辅助纬度与大地纬度间的无穷展开
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摘要
利用无穷级数理论和拉格朗日反演定理,详细推导了大地测量和制图学中常用的辅助纬度与大地纬度间的无穷展开,主要表现为参考椭球第一偏心率的幂级数形式。通过建立一系列严格的系数递推公式,得到了等量纬度反解展开式和等角纬度反解展开式;同时,推导了古德曼函数的泰勒展开式,进而得到了等角纬度正解展开式;利用级数除法公式,得到了等距离纬度正解展开式系数的行列式表示。通过比较本文方法与计算机代数系统Mathematica直接推导求得的辅助纬度正反解展开式e~0～e~(40)阶系数和相应的程序用时,表明本文算法是正确的、快速的。以CGCS2000参考椭球为例,对辅助纬度正反解进行了算例分析,也进一步验证了本文公式的正确性。
By using the theory of infinite series and the Lagrange inversion theorem, the infinite expansions of the auxiliary latitudes in geodesy and cartography with respect to the geodetic latitude are given, which are the power series of the first eccentricity of the reference ellipsoid. The inverse solutions of the isometric and the conformal latitudes can be obtained from the recurrences of a number of known coefficients. The forward expansion of the conformal latitude is proposed on the basis of the Taylor expansion of the Gudermannian function, and the forward expansion of the rectifying latitude is given in which the coefficient can be expressed as a explicit determinant by the division of power series. Compared the coefficients of the auxiliary latitudes expansions at order e~0 to e~(40) with the results by the computer algebra system Mathematica and the corresponding cost time, the algorithm of this paper is verified to be correct and fast. By taking CGCS2000 reference ellipsoid, the numerical experiment is made for the expansions and then the correctness is further proved.
By using the theory of infinite series and the Lagrange inversion theorem, the infinite expansions of the auxiliary latitudes in geodesy and cartography with respect to the geodetic latitude are given, which are the power series of the first eccentricity of the reference ellipsoid. The inverse solutions of the isometric and the conformal latitudes can be obtained from the recurrences of a number of known coefficients. The forward expansion of the conformal latitude is proposed on the basis of the Taylor expansion of the Gudermannian function, and the forward expansion of the rectifying latitude is given in which the coefficient can be expressed as a explicit determinant by the division of power series. Compared the coefficients of the auxiliary latitudes expansions at order e~0 to e~(40) with the results by the computer algebra system Mathematica and the corresponding cost time, the algorithm of this paper is verified to be correct and fast. By taking CGCS2000 reference ellipsoid, the numerical experiment is made for the expansions and then the correctness is further proved.
引文
[1] GRAFAREND E W,KRUMM F W.Map projections:cartographic information systems[M].Berlin:Springer,2006.
[2] 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 latitudes used in geodesy and cartology and their transformation relations[J].Geomatics Technology and Equipment,1995,18(3):14-19.
[3] 陈成,边少锋,李厚朴.一种解算椭球大地测量学反问题的方法及应用[J].海洋测绘,2015,35(6):8-13.CHEN Cheng,BIAN Shaofeng,LI Houpu.A method for solving inverse problems in ellipsoidal geodesy and its application[J].Hydrographic Surveying and Charting,2015,35(6):8-13.
[4] OSBORNE P.The mercator projections[M].Edinburgh:Edinburgh University Press,2008.
[5] 方俊.等面积纬度和等量纬度与地理纬度的关系[J].地理学报,1957,23(4):379-388.FANG Jun.Formulae and tables for conversion of the geographical latitude to the authalic and isometric latitudes[J].Acta Geographica Sinica,1957,23(4):379-388.
[6] YANG Q H,SNYDER J P,TOBLER W R.Map projection transformation:principles and applications[M].London:Taylor & Francis,2000.
[7] 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[M].Beijing:National Defend Industry Press,2015.
[8] 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:The People’s Liberation Army Press,1988.
[9] 华棠.海图数学基础[M].北京:海潮出版社,1985.HUA Tang.The mathematical basis of chart[M].Beijing:Haichao Press,1985.
[10] DAY J W R.The formula for finding the ordinary latitude from the isometric latitude[J].Survey Review,1988,29(230):383-385.
[11] ADAMS O S.Latitude developments connected with geodesy and cartography with tables,including a table for lambert equal-area meridional projection[M].Washington:U.S Coast and Geodetic Survey Special Publishing,1921.
[12] 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.
[13] BIAN Shaofeng,LI Houpu.Mathematical analysis in cartography by means of computer algebra system[M]//BATEIRA C.Cartography-A Tool for Spatial Analysis.Croatia:InTech,2012.
