The computation of long geodesics on the ellipsoid through Gaussian quadrature

Formulas for computing geodesics on the bi-axial ellipsoid through Gaussian quadrature are shown; the estimation of computational errors, truncation and roundoff errors, for the quadrature is carried out; and test examples found in [3] together with those which consist of near anti-podal points on the neighborhood of the equator, are computed with the evaluation of the computational errors.     

The computation of long geodesics on the ellipsoid by non-series expanding procedure

In this paper the author shows a procedure to settle the computation of very long geodesic lines on the ellipsoid without using the series expansion. The integration of elliptic integrals appearing in the procedure is numerically carried out by means of a mechanical quadrature-the method of Repeated Interval Halving. The author also devises formulae for the numerical solution of the problem, in order to make the amount of significance error least and determine the kind of quadrant for the computation of inverse trigonometric function. The anti-podal problem for the direct and inverse solution is rigorously solved by this method.     

Geodetic intersection on the ellipsoid

Through each of two known points on the ellipsoid a geodesic is passing in a known azimuth. We solve the problem of intersection of the two geodesics. The solution for the latitude is obtained as a closed formula for the sphere plus a small correction, of the order of the eccentricity of the ellipsoid, which is determined by numerical integration. The solution is iterative. Once the latitude is obtained, the longitude is determined without iteration.     

Intersections on the sphere and ellipsoid

 The problems of intersection on the sphere and ellipsoid are studied. On the sphere, the problem of intersection along great circles is explicitly solved. On the ellipsoid, each of the problems of intersection along arcs of constant azimuth, normal sections and geodesic lines is solved without any limitation on arc length. In the last case the solution is based on the Newton–Raphson method of iteration including numerical integration. Received: 11 April 2001 / Accepted: 3 September 2001     

Algorithms for geodesics

Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of geodesics to be computed.     

椭球面上绘制大地线的算法

针对传统二维线划图无法表达大区域地理要素的问题,该文提出了一种在椭球面上绘制大地线的算法。以等间距内插空间直线逼近空间曲线这一思想作为基础,通过Bessel大地主题正、反算及克莱劳方程,实现大地线内插与绘制的综合分析与解算。实验结果表明,该方法较好地解决了在参考椭球面上难以精确绘制大地线的问题,对传统地图向三维椭球面线划图的转换与应用具有一定的参考价值。     

参数椭球的密度分布研究

继“参数椭球”概念提出以及“参数椭球”的地球重力学性质和数学性质得到初步研究之后 ,本文研究了参数椭球的密度分布问题。结果表明 ,当参数椭球内的密度界面无限趋向表面时 ,两种“准等位条件”是完全等价的。在参数椭球的“准等位条件”下 ,匀质分层的密度分布并不符合地球的实际密度分布。因此 ,地球的密度分布与纬度密切相关     

椭球变换误差传播模型研究

针对在一些行业性地理信息处理中,基于各种坐标系的数据混用现象普遍存在,而椭球变换误差常被忽略,造成椭球变换对数据产生的影响缺乏科学评价的问题,文章以观测误差传播定律为基本研究工具,推导出椭球变换误差在大地坐标转换误差、地图投影误差、平面距离计算误差中的传播模型,最后以椭球变换误差对空间插值影响为例,对以上模型进行实践应用,研究结果表明,论文推导出的椭球变换误差模型能够成为行业空间信息处理中的椭球变换误差影响评价模型。     

大地高误差对坐标系转换精度的影响

卫星大地测量直接获取地面点的三维坐标,解决了传统作业中平面和高程分开建网的问题。但获得的三维坐标基于全球大地坐标框架(WGS84或IRTF),不能直接应用于实际的工程项目中,需要通过椭球变换实现坐标系转换。由于基于本地坐标框架(局部椭球)的大地高较难得到,椭球变换参数只能近似求得。分析大地高误差对坐标系转换精度的影响,并用实例证实此影响在局部范围内是可以忽略的。     

The Meissl scheme for the geodetic ellipsoid

We present a variant of the Meissl scheme to relate surface spherical harmonic coefficients of the disturbing potential of the Earth’s gravity field on the surface of the geodetic ellipsoid to surface spherical harmonic coefficients of its first- and second-order normal derivatives on the same or any other ellipsoid. It extends the original (spherical) Meissl scheme, which only holds for harmonic coefficients computed from geodetic data on a sphere. In our scheme, a vector of solid spherical harmonic coefficients of one quantity is transformed into spherical harmonic coefficients of another quantity by pre-multiplication with a transformation matrix. This matrix is diagonal for transformations between spheres, but block-diagonal for transformations involving the ellipsoid. The computation of the transformation matrix involves an inversion if the original coefficients are defined on the ellipsoid. This inversion can be performed accurately and efficiently (i.e., without regularisation) for transformation among different gravity field quantities on the same ellipsoid, due to diagonal dominance of the matrices. However, transformations from the ellipsoid to another surface can only be performed accurately and efficiently for coefficients up to degree and order 520 due to numerical instabilities in the inversion.     

