地图投影变形球面大圆弧的度量指标

地图投影是地图学的重要研究内容。任何地图投影都不可避免地存在变形问题。针对地图投影的变形,本文提出球面大圆弧和互补比率均值相结合的地图投影面积变形与形状变形指标。通过算例验证和相关性分析,大圆指标一方面简化了小圆指标(即互补比率均值)的计算过程,并能与小圆指标的结果保持一致;另一方面,大圆指标与微分指标之间也具有较高的一致性(形状变形指标的皮尔森积矩相关系数大于0.988)。由于大圆指标不依赖于微分计算,且计算简捷,因此大圆指标更具通用性。本文进一步采用回归分析对大圆指标进行分析,结果表明,大圆指标与微分指标具有较好的线性关系(线性回归的平均误差小于1.10‰)。为了降低采样点数量和解决采样点不统一问题,本文还提出了基于随机采样的指标计算方法,并对随机方法进行了验证和分析。依据大圆指标与微分指标的一致性和线性关系,可以认为使用大圆指标能够有效地评估地图投影的变形情况。

Spherical great circle arcs based indicators for evaluating distortions of map projections

Map projection is an important research content of cartography. However, distortions are inevitable for any map projection. To evaluate the distortions of map projections, averaged ratio between complementary profiles and spherical great circle arcs based shape and area metrics are proposed. By using Bonne projection as an example and exploiting correlation analyses, it is indicated that great circle arcs based indicators simplify the calculation process of small circle arcs based indicators (i.e., the averaged ratio between complementary profiles),and both great and small circle arcs based indicators are highly correlated with each other. Great circle arcs based indicators are also highly correlated with classical differential calculation based indicators(the Pearson product-moment correlation coefficient between them is greater than 0.988), while the proposed metrics are independent on the differential calculation. By utilizing the method of regression analyses, small regression errors are also achieved(the mean error of linear regression is less than 1.10‰). Finally, in order to reduce the number of sampling points and avoid the inconsistency of sampling points for different map projections, this paper also proposes and verifies a random sampling based calculating process. In all word,great circle arcs based indicators could effectively evaluate the distortions of map projections.