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.