目录 第1章 绪论 1 1.1 计算机代数 1 1.2 计算机代数系统 2 1.2.1 Mathematica 计算机代数系统 3 1.2.2 Maple 计算机代数系统 6 1.2.3 Mathcad 计算机代数系统 7 1.2.4 MATLAB矩阵计算与代数系统 10 1.3 大地测量数学分析研究现状 13 1.3.1 椭球大地测量数学分析 14 1.3.2 物理大地测量数学分析 15 1.3.3 卫星大地测量数学分析 16 1.3.4 小结 18 1.4 计算机代数与计算机代数系统在大地测量中的应用现状 18 1.5 本书的主要内容 19 第2章 椭球大地测量计算机代数分析 21 2.1 子午线弧长正解问题 21 2.2 参考椭球上梯形图幅面积计算 23 2.3 幂级数展开法解子午线弧长反问题 24 2.3.1 类纬度变量 24 2.3.2 微分方程展开为变量x的正弦的幂级数 25 2.3.3 逐项积分得子午线弧长反解展开式 28 2.4 两点埃尔米特插值法解子午线弧长反问题 29 2.4.1 基本思路分析 29 2.4.2 计算机代数系统下的进一步展开 30 2.5 三点埃尔米特插值法解子午线弧长反问题 31 2.5.1 中点x=π/4，B（x=π/4）及其导数的确定 32 2.5.2 利用插值条件确定待定系数 32 2.6 拉格朗日级数法解子午线弧长反问题 34 2.6.1 拉格朗日级数法 34 2.6.2 基于拉格朗日级数法的子午线弧长反解展开式确定 35 2.7 子午线弧长反解展开式的精度分析 37 2.8 大地纬度、地心纬度和归化纬度间的关系式 38 2.8.1 大地纬度、地心纬度和归化纬度间的严密式 38 2.8.2 大地纬度、地心纬度和归化纬度间的展开式 40 2.9 大地线的参数方程 42 2.10 大地线勒让德级数展开和高斯平均引数展开 46 2.11 贝塞尔大地问题解算 47 2.11.1 大地线长与球面大圆弧长微分方程的解 49 2.11.2 椭球面经度与球面经度微分方程的解 50 2.11.3 大地线长与球面大圆弧长微分方程非迭代解 51 2.12 法截线方位角与贝塞尔大地问题反解直接法 55 2.12.1 椭球面两点法截线方位角 55 2.12.2 贝塞尔大地问题反解 57 2.13 大椭圆法解大地问题 60 2.14 等角航线的正反解问题 63 2.14.1 等角航线基本方程 63 2.14.2 等角航线的正解问题 65 2.14.3 等角航线的反解问题 66 2.14.4 等角航线正反解算例 66 2.15 符号迭代法解椭球大地测量学反问题 69 2.15.1 符号迭代法思想 70 2.15.2 符号迭代法应用举例 70 第3章 地图投影计算机代数分析 75 3.1 等量纬度正解问题 75 3.2 级数法解等量纬度反问题 78 3.2.1 大地纬度 和等角纬度 的微分方程 78 3.2.2 微分方程展开为变量 的正弦的幂级数 79 3.2.3 逐项积分得等量纬度反解展开式 81 3.3 两点埃尔米特插值法解等量纬度反问题 81 3.4 拉格朗日级数法解等量纬度反问题 84 3.5 等量纬度反解展开式的精度分析 85 3.6 等量纬度和子午线弧长变换的直接展开式 86 3.6.1 等量纬度变换至子午线弧长的直接展开式 87 3.6.2 子午线弧长变换至等量纬度的直接展开式 88 3.6.3 直接展开式的精度分析 90 3.7 高斯投影正解的非迭代复变函数表示 91 3.8 高斯投影反解的非迭代复变函数表示 93 3.9 高斯投影尺度比和子午线收敛角的复变函数表示 94 3.10 高斯投影的复变函数表示算例 95 3.11 高斯投影作图 99 第4章 大地测量数据处理计算机代数分析 101 4.1 阵列代数双二次、双三次插值 101 4.1.1 双二次多项式插值 101 4.1.2 双三次多项式插值 104 4.2 高次多项式插值龙格现象和切比雪夫多项式插值 107 4.3 非线性函数线性化与误差方程组成 109 4.4 误差传播定律与控制网优化设计 111 4.5 典型图形平差 113 4.6 三角锁网精度估算 117 4.7 矩阵特征值、特征向量和矩阵范数计算 120 4.8 确定协方差参数的代数准则 121 4.8.1 准则1——比值函数法 124 4.8.2 准则2——样条函数法 126 4.8.3 算例 127 4.9 确定点质量法质点埋藏深度的代数准则 128 4.10 GPS姿态测量误差分析 131 第5章 物理大地测量计算机代数分析 137 5.1 柱体积分和层间改正 137 5.1.1 柱体积分 137 5.1.2 层间改正 139 5.2 矩形棱柱引力和引力位 139 5.2.1 矩形棱柱引力 139 5.2.2 矩形棱柱引力位 141 5.3 球冠和球层冠引力和引力位 143 5.3.1 球冠引力和引力位 143 5.3.2 球层冠引力和引力位 145 5.4 均质旋转椭球体在内部的引力和引力位 146 5.4.1 基本积分式的导出 146 5.4.2 用计算机代数系统计算与旋转椭球引力有关的定积分 147 5.4.3 通过复变函数得到实变函数积分 149 5.5 地形校正中央区积分 149 5.5.1 矩形域上的8分片线性插值地形校正积分 150 5.5.2 菱形域上的4分片线性插值地形校正积分 153 5.6 地球重力场泛函中央区奇异积分 155 5.6.1 重力异常双二次多项式插值中央区奇异积分计算 155 5.6.2 重力异常双三次多项式插值中央区奇异积分计算 159 5.7 莫洛坚斯基公式和逆维宁?曼尼兹公式中央区奇异积分 163 5.7.1 垂线偏差双二次多项式插值中央区奇异积分计算 164 5.7.2 垂线偏差双三次多项式插值中央区奇异积分计算 166 5.8 扰动位、扰动引力向量和扰动引力梯度张量的三维直角坐标表示 170 5.8.1 扰动位二阶项的直角坐标表示 170 5.8.2 扰动引力二阶项的直角坐标表示 171 5.8.3 扰动位二阶项梯度张量的直角坐标表示 172 5.9 全球或局部大地水准面和重力异常等值线图绘制 175 5.9.1 全球大地水准面等值线图绘制 175 5.9.2 全球重力异常等值线图绘制 177 5.10 扁球体的水准面 177 5.10.1 布隆斯扁球和赫尔默特扁球 177 5.10.2 水准椭球面和扁球体水准面的方程 178 5.10.3 精度估计 181 5.10.4 扁球体水准面和水准椭球面的形状差异 181 5.11 勒让德多项式、连带勒让德函数和球面调和函数 187 5.11.1 勒让德多项式和连带勒让德函数定义与显示 187 5.11.2 