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Article(id=1281909276234977513, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1281909275651969257, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.12.421, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1764604800000, receivedDateStr=2025-12-02, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1783560588177, onlineDateStr=2026-07-09, pubDate=1781452800000, pubDateStr=2026-06-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1783560588177, onlineIssueDateStr=2026-07-09, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1783560588177, creator=13701087609, updateTime=1783560588177, updator=13701087609, issue=Issue{id=1281909275651969257, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='6', pageStart='662', pageEnd='789', issueExtLink='null', onlineDate='null', pubDate='1781452800000', pubDateStr='2026-06-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1783560588038, creator='13701087609', updateTime=1783566454347, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1281933881221812905, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1281909275651969257, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1281933881221812906, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1281909275651969257, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=662, endPage=667, 717, ext={EN=ArticleExt(id=1281909276444692714, articleId=1281909276234977513, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Coordinate Inverse Transformation for the Triaxial Geoid Model, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

By employing fundamental methods from the theory of curves and surfaces, the concepts of different types of latitude and longitude (reduced latitude and longitude, geometric latitude and longitude, and geodetic latitude and longitude) at an arbitrary point on a triaxial ellipsoidal surface are proposed. Theoretical relationship models are established between Cartesian coordinates (spatial rectangular coordinates) and geodetic coordinates (geodetic latitude, longitude, and height), as well as between Cartesian coordinates and geometric coordinates (geometric latitude, longitude, and height), for any point in space both inside and outside the ellipsoid. Based on the theoretical relationships among the different latitude and longitude types, approximate geodetic coordinates of the study point are derived. Using these approximations as initial values, a novel method for the inverse transformation of coordinates is presented via Newton's iterative approach. Extensive numerical calculations demonstrate that, for any arbitrary point in space, the proposed method accomplishes the transformation from Cartesian to geodetic coordinates nearly instantaneously; even for points near the Earth's surface, only three iterations are required to achieve convergence.

, authors=Hanwei ZHANG1, Pengfei XU1, *, Ke WANG1, authorsList=Hanwei ZHANG, Pengfei XU, Ke WANG, authorCompany=null, correspAuthors=Pengfei XU, authorNote=null, correspAuthorsNote=null, copyrightStatement=All rights reserved. Unauthorized reproduction is prohibited., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1281909279355539700, articleId=1281909276234977513, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=三轴地球椭球模型的坐标逆变换, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

借助曲线曲面论的基本方法, 提出了三轴椭球曲面上任意点不同经纬度(归化经纬度、几何经纬度和大地经纬度)的概念, 建立了椭球面内外空间任意一点的笛卡尔坐标(空间直角坐标)与大地坐标(大地经纬度和大地高), 以及笛卡尔坐标与几何坐标(几何经纬度和几何高)之间的理论关系模型。根据不同经纬度之间的理论关系, 给出了研究点对应的大地坐标近似值。以此近似值为基础, 采用牛顿迭代方法, 给出了坐标逆变换的一个新方法。大量数值计算表明, 对于空间任意点, 利用新方法可以基本瞬时完成由笛卡尔坐标到大地坐标的计算, 即使是地面附近的点也只需迭代3次即可完成。

, authors=张捍卫1, 徐鹏飞1, *, 王珂1, authorsList=张捍卫, 徐鹏飞, 王珂, authorCompany=null, correspAuthors=徐鹏飞, authorNote=

张捍卫, 博士, 教授, 博士生导师, 主要从事大地测量学教学与研究, E-mail:

, correspAuthorsNote=
徐鹏飞, 博士, 讲师, 主要从事大地测量学教学与研究, E-mail:
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张捍卫, 博士, 教授, 博士生导师, 主要从事大地测量学教学与研究, E-mail:

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Journal of Geodesy, 2012, 86 (4): 249- 256, articleTitle=Cartesian to Geodetic Coordinates Conversion on a Triaxial Ellipsoid, refAbstract=null), Reference(id=1281933702926155874, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1281909276234977513, doi=null, pmid=null, pmcid=null, year=2015, volume=2, issue=2, pageStart=609, pageEnd=616, url=null, language=null, rfNumber=17, rfOrder=17, authorNames=Bektas S, journalName=International Multidisciplinary Scientific GeoConference: SGEM, refType=null, unstructuredReference= Bektas S . Which Reference Surface? Rotational or Triaxial Ellipsoid[J]. International Multidisciplinary Scientific GeoConference: SGEM, 2015, 2 (2): 609- 616, articleTitle=Which Reference Surface? Rotational or Triaxial Ellipsoid, refAbstract=null), Reference(id=1281933703010041955, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1281909276234977513, doi=null, pmid=null, pmcid=null, year=2017, volume=120, issue=null, pageStart=192, pageEnd=207, url=null, language=null, rfNumber=18, rfOrder=18, authorNames=Husár L, Švaral P, Janák J, journalName=Journal of Geometry and Physics, refType=null, unstructuredReference= Husár L , Švaral P , Janák J . About the Geometry of the Earth Geodetic Reference Surfaces[J]. 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Parameters of the Earth's mean ellipsoid (biaxial ellipsoid)

