椭球面三角形外心的地图代数解法

椭球面三角形外心到3个相邻顶点的大地线距离都相等。面向椭球面空间的外心大地坐标的求解对于椭球面Voronoi图的生成和椭球面Delaunay三角网的构造具有重要作用。利用基于地图代数理论的矢栅结合方法,首先基于地图代数测地变换建立高精度椭球面空间距离场,再通过边界跟踪配对确定外心所在的栅格范围,最后通过数值计算内插生成初始等距点并不断逼近外心的精确大地坐标。试验结果表明,采用本文方法求解的椭球面三角形外心大地坐标,在103~104 km跨度内其定位误差小于0.001m,且算法非常适用于海量空间数据的高精度快速计算。

Determination of Circumcenter of Triangle on Ellipsoidal Surface Based on Map Algebra

The geodesic distances from the circumcenter to 3 vertexes of the triangle on ellipsoidal surface are equal.The ellipsoid-oriented determination of circumcenter of triangle on ellipsoidal surface is applicable when it comes to generation of the Voronoi diagram and construction of the Delaunay triangulation net on the ellipsoidal earth,which can be considered as a solution of significance in computation of geometries and spatial analysis on the ellipsoid.Based on the idea of combining the raster and vector methods and the theory of map algebra,the working process can be described as below:firstly,initiate the geographical distance transformation and create the distance field with a high degree of accuracy;secondly,conduct boundary tracking and matching and then determinate the range of grids where the circumcenter of triangle locates;thirdly,interpolate the initial equidistant point;finally, approximate the circumcenter of triangle on earth ellipsoidal surface by means of numeric calculation.The positioning error of this algorithm is controlled less than 0.001 m within several thousand kilometers range of span.As regards the method proposed in the present paper,its computational efficiency isO(m)wherem is the number of pixels in the image,i.e.,grid resolution.In conclusion,this algorithm can be considered as both ellipsoid-oriented and not content-related,which is especially appropriate for complex geocomputation globally.