球面调和函数 189 5.12 计算机代数系统解物理大地测量各问题举例 190 第6章 卫星大地测量计算机代数分析 193 6.1 椭圆运动的级数展开 193 6.1.1 贝塞尔函数 193 6.1.2 偏近点角、真近点角和平近点角几何关系 194 6.1.3 偏近点角的傅里叶级数展开 195 6.1.4 真近点角的傅里叶级数展开 196 6.1.5 汉申函数 197 6.2 卫星运动近点角间的差异极值符号表达式 198 6.2.1 偏近点角与平近点角差值关于平近点角的极值符号表达式 198 6.2.2 真近点角与偏近点角差值关于平近点角的极值符号表达式 199 6.2.3 真近点角与偏近点角差值关于偏近点角的极值符号表达式 200 6.2.4 真近点角与平近点角差值关于平近点角的极值符号表达式 201 6.2.5 算例分析 203 6.3 三角多项式两种形式的转换、勒让德函数向三角多项式的转换 205 6.3.1 三角函数倍角形式化为幂形式 205 6.3.2 三角函数幂形式化为倍角形式 206 6.3.3 连带勒让德函数转化为三角多项式 206 6.4 一些函数的平均值 207 6.4.1 一类含有三角函数特殊积分的计算 207 6.4.2 平均值 确定 208 6.5 一阶项摄动和J2项 209 6.5.1 部分轨道根数一阶项摄动 209 6.5.2 部分轨道根数一阶长周期项摄动 211 6.6 Kaula理论与摄动函数展开 211 6.7 倾角函数确定 213 6.8 偏心率函数确定 214 6.9 改型拉格朗日线性解中的偏心率函数 215 6.10 微分方程数值解法 216 第7章 结论与展望 219 7.1 研究结论 219 7.2 展望 220 参考文献 222 附录 双曲函数和若干初等复变函数定义 227 后记 228 Contents Chapter I Introduction 1 1.1 Computer Algebra 1 1.2 Computer Algebra System 2 1.2.1 Computer Algebra System-Mathematica 3 1.2.2 Computer Algebra System-Maple 6 1.2.3 Computer Algebra System-Mathcad 7 1.2.4 Matrix Computation and Computer Algebra System-MATLAB 10 1.3 The Research Status of Mathematical Analysis on Geodesy 13 1.3.1 Mathematical Analysis on Ellipsoidal Geodesy 14 1.3.2 Mathematical Analysis on Physical Geodesy 15 1.3.3 Mathematical Analysis on Satellite Geodesy 16 1.3.4 Summary 18 1.4 The Application Status of Computer Algebra and Computer Algebra System on Geodesy 18 1.5 Main Contents of This Book 19 Chapter II Computer Algebra Analysis on Ellipsoidal Geodesy 21 2.1 The Forward Solution of Meridian Arc 21 2.2 The Determination of a Trapezoidal Map Area on an Ellipsoid 23 2.3 The Inverse Problem of Meridian Arc in terms of Power Series 24 2.3.1 Quasi-Latitude Variable 24 2.3.2 Expanding Differential Equation as a Power Series of Sine Function of Variable x 25 2.3.3 Getting the Inverse Expansion of Meridian Arc through Integration Term by Term 28 2.4 Solving the Inverse Problem of Meridian Arc by Two-point Hermite Interpolation Method 29 2.4.1 Basic Concept 29 2.4.2 Expansion in Terms of Computer Algebra System 30 2.5 Solving the Inverse Problem of Meridian Arc by Three-point Hermite Interpolation Method 31 2.5.1 Determination of Functional and Derivative Values at Mid-point x=π/4, B(x=π/4) 32 2.5.2 Determination of Unknown Coefficients Using Interpolation Conditions 32 2.6 Solving the Inverse Problem of Meridian Arc by Lagrange Series Method 34 2.6.1 Lagrange Series Method 34 2.6.2 Determination of the Inverse Expansion of Meridian Arc Based on Lagrange Series Method 35 2.7 Accuracy Analysis on the Inverse Expansion of Meridian Arc 37 2.8 The Relationship between Geodetic, Geocentric and Reduced Latitudes 38 2.8.1 The Rigorous Relationship between Geodetic, Geocentric and Reduced Latitudes 38 2.8.2 Expansions between Geodetic, Geocentric and Reduced Latitudes 40 2.9 Parametric Equations for Geodetic