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双轴椭球名称 平均赤道半径/m 极扁率e
1841贝塞尔椭球 6 377 397 1/299.152
1866克拉克椭球 6 378 206 1/294.978
1910海福特椭球 6 378 388 1/297.000
1940克拉索夫斯基椭球 6 378 245 1/298.300
1967鲁塞恩椭球(14届IAG推荐值) 6 378 160 1/298.247
1975格勒诺布尔椭球(16届IAG推荐值) 6 378 140 1/298.257
1979堪培拉椭球(17届IAG推荐值) 6 378 137 1/298.257
1983汉堡椭球(18届IAG和IERS1989推荐值) 6 378 136 1/298.257
IERS1992推荐值 6 378 136.3 1/298.257
IERS1996推荐值 6 378 136.49 1/298.256 45
IERS2003和IERS2010推荐值 6 378 136.6 1/298.256 42
Hadi Amin椭球[13] 6 378 137.678 1/298.256 86
), ArticleFig(id=1281933699285499978, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1281909276234977513, language=CN, label=表1, caption=

地球平均椭球(双轴椭球)参数

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双轴椭球名称 平均赤道半径/m 极扁率e
1841贝塞尔椭球 6 377 397 1/299.152
1866克拉克椭球 6 378 206 1/294.978
1910海福特椭球 6 378 388 1/297.000
1940克拉索夫斯基椭球 6 378 245 1/298.300
1967鲁塞恩椭球(14届IAG推荐值) 6 378 160 1/298.247
1975格勒诺布尔椭球(16届IAG推荐值) 6 378 140 1/298.257
1979堪培拉椭球(17届IAG推荐值) 6 378 137 1/298.257
1983汉堡椭球(18届IAG和IERS1989推荐值) 6 378 136 1/298.257
IERS1992推荐值 6 378 136.3 1/298.257
IERS1996推荐值 6 378 136.49 1/298.256 45
IERS2003和IERS2010推荐值 6 378 136.6 1/298.256 42
Hadi Amin椭球[13] 6 378 137.678 1/298.256 86
), ArticleFig(id=1281933699386163275, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1281909276234977513, language=EN, label=Tab. 2, caption=

Parameters of the Earth's mean ellipsoid (triaxial ellipsoid)

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三轴椭球名称 长半轴/m 长半轴方向 赤道扁率f 极扁率e
Schliephake1955模型 6 378 245 15°E 1/30 001 1/298.299 7
Eitschberger1978模型 6 378 173.435 14°53′42″W 1/91 726 1/297.780 6
第18届IAG推荐值 6 378 172 14°54′W 1/92 000 1/297.776 0
GEM-T3引力场模型采用值 6 378 171.55 14°55′42.6″W 1/91 035 1/297.766 2
JGM-3引力场模型采用值 6 378 171.55 14°55′44.8″W 1/91 026 1/297.766 1
Ligas2012模型[16] 6 378 172 1/92 437 1/297.781 0
), ArticleFig(id=1281933699465855052, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1281909276234977513, language=CN, label=表2, caption=

地球平均椭球(三轴椭球)参数

, figureFileSmall=null, figureFileBig=null, tableContent=
三轴椭球名称 长半轴/m 长半轴方向 赤道扁率f 极扁率e
Schliephake1955模型 6 378 245 15°E 1/30 001 1/298.299 7
Eitschberger1978模型 6 378 173.435 14°53′42″W 1/91 726 1/297.780 6
第18届IAG推荐值 6 378 172 14°54′W 1/92 000 1/297.776 0
GEM-T3引力场模型采用值 6 378 171.55 14°55′42.6″W 1/91 035 1/297.766 2
JGM-3引力场模型采用值 6 378 171.55 14°55′44.8″W 1/91 026 1/297.766 1
Ligas2012模型[16] 6 378 172 1/92 437 1/297.781 0
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三轴地球椭球模型的坐标逆变换
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张捍卫 1 , 徐鹏飞 1, * , 王珂 1
大地测量与地球动力学 | 大地测量学 2026,46(6): 662-667, 717
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大地测量与地球动力学 |大地测量学 2026 , 46 (6) : 662 -667, 717
三轴地球椭球模型的坐标逆变换
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张捍卫1 , 徐鹏飞1, * , 王珂1
作者信息
  • 1 山东理工大学建筑工程与空间信息学院, 淄博, 255000
通讯作者:
徐鹏飞, 博士, 讲师, 主要从事大地测量学教学与研究, E-mail:
作者简介:

