複雑な線図形の長さを積分幾何学を用いて測定する方法に関する理論的・実験的解析

抄録

The length of a linear figure drawn on a plane, such as an urban road map or a river, is usually measured by tracing the figure with a curvimeter. But when the length of a complex figure is measured, the result often varies widely. One of the reasons for this error is that some part of the figure has been traced more than once or not traced at all.
On the other hand, an alternative method based on integral geometry was proposed by Steinhaus⁵⁾ which is to count the number of intersection points between the figure and parallel lines superposed on it. The counting is done several times changing the orientation of the parallel lines. By this method, the length can be determined with high accuracy by decreasing the interval d of parallel lines and/or increasing the number of directions m, and when only a rough estimate is needed the time necessary for measuring can be reduced.
In this paper, we derive an estimate of the error in measuring the length of a complex figure by Steinhaus' method by theoretical analysis as well as by numerical experiments. Based on this estimate, we show how to choose the appropriate values of d and m in order to achieve the required accuracy. In addition, we propose several conditions in order to minimize the human error in counting the number of intersection points.
The experimental results show that the method based on integral geometry is better than the method using a curvimeter both in accuracy and in time for some complex figures that are likely to appear in practice.