Line 42 2.10 Legendre Series Expansion for Geodetic Line and Gauss Average Argument Expansion 46 2.11 Bessel’s Solution of Geodetic Problem 47 2.11.1 Solution of the Differential Equation for Geodetic Length and Spherical Arc 49 2.11.2 Solution of the Differential Equation for Ellipsoidal and Spherical Longitudes 50 2.11.3 Non-iterative Solution of the Differential Equation for Geodetic Length and Spherical Arc 51 2.12 Azimuth of Normal Section and Direct Formula for Bessel’s Inverse Solution of Geodetic Problem 55 2.12.1 Azimuth of Normal Section between Two Points on an Ellipsoid 55 2.12.2 Bessel Inverse Solution of Geodetic Problem 57 2.13 Solution of Geodetic Problem by Great Ellipse Method 60 2.14 Forward and Inverse Solutions of Rhumb Line 63 2.14.1 Basic Equation for Rhumb Line 63 2.14.2 Forward Solution of Rhumb Line 65 2.14.3 Inverse Solution of Rhumb Line 66 2.14.4 Computation Examples for Forward and Inverse Solutions of Rhumb Line 66 2.15 Solution of Inverse Problems in Ellipsoidal Geodesy by Symbolical Iterative Method 69 2.15.1 Basic Concept of Symbolical Iterative Method 70 2.15.2 Application Examples of Symbolical Iterative Method 70 Chapter III Computer Algebra Analysis on Map Projection 75 3.1 The Forward Solution of Isometric Latitude 75 3.2 Solving the Inverse Problem of Isometric Latitude by Power Series Method 78 3.2.1 The Differential Equation for Geodetic Latitude B and Conformal Latitude 78 3.2.2 Expanding Differential Equation as a Power Series of Sine Function of Variable 79 3.2.3 Getting the Inverse Expansion of Isometric Latitude through Integration Term by Term 81 3.3 Solving the Inverse Problem of Isometric Latitude by Two-point Hermite Interpolation Method 81 3.4 Solving the Inverse Problem of Isometric Latitude by Lagrange Series Method 84 3.5 Accuracy Analysis on the Inverse Expansion of Isometric Latitude 85 3.6 Direct Expansions of Transformations between Isometric Latitude and Meridian Arc 86 3.6.1 The Direct Expansion of the Transformation from Isometric Latitude to Meridian Arc 87 3.6.2 The Direct Expansion of the Transformation from Meridian Arc to Isometric Latitude 88 3.6.3 Accuracy Analysis on Direct Expansions 90 3.7 The Non-iterative Formulas for the Forward Solution of Gauss Projection by Complex Function 91 3.8 The Non-iterative Formulas for the Inverse Solution of Gauss Projection by Complex Function 93 3.9 Formulas for Scale and Meridian Convergence Angle of Gauss Projection by Complex Function 94 3.10 Computation Examples for Gauss Projection by Complex Function 95 3.11 Gauss Projection Plotting 99 Chapter IV Computer Algebra Analysis on Geodetic Data Processing 101 4.1 Bi-quadratic and Bi-cubic Polynomial Interpolation 101 4.1.1 Bi-quadratic Polynomial Interpolation 101 4.1.2 Bi-cubic Polynomial Interpolation 104 4.2 Runge Phenomena