张捍卫, 博士, 教授, 博士生导师, 主要从事大地测量学教学与研究, E-mail:

Coordinate Inverse Transformation for the Triaxial Geoid Model
Hanwei ZHANG1 , Pengfei XU1, * , Ke WANG1
Affiliations
  • 1 School of Civil Engineering and Geomatics, Shandong University of Technology, Zibo 255000, China
出版时间: 2026-06-15 doi: 10.14075/j.jgg.2025.12.421
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摘要
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借助曲线曲面论的基本方法, 提出了三轴椭球曲面上任意点不同经纬度(归化经纬度、几何经纬度和大地经纬度)的概念, 建立了椭球面内外空间任意一点的笛卡尔坐标(空间直角坐标)与大地坐标(大地经纬度和大地高), 以及笛卡尔坐标与几何坐标(几何经纬度和几何高)之间的理论关系模型。根据不同经纬度之间的理论关系, 给出了研究点对应的大地坐标近似值。以此近似值为基础, 采用牛顿迭代方法, 给出了坐标逆变换的一个新方法。大量数值计算表明, 对于空间任意点, 利用新方法可以基本瞬时完成由笛卡尔坐标到大地坐标的计算, 即使是地面附近的点也只需迭代3次即可完成。

地球椭球  /  归化经纬度  /  几何经纬度  /  大地经纬度  /  大地坐标

By employing fundamental methods from the theory of curves and surfaces, the concepts of different types of latitude and longitude (reduced latitude and longitude, geometric latitude and longitude, and geodetic latitude and longitude) at an arbitrary point on a triaxial ellipsoidal surface are proposed. Theoretical relationship models are established between Cartesian coordinates (spatial rectangular coordinates) and geodetic coordinates (geodetic latitude, longitude, and height), as well as between Cartesian coordinates and geometric coordinates (geometric latitude, longitude, and height), for any point in space both inside and outside the ellipsoid. Based on the theoretical relationships among the different latitude and longitude types, approximate geodetic coordinates of the study point are derived. Using these approximations as initial values, a novel method for the inverse transformation of coordinates is presented via Newton's iterative approach. Extensive numerical calculations demonstrate that, for any arbitrary point in space, the proposed method accomplishes the transformation from Cartesian to geodetic coordinates nearly instantaneously; even for points near the Earth's surface, only three iterations are required to achieve convergence.

