等角圆锥投影基准纬度非迭代算法

为简化传统正轴等角圆锥投影求解基准纬度时繁琐的迭代算法，引入平均纬度和平均纬差的概念，借助计算机代数系统Mathematica，在平均纬差处级数展开，导出了基于球体模型的正轴等角圆锥投影求解基准纬度的非迭代算法。以全国和不同纬差的省区为例，将其与传统椭球迭代算法进行对比分析。结果表明，推导的基于球体模型的非迭代公式计算基准纬度$ {B}_{0} $、$ {B}_{1} $、$ {B}_{2} $的相对误差最大值为2.011%，长度变形的相对误差小于1×$ {10}^{-6} $，基本可满足全国以及各省区地图制图的精度要求，从而验证了所研究算法的精确性与实用性。

Non?iterative Algorithm for Calculating the Reference Latitude of Conformal Conic Projection

  Objectives  For the traditional conformal conic projection, Newton's iterative method is generally used to solve the reference latitude. However, the method requires to be iterated repeatedly, which leads to relatively low computational efficiency and is not easy to be extended and utilized. Therefore, non-iterative expressions for the reference latitudes $ {B}_{0} $, $ {B}_{1} $, $ {B}_{2} $ of the spherical conformal conic projection are derived in this paper to facilitate theoretical analysis and numerical computation.  Methods  In order to simplify the fussy iterative algorithm in calculating the reference latitude of the traditional conformal conic projection, the average latitude and the average latitude difference were introduced. With the help of the powerful computer algebra system Mathematica, the non-iterative algorithm of the conformal conic projection based on the sphere model is proposed. Compared with the traditional ellipsoid iterative algorithm, the country and provinces with different latitude differences are taken as examples.  Results  Numerical examples show that the maximum relative error of the reference latitudes $ {B}_{0} $, $ {B}_{1} $, $ {B}_{2} $ calculated by the non-iterative formula of the sphere model derived is less than 2.011% and the relative error of length deformation is less than $ 1\times {10}^{-6} $. For provinces with different latitude differences, the smaller the latitude difference, the closer the reference latitude calculated based on the sphere model to that based on the ellipsoid model, the smaller the relative error of length, and the closer the length deformation.  Conclusions  The derived non-iterative expressions overcome the tedious iterative process and make the theoretical analysis more convenient, which enriches the map projection theory to a certain extent. The established mathematical model can be extended to the non-iterative algorithm for solving the reference latitude for rectifying and authalic conic projection under the sphere model and each conic projection under the ellipsoidal model, and the algorithm is applicable to the whole map of China and all provinces. Therefore, the non-iterative algorithm of spherical conic projection derived can be used to replace the traditional complex iterative algorithm in the process of making small and medium scale maps in China.