Abstract The transformation of Cartesian coordinates (xp, yp, zp) of a point (P) into their geodetic equivalent (ϕ, λ, h) in reference to the geodetic ellipsoid, is an essential requirement in geodesy. There are many well-known algorithms solving this transformation in closed-form, approximate or iterative approaches. This paper presents a new algorithm named “Trilateration Algorithm” for this transformation. It is based on the new “Seta-Point Theorem” in the meridian plan, which defines a new deterministic Twin-Point (P0) for the point (P). From the Twin-Point (P0), a single iteration solution is processed to achieve highly-accurate values for (ϕ, h) in a relatively simple and deterministic computation algorithm which is valid and stable for all values of (ϕ, h). The proposed solution was tested on a sample of 4277 points that cover all possible cases of point (P). The produced maximum absolute error in latitude is (0.000 0026″) and (0.000 476 mm) in height, computed from first iteration.

中文翻译：

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将笛卡尔坐标转换为大地坐标的三边测量算法

摘要 将点 (P) 的笛卡尔坐标 (xp, yp, zp) 转换为其相对于大地测量椭球的大地等效值 (ϕ, λ, h) 是大地测量学的基本要求。有许多众所周知的算法以封闭形式、近似或迭代方法解决这种变换。本文针对这种变换提出了一种名为“三边测量算法”的新算法。它基于子午线计划中新的“Seta-Point Theorem”，它为点 (P) 定义了一个新的确定性孪生点 (P0)。从双点 (P0) 开始，处理单个迭代解以在相对简单和确定性的计算算法中获得 (ϕ, h) 的高精度值，该算法对 (ϕ, h) 的所有值均有效且稳定。建议的解决方案在 4277 个点的样本上进行了测试，这些样本涵盖了点 (P) 的所有可能情况。产生的最大纬度绝对误差为 (0.000 0026") 和 (0.000 476 mm) 高度，从第一次迭代计算。