Earth ellipsoid  /  reduced latitude and longitude  /  geometric latitude and longitude  /  geodetic latitude and longitude  /  geodetic coordinates
张捍卫, 徐鹏飞, 王珂. 三轴地球椭球模型的坐标逆变换. 大地测量与地球动力学, 2026 , 46 (6) : 662 -667, 717 . DOI: 10.14075/j.jgg.2025.12.421
Hanwei ZHANG, Pengfei XU, Ke WANG. Coordinate Inverse Transformation for the Triaxial Geoid Model[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (6) : 662 -667, 717 . DOI: 10.14075/j.jgg.2025.12.421
正文
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最接近于大地体(大地水准面闭合形体)的椭球称为平均椭球。平均椭球曲面不但是大地测量计算的基准面，而且也是确定大地水准面形状和地图投影的参考面[1]。在大地测量计算中，经常涉及空间任意点的大地坐标与笛卡尔坐标之间的转换。其中，由大地坐标(大地经纬度和大地高)计算笛卡尔坐标(空间直角坐标)称为正变换，由笛卡尔坐标求解大地坐标称为逆变换。Featherstone等[2]和Panou[3]系统性地总结了历年来有关逆变换的理论和方法。Fok等[4]依据逆变换计算过程的稳定性、准确性和速度等指标，对不同逆变换方法进行了比较。逆变换方法主要分为2类：一是解析方法[5]；二是数值方法[3, 6]。Seemkooei[7]指出，解析方法不如数值方法准确，而且运算速度也比较慢。如果平均椭球是非旋转对称的，那么坐标逆变换过程会变得更加困难。近年来，很多学者针对三轴椭球模型，研究由笛卡尔坐标求解大地坐标的新理论和新方法[8-10]。Kopeilin[11]研究在相对论框架下三轴参考椭球体和正常重力场表述方法。可确信未来的椭球大地测量学应该是三轴椭球大地测量学。
在三轴椭球情况下，大地坐标与笛卡尔坐标之间的理论关系是什么？如何进行逆变换？是亟待研究的问题。假设椭球中心到椭球面上任意点的向径为r，椭球体的短轴方向为e3，椭球面上任意点切平面的法线方向由单位向量n表示。对于双轴椭球体而言，n始终位于向径r和短轴e3所决定的平面内；而对于三轴椭球体，n则不再处于该平面内。这就是三轴椭球与双轴椭球的本质差别，也是用三轴椭球表示大地体给测量计算带来不便的原因。利用微分几何学中的曲线和曲面描述方法，能够给出三轴椭球模型的大地坐标与笛卡尔坐标之间的理论关系；通过引入归化经纬度、几何经纬度和大地经纬度的概念，可给出三轴椭球模型坐标逆变换的一个新方法。
1 地球平均椭球模型
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以椭球几何中心为坐标原点，3个坐标轴分别与椭球的长轴、中轴和短轴重合的直角坐标系称为几何主轴直角坐标系。在此坐标系下，三轴椭球的曲面方程可表述为[1]：
$\begin{equation*}\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \end{equation*}$
式中，a、b、c分别称为三轴椭球体的长半轴、中半轴和短半轴，且a≥b≥c>0。
定义极扁率e和赤道扁率f分别为：
$\begin{equation*}e=\frac{a-c}{a}, f=\frac{a-b}{a} \end{equation*}$
关于双轴地球椭球的几何参数见表 1。表 1给出的是地球椭球的平均赤道半径，并不是地球椭球的长半轴，它们之间有严格的转换关系[12]。自从IERS1996规范发布以来，虽然考虑了不同潮汐约定，但是仍然假设平均椭球的短轴与国际地球参考系(ITRS)第三轴重合。
通过卫星轨道动力学得到的赤道扁率约为1/90 000。此数值大约为卫星出现前计算的数值的1/3(表 2)。Burša[14]解释了出现这种差异的具体原因。表 2列举了几个主要的三轴地球椭球模型的几何参数。其中，Burša等[12]给出的地球、月球和火星形状的几何参数被第十八届国际大地测量协会(IAG)采纳。关于潮汐和时间系统的约定请参阅IERS规范和文献[15]。
Bektas[17]指出，在计算大地高时，双轴椭球和三轴椭球模型的差异是：赤道处是70 m，在纬度45°处是35 m；大地纬度差异是2.267″。Husár等[18]指出，赤道扁率可导致经纬线偏离正交性几个毫角秒；三轴椭球比双轴椭球更接近于大地体，接近程度提高了26%以上。因此，建议在未来的大地测量计算中，采用三轴椭球面或更高级的曲面。
2 各种经纬度的定义
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假设$\hat{\boldsymbol{e}}_{1} $、$\hat{\boldsymbol{e}}_{2} $和$\hat{\boldsymbol{e}}_{3} $分别是几何主轴直角坐标系的单位坐标基，且在此坐标系中，椭球曲面方程通过式(1)表示。本节给出各种经纬度的定义。
2.1 归化经纬度
根据式(1)，可设椭球面上任意点P的向径为：
$\begin{gather*}\boldsymbol{r}=(a \sin u \cos v) \hat{\boldsymbol{e}}_{1}+(b \sin u \sin v) \hat{\boldsymbol{e}}_{2}+ \\(c \cos u) \hat{\boldsymbol{e}}_{3}=x \hat{\boldsymbol{e}}_{1}+y \hat{\boldsymbol{e}}_{2}+z \hat{\boldsymbol{e}}_{3}=\rho \hat{\boldsymbol{r}} \end{gather*}$