of High Order Polynomial Interpolation and Chebshev Polynomial Interpolation 107 4.3 Linearization of Non-linear Function and Error Equations Formation 109 4.4 Law of Error Propagation and Optimization of Control Networks 111 4.5 Typical Figure Adjustments 113 4.6 Precision Estimation of Triangular Networks 117 4.7 Matrix Eigenvalues, Eigenvectors and Error Equations 120 4.8 Algebra Rules for Parameters of Covariance Functions 121 4.8.1 Criteria I-The Ratio Function Method 124 4.8.2 Criteria II-Spline Method 126 4.8.3 Computation Examples 127 4.9 Algebra Rules for the Determination of Mass Points 128 4.10 Error Analysis of Attitude Determination by GPS 131 Chapter V Computer Algebra Analysis on Physical Geodesy 137 5.1 Cylinder Integration and Plate Correction 137 5.1.1 Cylinder Integration 137 5.1.2 Plate Correction 139 5.2 Gravitation and Gravitational Potential of Rectangular Prism 139 5.2.1 Gravitation of Rectangular Prism 139 5.2.2 Gravitational Potential of Rectangular Prism 141 5.3 Gravitational Potential and Gravitation of Spherical Cap and Spherical Layer Cap 143 5.3.1 Gravitational Potential and Gravitation of Spherical Cap 143 5.3.2 Gravitational Potential and Gravitation of Spherical Layer Cap 145 5.4 Inner Gravitational Potential and Gravitation of Homogeneous Rotating Ellipsoid 146 5.4.1 Derivation of Basic Integral Formula 146 5.4.2 Determining the Related Definite Integrals in Terms of Computer Algebra System 147 5.4.3 Real Function Integral through Complex Function 149 5.5 Innermost Integration of Terrain Corrections 149 5.5.1 8 Piece Linear Interpolation and Integration in a Rectangular Area 150 5.5.2 4 Piece Linear Interpolation and Integration in a Diamond Area 153 5.6 The Singular Integration of Gravity Field Functional in the Innermost Area 155 5.6.1 Bi-quadratic Polynomial Interpolation of Gravity Anomaly and Singular Integration Calculation 155 5.6.2 Bi-cubic Polynomial Interpolation of Gravity Anomaly and Singular Integration Calculation 159 5.7 The Singular Integration of Molodensky and Inverse Vening-Meinesz Formulas 163 5.7.1 Bi-quadratic Polynomial Interpolation of the Deflection of the Vertical and Singular Integration Calculation 164 5.7.2 Bi-cubic Polynomial Interpolation of The Deflection of the Vertical and Singular Integration Calculation 166 5.8 Expressions of Disturbing Potential, Gravity Vector and Gravity Gradient Tensor in the Rectangular Coordinates 170 5.8.1 Expressions of the Second Order Disturbing Potential in the Rectangular Coordinates 170 5.8.2 Expressions of the Second Order Disturbing Gravity in the Rectangular Coordinates 171 5.8.3 Expressions of the Second Order Disturbing Gravity Gradient Tensor in the Rectangular Coordinates 172 5.9 Contour Maps of Global Geoid and Gravity Anomaly 175 5.9.1 Contour Map of Global Geoid 175 