数字曲线拐点的自动确定

对于由离散点表示的数字地图与GIS图形数据，本文首先利用两相邻矢量叉积乘积的原理来判定拐点所在的折线边；然后利用曲线光滑原理，在已确定的折线边的两个端点之间，建立一条光滑加密了的S形曲线，把后者看作是原始折线的精确曲线，对它进行曲线段凹向改变点(拐点)的定位计算。对于离散数据来说，此处不是采用通常的数值微分方法，而是多次应用矢量叉积乘积的原理，求出最或然拐点，并看作是理论拐点。为了简化计算量，探讨了如何避免为求拐点而进行光滑加密的辅助计算过程。对此，研究分析了拐点在折线边上的移动规律与其前后的曲线转角之间的相关关系，借此可直接根据原始离散数据作简单计算，在足够精确的程度上得出拐点的位置。

Automatic Determination of Inflection Point and Its Applications

For digital maps and GIS data , locating the inflection point can be performed in two steps. One is to sweep the original polyline to find a side where there exist inflection points. To perform this task, we use the every four consecutive points forming two cross products of two vectors. If the multiplication of these two cross products gives a value with positive sign , it means that these four consecutive points have the same concavity, otherwise, the four points form two curve segments, with signs opposed each other. The other step is to locate the inflection point on the side found before. For this purpose, an auxiliary smooth interpolation is needed as a bridge. Our task is to find a simplified method to locate the inflection point without bridging interpolated and smoothed curve. Through regression analysis it is observed that the inflection point is located on the side of original polyline approximately and moves along this side in dependence on the angle ratio at rotation angles at the incident vertices. Therefore, we can calculate the rotation angles at vertices of relevant original side found in the first step. Using the exponent value obtained from the regression analysis and calculating the angle ratio at two vertices in advance, a desired inflection point can be calculated directly by the original data without interpolation procedure. In the end, this paper presents several main application aspects of inflection point.