The paper presents a coherent, original theory and an effective algorithm to calculate the surface area of a geodesic polygon, i.e. the area on an ellipsoid of revolution limited by geodesics, connecting pairs of subsequent vertices of known geographical coordinates. Using the cylindrical coordinate system and introducing an equatorial geodesic triangle, the novel strict recursive formulas were derived for the longitude on the ellipsoid and the distance along the geodesic and the area of a geodesic polygon. The area is determined by constant coefficients, not the values of finite series. The orthodromic properties of geodesics, which led to the detection of bifurcation lines composed of the double solutions of the inverse geodetic problem were analysed. The novel algorithm presented, integrating the problem with geodesic polygon area calculation, is based on the newly proposed parameterisation with two alternative working unknowns and a new formula to obtain the initial value of the geodesic C parameter. It allows solutions for all cases, including near-antipodal or near-vertices, and offers two solutions if bifurcation occurs. The algorithm presented was tested and verified using numerical examples from known publications. The numerical tests proved the accuracy of 1m² of the computed geodesic area using standard IEEE 754 Double Floating Point PC arithmetic, for geodesic lengths up to ten thousand km.

本文提出了一种连贯的、原创的理论和有效的算法来计算测地线多边形的表面积，即受测地线限制的旋转椭球上的面积，连接已知地理坐标的后续顶点对。利用圆柱坐标系，引入赤道测地三角形，推导出椭球经度、测地线距离和测地多边形面积的严格递推公式。面积由常数系数确定，而不是有限级数的值。分析了测地线的正交特性，这导致了由逆测地线问题的双解组成的分岔线的检测。提出的新算法将问题与测地线多边形面积计算相结合，C参数。它允许所有情况的解决方案，包括接近对映点或接近顶点，并在发生分叉时提供两种解决方案。使用已知出版物中的数值示例对所提出的算法进行了测试和验证。数值测试证明了使用标准 IEEE 754 双浮点 PC 算法计算的测地线区域的精度为 1m ^{2 ，测地线长度可达一万公里。}