式中，u∈[0, π]称为归化余纬度；v∈[0，2π)称为归化经度。ρ和$\hat{\boldsymbol{r}} $分别是椭球面上任意点P的向径长度和单位向量。其中：
$\begin{equation*}\rho^{2}=(a \sin u \cos v)^{2}+(b \sin u \sin v)^{2}+(c \cos u)^{2} \end{equation*}$
而单位向量$\hat{\boldsymbol{r}} $为：
$\begin{gather*}\hat{\boldsymbol{r}}=\rho^{-1}\left[(a \sin u \cos v) \hat{\boldsymbol{e}}_{1}+\right. \\\left.(b \sin u \sin v) \hat{\boldsymbol{e}}_{2}+(c \cos u) \hat{\boldsymbol{e}}_{3}\right] \end{gather*}$
可见，利用归化经纬度描述曲面是最方便的，但不具有直观的几何意义。
2.2 几何经纬度
如果将椭球面上任意点P的单位向量写为：
$\begin{equation*}\hat{\boldsymbol{r}}=(\sin \theta \cos \lambda) \hat{\boldsymbol{e}}_{1}+(\sin \theta \sin \lambda) \hat{\boldsymbol{e}}_{2}+(\cos \theta) \hat{\boldsymbol{e}}_{3} \end{equation*}$
那么，θ∈[0, π]称为几何余纬度；λ∈[0, 2π)称为几何经度。
根据式(6)可得：
$\begin{gather*}\hat{\boldsymbol{e}}_{3} \times \hat{\boldsymbol{r}}=-(\sin \theta \sin \lambda) \hat{\boldsymbol{e}}_{1}+(\sin \theta \cos \lambda) \hat{\boldsymbol{e}}_{2}= \\{\left[-(\sin \lambda) \hat{\boldsymbol{e}}_{1}+(\cos \lambda) \hat{\boldsymbol{e}}_{2}\right] \sin \theta=\hat{\boldsymbol{k}} \sin \theta} \end{gather*}$
式中，单位向量$\hat{\boldsymbol{k}} $称为点P几何子午面的法线向量。
根据式(6)和式(7)可得：
$\left\{\begin{array}{l}\hat{\boldsymbol{e}}_{3} \cdot \hat{\boldsymbol{r}}=\cos \theta \\\hat{\boldsymbol{e}}_{3} \times \hat{\boldsymbol{r}}=\hat{\boldsymbol{k}} \sin \theta \\\hat{\boldsymbol{e}}_{2} \cdot \hat{\boldsymbol{k}}=\cos \lambda \\\hat{\boldsymbol{e}}_{2} \times \hat{\boldsymbol{k}}=\hat{\boldsymbol{e}}_{3} \sin \lambda\end{array}\right.$
式(8)也可以理解为关于几何经纬度的定义式。
如果令某空间点沿着椭球面上点的径向方向高出椭球面h(几何高)，那么就有：
$\begin{equation*}\boldsymbol{R}=\boldsymbol{r}+h \hat{\boldsymbol{r}}_{0} \end{equation*}$
利用式(3)和式(6)，可将式(9)表示为：
$\begin{gather*}\boldsymbol{R}=X \hat{\boldsymbol{e}}_{1}+Y \hat{\boldsymbol{e}}_{2}+Z \hat{\boldsymbol{e}}_{3}=(\rho+h) \hat{\boldsymbol{r}}= \\(\rho+h)\left[(\sin \theta \cos \lambda) \hat{\boldsymbol{e}}_{1}+(\sin \theta \sin \lambda) \hat{\boldsymbol{e}}_{2}+(\cos \theta) \hat{\boldsymbol{e}}_{3}\right] \end{gather*}$
这就是用几何经纬度表示的椭球面外某点的直角坐标。
2.3 大地经纬度
令椭球曲面上任意点P的切平面的法线向量为$\hat{\boldsymbol{n}} $，其表达式为：
$\begin{equation*}\hat{\boldsymbol{n}}=\left|\frac{\partial \boldsymbol{r}}{\partial u} \times \frac{\partial \boldsymbol{r}}{\partial v}\right|^{-1}\left(\frac{\partial \boldsymbol{r}}{\partial u} \times \frac{\partial \boldsymbol{r}}{\partial v}\right) \end{equation*}$
将式(3)代入式(11)可得：
$\begin{gather*}\hat{\boldsymbol{n}}=\frac{1}{\eta}\left[(b c \sin u \cos v) \hat{\boldsymbol{e}}_{1}+\right. \\\left.(a c \sin u \sin v) \hat{\boldsymbol{e}}_{2}+(a b \cos u) \hat{\boldsymbol{e}}_{3}\right] \end{gather*}$
其中，