5.9.2 Contour Map of Global Gravity Anomaly 177 5.10 The Level Surface of the Oblate Spheroid 177 5.10.1 Bruns’ Spheroid and Helmert’s Spheroid 177 5.10.2 Level Surface Equations of the Ellipsoid and the Oblate Spheroid 178 5.10.3 Accuracy Estimation 181 5.10.4 The Shape Difference between the Level Surfaces of the Oblate Spheroid and the Ellipsoid 181 5.11 Legendre Polynomial, Associated Legendre Function and Spherical Harmonic Function 187 5.11.1 The Definition and Display of Legendre Polynomial and Associated Legendre Function 187 5.11.2 Spherical Harmonic Function 189 5.12 Other Application of Computer Algebra System in Physical Geodesy 190 Chapter VI Computer Algebra Analysis on Satellite Geodesy 193 6.1 Series Expansions of Elliptical Motion 193 6.1.1 Bessel Function 193 6.1.2 The Geometric Relationship between Eccentric, True and Mean Anomalies 194 6.1.3 Fourier Series Expansion of Eccentric Anomaly 195 6.1.4 Fourier Series Expansion of True Anomaly 196 6.1.5 Hansen Function 197 6.2 Symbolical Expressions of Difference Extrema between Anomalies for Satellite Motion 198 6.2.1 Symbolical Expressions of Difference Extrema between Eccentric and Mean Anomalies about Mean Anomaly 198 6.2.2 Symbolical Expressions of Difference Extrema between True and Eccentric Anomalies about Mean Anomaly 199 6.2.3 Symbolical Expressions of Difference Extrema between True and Eccentric Anomalies about Eccentric Anomaly 200 6.2.4 Symbolical Expressions of Difference Extrema between True and Mean Anomalies about Mean Anomaly 201 6.2.5 Computation Examples 203 6.3 Conversions of Two Kinds of Trigonometric Polynomials, Conversions of Legendre Function to Trigonometric Polynomial 205 6.3.1 Conversions of Trigonometric Multiple Angles to Power Formulas 205 6.3.2 Conversions of Trigonometric Power to Multiple Angles Formulas 206 6.3.3 Conversions of Associated Legendre Function to Trigonometric Polynomial 206 6.4 Average Values for Some Functions 207 6.4.1 The Computation of Some Special Integrals Including Trigonometric Function 207 6.4.2 The Determination of Average Value 208 6.5 The First Order Perturbation and Item 209 6.5.1 The First Order Perturbation for Partial Orbital Elements 209 6.5.2 The First Order Long Periodical Perturbation for Partial Orbital Elements 211 6.6 Kaula Theory and Expansions of Perturbation Functions 211 6.7 Determination of Inclination Functions 213 6.8 Determination of Eccentricity Functions 214 6.9 Eccentricity Functions in the Linear Retrofitted Lagrange Solutions 215 6.10 Numerical Methods for Differential Equations 216 Chapter VII Conclusions and Prospect 219 7.1 Research Conclusions 219 7.2 Prospect 220 References 222 Appendix Definitions of Hyperbolic Functions and Some Elementary Complex Functions 227 Postscript 228
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