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.