$\begin{equation*}\eta=\left[(b c \sin u \cos v)^{2}+(a c \sin u \sin v)^{2}+(a b \cos u)^{2}\right]^{\frac{1}{2}} \end{equation*}$
根据式(12)，可令：
$\left\{\begin{array}{l}\hat{\boldsymbol{e}}_{3} \cdot \hat{\boldsymbol{n}}=\cos B \\\hat{\boldsymbol{e}}_{3} \times \hat{\boldsymbol{n}}=\hat{\boldsymbol{l}} \sin B \\\hat{\boldsymbol{e}}_{2} \cdot \hat{\boldsymbol{l}}=\cos L \\\hat{\boldsymbol{e}}_{2} \times \hat{\boldsymbol{l}}=\hat{\boldsymbol{e}}_{3} \sin L\end{array}\right.$
式中，B∈[0, π]称为大地余纬度；L∈[0, 2π)称为大地经度。单位向量$\hat{\boldsymbol{l}} $称为点P的大地子午面的法线向量。
如果空间点Q沿着椭球面上某点的法线方向高出椭球面H(大地高)，那么就有：
$\begin{equation*}\boldsymbol{R}=\boldsymbol{r}+H \hat{\boldsymbol{n}} \end{equation*}$
根据式(3)和式(12)，式(15)可表示为：
$\begin{gathered}\boldsymbol{R}=X \hat{\boldsymbol{e}}_1+Y \hat{\boldsymbol{e}}_2+Z \hat{\boldsymbol{e}}_3=\left[\left(\frac{H}{\eta} b c+a\right) \sin u \cos v\right] \hat{\boldsymbol{e}}_1+ \\{\left[\left(\frac{H}{\eta} a c+b\right) \sin u \sin v\right] \hat{\boldsymbol{e}}_2+\left[\left(\frac{H}{\eta} a b+c\right) \cos u\right] \hat{\boldsymbol{e}}_3}\end{gathered}$
式(16)即为用归化经纬度表示的空间点Q的直角坐标。
3 大地经纬度和归化经纬度之间的理论关系
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基于式(12)和式(13)，利用式(14)可得：
$\left\{\begin{array}{l}\cos B=\frac{a b \cos u}{\sqrt{(b c \sin u \cos v)^{2}+(a c \sin u \sin v)^{2}+(a b \cos u)^{2}}} \\\sin B=\frac{\sqrt{(b c \sin u \cos v)^{2}+(a c \sin u \sin v)^{2}}}{\sqrt{(b c \sin u \cos v)^{2}+(a c \sin u \sin v)^{2}+(a b \cos u)^{2}}}\end{array}\right.$
以及，
$\left\{\begin{array}{l}\cos L=\frac{b \cos v}{\left[(b \cos v)^{2}+(a \sin v)^{2}\right]^{\frac{1}{2}}} \\\sin L=\frac{a \sin v}{\left[(b \cos v)^{2}+(a \sin v)^{2}\right]^{\frac{1}{2}}}\end{array}\right.$
以上两式即为利用归化经纬度计算大地经纬度的理论公式。
式(17)的反变换为：
$\left\{\begin{array}{l}\cos v=\frac{a \cos L}{\left[(a \cos L)^{2}+(b \sin L)^{2}\right]^{\frac{1}{2}}} \\\sin v=\frac{b \sin L}{\left[(a \cos L)^{2}+(b \sin L)^{2}\right]^{\frac{1}{2}}}\end{array}\right.$
利用式(17)，并考虑到式(19)，可得：
$\left\{\begin{array}{l}\cos u=\frac{c \cos B}{\sqrt{(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}+(c \cos B)^{2}}} \\\sin u=\frac{\sqrt{(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}}}{\sqrt{(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}+(c \cos B)^{2}}}\end{array}\right.$
以上两式即为利用大地经纬度计算归化经纬度的理论公式。
4 三轴椭球下的笛卡尔坐标与大地坐标
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将式(19)和式(20)代入式(13)可得：
$\begin{equation*}\eta^{2}=\frac{(a b c)^{2}}{(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}+(c \cos B)^{2}} \end{equation*}$
将式(21)、式(19)和式(20)代入式(16)可得：
