for Navigation in the Arctic[J]. Marine Geodesy, 2013, 36(2):234-259.
[93] 李树军, 张哲, 李惠雯, 等. 编制北极地区航海图有关问题的探讨[J]. 海洋测绘, 2012, 32(1):58-60. LI Shujun, ZHANG Zhe, LI Huiwen, et al. Research on Compilation of Nautical Charts of Arctic Regions[J]. Hydrographic Surveying and Charting, 2012, 32(1):58-60.
[94] 温朝江, 卞鸿巍, 王荣颖, 等. 极区极球面投影的可用性及误差分析[J]. 海军工程大学学报, 2014, 26(3):42-47. WEN Chaojiang, BIAN Hongwei, WANG Rongying, et al. Availability and Error Analysis of Polar Stereographic Projection in Polar Navigation[J]. Journal of Naval University of Engineering, 2014, 26(3):42-47.
[95] 温朝江, 卞鸿巍, 陈秋, 等. 双重极球面投影的极区投影误差分析[J]. 海洋测绘, 2015, 35(1):34-37. WEN Chaojiang, BIAN Hongwei, CHEN Qiu, et al. Error Analysis of Double Polar Stereographic Projection in Polar Regions[J]. Hydrographic Surveying and Charting, 2015, 35(1):34-37.
[96] 温朝江, 卞鸿巍, 边少锋, 等. 基于等距圆的极球面投影距离量测方法[J]. 武汉大学学报(信息科学版), 2015, 40(11):1504-1508, 1512. WEN Chaojiang, BIAN Hongwei, BIAN Shaofeng, et al. A Distance Measuring Method on Polar Stereographic Charts Based on Equidistant Circles[J]. Geomatics and Information Science of Wuhan University, 2015, 40(11):1504-1508, 1512.
[97] 张志衡, 彭认灿, 董箭, 等. 极地海区等距离正圆柱投影平面上等角航线的展绘方法[J]. 测绘科学技术学报, 2015, 32(5):535-538, 544. ZHANG Zhiheng, PENG Rencan, DONG Jian, et al. The Method for Describing Rhumb Line in Polar Sea on Equidistance Cylindrical Projection Plane[J]. Journal of Geomatics Science and Technology, 2015, 32(5):535-538, 544.
[98] 李忠美, 李厚朴, 边少锋. 极区横墨卡托投影非奇异公式及投影变形分析[C]//第二十四届海洋测绘综合性学术研讨会. 义乌:中国测绘学会海洋测绘专业委员会, 2012:189-195. LI Zhongmei, LI Houpu, BIAN Shaofeng. The Nonsingular Formulas and Deformation Analysis for Transverse Mercator Projection in Polar Regions[C]//Hydrographic Surveying and Charting Symposium of 24th General Meeting. Yiwu:CSGPC, 2012:189-195.
[99] 张晓平, 边少锋, 李忠美. 极区高斯投影与日晷投影的比较[J]. 武汉大学学报(信息科学版), 2015, 40(5):667-672. ZHANG Xiaoping, BIAN Shaofeng, LI Zhongmei. Comparisons Between Gauss and Gnomonic Projections in Polar Regions[J]. Geomatics and Information Science of Wuhan University, 2015, 40(5):667-672.
[100] 边少锋, 李忠美, 李厚朴. 极区非奇异高斯投影复变函数表示[J]. 测绘学报, 2014, 43(4):348-352, 359. DOI:10.13485/j.cnki.11-2089.2014.0052. BIAN Shaofeng, LI Zhongmei, LI Houpu. The Non-Singular Formula of Gauss Projection in Polar Regions by Complex Numbers[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(4):348-352, 359. DOI:10.13485/j.cnki.11-2089.2014.0052.
[101] 李忠美, 边少锋, 金立新, 等. 极区不分带高斯投影的正反解表达式[J]. 测绘学报, 2017, 46(6):780-788. DOI:10.11947/j.AGCS.2017.20170009. LI Zhongmei, BIAN Shaofeng, JIN Lixin, et al. Forward and Inverse Expressions of Polar Gauss Projection without Zoning Limitations[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(6):780-788. DOI:10.11947/j.AGCS.2017.20170009.
[102] 李厚朴, 边少锋, 刘强, 等. 常用极区海图投影直接变换的闭合公式[J]. 海洋测绘, 2017, 37(2):32-34, 38. LI Houpu, BIAN Shaofeng, LIU Qiang, et al. Closed Formulae of Direct Transformations Between Commonly Used Polar Chart Projections[J]. Hydrographic Surveying and Charting, 2017, 37(2):32-34, 38.
[103] 陈成. 极区海图投影及其变换研究[D]. 武汉:海军工程大学, 2015. CHEN Cheng. The Research on Polar Chart Projection with Its Transformation[D]. Wuhan:Naval University of Engineering, 2015.
[104] COLVOCORESSES A P. Space Oblique Mercator[J]. Photogrammetric Engineering and Remote Sensing, 1974, 40(8):921-926.
[105] SNYDER J P. The Space Oblique Mercator Projection[J]. Photogrammetric Engineering and Remote Sensing, 1978, 44(5):585-596.
[106] REN Liucheng, CLARKE K C, ZHOU Chenghu, et al. Geometric Rectification of Satellite Imagery with Minimal Ground Control Using Space Oblique Mercator Projection Theory[J]. Cartography and Geographic Information Science, 2010, 37(4):261-272.