[14] 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.
[15] PETERS J M H.The gudermannian[J].The Mathematical Gazette,1984,68(445):192-196.
[16] GRADSHTEYN I S,RYZHIK I M.Table of integrals,series,and products[M].Amsterdam:Academic Press,2015.
[17] 王竹溪,郭敦仁.特殊函数概论[M].北京:北京大学出版社,2000.WANG Zhuxi,GUO Dunren.Introduction to special function[M].Beijing:Peking University Press,2000.
[18] LICHTENEGGER H.Transformation of geodetic and isometric latitude by numerical integration[J].Survey Review,1990,30(236):294-296.
[19] BOWRING D R.New ideas on isometric latitude[J].Survey Review,1990,30(236):270-280.
[20] KAYA A.An alternative formula for finding the geodetic latitude from the isometric latitude[J].Survey Review,1994,32(253):450-452.
[21] BERMEJO-SOLERA M,OTERO J.Simple and highly accurate formulas for the computation of transverse Mercator coordinates from longitude and isometric latitude[J].Journal of Geodesy,2009,83(1):1-12.
[22] 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.Research on polar nautical chart projections and their transformations[D].Wuhan:Naval University of Engineering,2015.
[23] KUZMINA R P.Asymptotic methods for ordinary differential equations[M].Dordrecht:Springer,2000.
[2] 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 latitudes used in geodesy and cartology and their transformation relations[J].Geomatics Technology and Equipment,1995,18(3):14-19.
[3] 陈成,边少锋,李厚朴.一种解算椭球大地测量学反问题的方法及应用[J].海洋测绘,2015,35(6):8-13.CHEN Cheng,BIAN Shaofeng,LI Houpu.A method for solving inverse problems in ellipsoidal geodesy and its application[J].Hydrographic Surveying and Charting,2015,35(6):8-13.
[4] OSBORNE P.The mercator projections[M].Edinburgh:Edinburgh University Press,2008.
[5] 方俊.等面积纬度和等量纬度与地理纬度的关系[J].地理学报,1957,23(4):379-388.FANG Jun.Formulae and tables for conversion of the geographical latitude to the authalic and isometric latitudes[J].Acta Geographica Sinica,1957,23(4):379-388.
[6] YANG Q H,SNYDER J P,TOBLER W R.Map projection transformation:principles and applications[M].London:Taylor & Francis,2000.
[7] 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[M].Beijing:National Defend Industry Press,2015.
[8] 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:The People’s Liberation Army Press,1988.
[9] 华棠.海图数学基础[M].北京:海潮出版社,1985.HUA Tang.The mathematical basis of chart[M].Beijing:Haichao Press,1985.
[10] DAY J W R.The formula for finding the ordinary latitude from the isometric latitude[J].Survey Review,1988,29(230):383-385.
[11] ADAMS O S.Latitude developments connected with geodesy and cartography with tables,including a table for lambert equal-area meridional projection[M].Washington:U.S Coast and Geodetic Survey Special Publishing,1921.
[12] 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.
[13] BIAN Shaofeng,LI Houpu.Mathematical analysis in cartography by means of computer algebra system[M]//BATEIRA C.Cartography-A Tool for Spatial Analysis.Croatia:InTech,2012.
[14] 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.
[15] PETERS J M H.The gudermannian[J].The Mathematical Gazette,1984,68(445):192-196.
[16] GRADSHTEYN I S,RYZHIK I M.Table of integrals,series,and products[M].Amsterdam:Academic Press,2015.
[17] 王竹溪,郭敦仁.特殊函数概论[M].北京:北京大学出版社,2000.WANG Zhuxi,GUO Dunren.Introduction to special function[M].Beijing:Peking University Press,2000.
[18] LICHTENEGGER H.Transformation of geodetic and isometric latitude by numerical integration[J].Survey Review,1990,30(236):294-296.
[19] BOWRING D R.New ideas on isometric latitude[J].Survey Review,1990,30(236):270-280.
[20] KAYA A.An alternative formula for finding the geodetic latitude from the isometric latitude[J].Survey Review,1994,32(253):450-452.
[21] BERMEJO-SOLERA M,OTERO J.Simple and highly accurate formulas for the computation of transverse Mercator coordinates from longitude and isometric latitude[J].Journal of Geodesy,2009,83(1):1-12.
[22] 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.Research on polar nautical chart projections and their transformations[D].Wuhan:Naval University of Engineering,2015.
[23] KUZMINA R P.Asymptotic methods for ordinary differential equations[M].Dordrecht:Springer,2000.