一种表示地球椭球体上椭圆的算法

采用常规的投影算法表示地球椭球体上的椭圆时,所得椭圆与实际椭圆存在一定误差。针对上述问题,阐述了一种通过解算大地主题表示地球椭球体上椭圆的算法。通过几个实例,对两种算法进行了比较和分析,结果表明,通过解算大地主题表示地球椭球体上椭圆的算法具有非常高的精度。     

基于椭球不确定性的平差模型与算法

测量平差模型中的参数通常存在一些不确定的附加信息或先验信息,充分利用它们可以对部分参数进行约束,从而保证参数解的唯一性和稳定性。本文利用椭球集合描述不确定性,建立了一个新的带有椭球不确定性的平差模型。以两个椭球交集的外接椭球的特征矩阵的迹最小平差准则,分析了不确定度的传播规律,给出了带有椭球不确定性的平差方法。最后,通过算例验证了算法的有效性,说明了平差解与带权混合估计的关系。     

基于地球椭球面模型的海上划界方法

目前,世界上还有许多国家的海洋边界没得到确定,由此引起的国家间海上争端事件不断。因此,开展海洋划界技术和方法研究,科学精确地确定两国间海洋边界显得十分必要。本文阐述了基于墨卡托投影的海图进行海上划界方法的不足,研究了利用高斯平均引数大地主题解算计算点到大地线的球面距离模型,最后设计了基于地球椭球面模型的等比例海上划界算法,并基于ArcGIS Engine实现划界功能。     

The generalized Mollweide projection of the biaxial ellipsoid

Summary The standard Mollweide projection of the sphere S _R ² which is of type pseudocylindrical — equiareal is generalized to the biaxial ellipsoid E _A,B ² .Within the class of pseudocylindrical mapping equations (1.8) of E _A,B ² (semimajor axis A, semiminor axis B) it is shown by solving the general eigenvalue problem (Tissot analysis) that only equiareal mappings, no conformal mappings exist. The mapping equations (2.1) which generalize those from S _R ² to E _A,B ² lead under the equiareal postulate to a generalized Kepler equation (2.21) which is solved by Newton iteration, for instance (Table 1). Two variants of the ellipsoidal Mollweide projection in particular (2.16), (2.17) versus (2.19), (2.20) are presented which guarantee that parallel circles (coordinate lines of constant ellipsoidal latitude) are mapped onto straight lines in the plane while meridians (coordinate lines of constant ellipsoidal longitude) are mapped onto ellipses of variable axes. The theorem collects the basic results. Six computer graphical examples illustrate the first pseudocylindrical map projection of E _A,B ² of generalized Mollweide type.     

Cartesian to geodetic coordinates conversion on a triaxial ellipsoid

A new method of transforming Cartesian to geodetic (or planetographic) coordinates on a triaxial ellipsoid is presented. The method is based on simple reasoning coming from essentials of vector calculus. The reasoning results in solving a nonlinear system of equations for coordinates of the point being the projection of a point located outside or inside a triaxial ellipsoid along the normal to the ellipsoid. The presented method has been compared to a vector method of Feltens (J Geod 83:129–137, 2009) who claims that no other methods are available in the literature. Generally, our method turns out to be more accurate, faster and applicable to celestial bodies characterized by different geometric parameters. The presented method also fits to the classical problem of converting Cartesian to geodetic coordinates on the ellipsoid of revolution.     

地球椭球面图斑面积计算方法探讨

针对地球椭球面图斑连线类型的非唯一性、椭球面梯形面积计算公式的多样性以及任意图斑面积计算精度控制的复杂性等问题,该文基于理论探讨与实例计算相结合的方法,明确了土地面积量算中适宜采用的椭球面图斑类型,测定了各椭球面梯形面积公式的计算精度,给出了任意图斑面积计算精度控制的新思路。研究表明:采用等角航线作为地球椭球面两界址点之间连线的等角航线图斑更适宜于土地面积量算工作;取至e~(10)项的乘积项椭球面梯形面积公式不仅精度高于其他两个理论上等价的近似公式,而且也高于精确计算公式;把割、补三角形看做特殊的椭球面大梯形,借助递归算法,提出了椭球面大梯形面积计算的新公式,这为椭球面上任意图斑面积计算及精度控制提供了新思路。     

Supplement to inverse solution of long geodesics

Alternate formulas are given for the rigorous non-iterative solution of very long (as well as medium and very short) geodesics. They are not only shorter and simpler than the author’s original version published in the Bulletin Géodésique, but the powers of the spheroid parameter can be factored out in the same manner as in the corresponding solution of the Direct Geodetic Problem. Theoretical, as well as practical, significance is noted.