$\left\{\begin{array}{l}X=\left[\frac{a^{2}}{\sqrt{\left[(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}\right]+(c \cos B)^{2}}}+H\right] \sin B \cos L \\Y=\left[\frac{b^{2}}{\sqrt{\left[(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}\right]+(c \cos B)^{2}}}+H\right] \sin B \sin L \\Z=\left[\frac{c^{2}}{\sqrt{\left[(a \sin B \cos L)^{2}+(b \sin B \sin L)^{2}\right]+(c \cos B)^{2}}}+H\right] \cos B\end{array}\right.$
上式即为在三轴椭球模型情况下，空间点Q的大地坐标与笛卡尔坐标之间的关系。值得注意的是，大地经度L指的是椭球面上任意点大地子午面与xz平面(椭球半长轴和半短轴)的二面角，而非与格林尼治大地子午面的二面角。大地高的取值范围为：H∈[-c，+∞)。
在双轴椭球情况下(a=b>c)，根据式(22)，有：
$\left\{\begin{array}{l}X=\left(\frac{a^{2}}{\sqrt{(a \sin B)^{2}+(c \cos B)^{2}}}+H\right) \sin B \cos L \\Y=\left(\frac{b^{2}}{\sqrt{(a \sin B)^{2}+(c \cos B)^{2}}}+H\right) \sin B \sin L \\Z=\left(\frac{c^{2}}{\sqrt{(a \sin B)^{2}+(c \cos B)^{2}}}+H\right) \cos B\end{array}\right.$
由于旋转对称，可设置格林尼治子午面作为经度起算面。
大地坐标与笛卡尔直角坐标之间的逆变换主要有解析方法[5]、数值方法[3, 6]和三轴椭球的逆变换方法[10-11]等。本文通过引入归化经纬度、几何经纬度和大地经纬度概念，基于不同经纬度之间的理论关系，利用迭代方法进行逆变换。
5 由笛卡尔坐标求解大地坐标的新方法
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假设空间任意点Q在几何主轴直角坐标系下的直角坐标是已知的，且直线OQ与椭球曲面的交点是G(图 1)。
第一步，利用式(10)计算点Q(或点G)的几何经纬度：
$\left\{\begin{array}{l}\cos ^{2} \theta=\frac{Z^{2}}{X^{2}+Y^{2}+Z^{2}} \\\cos ^{2} \lambda=\frac{X^{2}}{X^{2}+Y^{2}}\end{array}\right.$
在此基础上，再利用式(10)和式(4)计算Q点的几何高：
$\begin{gather*}h=\left[\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}+\frac{Z^{2}}{c^{2}}\right]^{-\frac{1}{2}} .\\{\left[\left(\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}+\frac{Z^{2}}{c^{2}}\right)^{\frac{1}{2}}-1\right]\left(X^{2}+Y^{2}+Z^{2}\right)^{\frac{1}{2}}} \end{gather*}$
值得注意的是，在此用到几何经纬度和归化经纬度之间的关系，即：
$\left\{\begin{array}{l}\cos ^{2} v=\frac{(b \cos \lambda)^{2}}{(a \sin \lambda)^{2}+(b \cos \lambda)^{2}} \\\cos ^{2} u=\frac{(a b \cos \theta)^{2}}{(a b \cos \theta)^{2}+\left((a \sin \lambda)^{2}+(b \cos \lambda)^{2}\right)(c \sin \theta)^{2}}\end{array}\right.$
上式类似于本文第3节中式(17)~(20)的推导，具体过程略。
第二步，基于式(24)，计算G点的归化经纬度：
$\left\{\begin{array}{l}\cos ^{2} u_{G}=\left[\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}+\frac{Z^{2}}{c^{2}}\right]^{-1} \frac{Z^{2}}{c^{2}} \\\cos ^{2} v_{G}=\left[\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}\right]^{-1} \frac{X^{2}}{a^{2}}\end{array}\right.$
值得注意的是，在此用到几何经纬度和归化经纬度之间的关系，即式(26)。
第三步，基于式(27)，利用式(17)和式(18)计算G点的大地经纬度：
$\left\{\begin{array}{l}\cos ^{2} B_{G}=\left[\left(\frac{b c X}{a}\right)^{2}+\left(\frac{a c Y}{b}\right)^{2}+\left(\frac{a b Z}{c}\right)^{2}\right]^{-1}\left(\frac{a b Z}{c}\right)^{2} \\\cos ^{2} L_{G}=\left[\left(\frac{b X}{a}\right)^{2}+\left(\frac{a Y}{b}\right)^{2}\right]^{-1}\left(\frac{b X}{a}\right)^{2}\end{array}\right.$
至此，根据空间点Q的直角坐标(X, Y, Z)，得到了椭球面上G点的大地坐标(BG, LG)。在以上计算过程中，需要根据点Q的直角坐标判断式(24)中的几何经纬度的象限。由于不同经纬度永远在同一个象限内，因此可得出椭球面上点G的大地经纬度所在象限。
实际上，我们需要求出椭球面上点P的大地经纬度(BP, LP)和大地高H。由于地球几何扁率的微小性，点P和点G的大地经纬度差别很小。因此，以Q点的几何高h=$\overline{GP}$作为Q点大地高H的近似值，以点G的(BG, LG)作为点P的(BP, LP)的近似值，代入式(22)求解空间直角坐标的变化量，然后采用牛顿迭代方法求解大地坐标的变化量。直到计算的大地坐标在亚毫米量级上满足式(22)即可。
6 数值计算
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利用符号gBP、gLP和gHP分别表示式(22)中的B、L和H。这样，利用式(22)计算空间点Q的直角坐标，记为XQ、YQ和ZQ。然后，根据点Q的直角坐标，利用§5给出的逆变换方法，计算椭球面上点P的大地坐标(大地余纬度GeoB和大地经度GeoL)，以及空间点Q的大地高GeoH。因此，本文将gBP、gLP和gHP称为理论值或真值；将GeoB、GeoL和GeoH称为计算值或者观测值。3个差值(GeoB-gBP，GeoL-gLP，GeoH-gHP)反映的是式(22)逆变换方法的准确度。
1) 设定gLP=45°和gHP=750 m，图 2反映的是3个差值随大地余纬度的变化情况。
2) 设定gBP和gLP都是45°，图 3反映的是3个差值随大地高的变化情况。
3) 设定gBP=15°和gHP=700 m，图 4反映的是3个差值随大地经度的变化情况。
图 2表示的是：当已知大地经度L=45°，大地高H=750 m，以及大地余纬度B∈[0°, 180°]的情况下，逆变换误差随大地余纬度的变化情况。由图 2可见，大地余纬度的逆变换误差约为3 nas，大地经度的逆变换误差约为0.3 nas，大地高的逆变换误差约为0.5 μm。当大地经度和大地高取其他数值时，误差随余纬度的变化曲线和误差大小基本相同。
图 3表示的是：当已知大地经度L=45°，大地余纬度B=45°，以及大地高H∈[-70 km, +70 km]的情况下，逆变换误差随大地高的变化情况。由图 3可见，大地余纬度的逆变换误差约为100 nas，大地经度的逆变换误差约为30 nas，大地高的逆变换误差约为150 μm。当大地经度和大地余纬度取其他数值时，误差随大地高的变化曲线和误差大小基本相同。
图 4表示的是：当已知大地余纬度L=15°，大地高H=700 m，以及大地经度L∈[0, 360°]的情况下，逆变换误差随大地经度的变化情况。由图 4可见，大地余纬度的逆变换误差大小约为1 nas，大地经度的逆变换误差约为0.2 nas，大地高的逆变换误差约为0.1 μm。当大地余纬度和大地高取其他数值时，误差随大地经度的变化曲线和误差大小基本相同。
总之，无论何种情况(利用已知的大地坐标计算空间直角坐标，以及利用逆变换方法由这些空间直角坐标反演大地坐标)，逆变换误差的大小均在纳角秒和微米量级上，远低于当前空间大地测量技术的观测精度。因此，本文的三轴椭球模型的坐标逆变换方法是可行的。
7 结语
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本文建立了三轴椭球大地坐标与笛卡尔坐标之间的理论模型，引入了归化经纬度、几何经纬度和大地经纬度的概念。其中，归化经纬度是表示所有规则曲面的2个独立变量，在大地测量学中是联系几何经纬度和大地经纬度的桥梁。本文基于归化经纬度概念并利用牛顿迭代算法，给出了坐标逆变换的一个新方法。
当给定大地经纬度和大地高后，可计算此点的笛卡尔坐标。利用坐标逆变换新方法，可反求大地经纬度和大地高。大量数值计算表明，对于空间任意点，大地余纬度的计算值与其真值之差，以及大地经度的计算值与其真值之差是纳角秒量级；大地高的计算值与其真值之差是微米量级。针对空间任意点，利用新方法基本能瞬时完成计算，最多迭代3次。
基金
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  • 国家自然科学基金(12473068)
  • 国家自然科学基金(42074002)
文献
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参考文献 引证文献
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Ligas M . Various Parameterizations of "Latitude" Equation-Cartesian to Geodetic Coordinates Transformation[J]. Journal of Geodetic Science, 2013, 3 (2): 87- 110
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2026年第46卷第6期
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doi: 10.14075/j.jgg.2025.12.421
  • 接收时间：2025-12-02
  • 首发时间：2026-07-09
  • 出版时间：2026-06-15
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  • 收稿日期：2025-12-02
基金
国家自然科学基金(12473068)
国家自然科学基金(42074002)
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    1 山东理工大学建筑工程与空间信息学院, 淄博, 255000

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徐鹏飞, 博士, 讲师, 主要从事大地测量学教学与研究, E-mail:
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