Online Searching A Line With Three Possible Slope In The Plane

In this paper,we discuss online searching for a line with three possible slopes in the plane from a fixed point by using competitive algorithm theory about online problems.If the line has a fixed distance,we consider the line as one side of a triangle,and the fixed distance is the triangle inscribed circle radius.We give an optimal search method with the shortest search path from the center of a circle to the line,thus the optimal competition ration is obtained.If the line's distance is unknown,we see the line is one side of a gens triangle,and the-linear-spiral-algorithm is presented.First,we study a special case,three possible slopes of the line form a regular triangle,that is,the line is one side of a gens regular triangle,and we obtain the competitive ration of the linear spiral algorithm is 12.1074.Second,when the line is one side of a gens general triangle,we consider its online path of the-linear-spiral-algorithm in three cases respectively,under the competitive framework,the optimal searching path of the-linear-spiral-algorithm in the worst case is discussed.We obtain the optimal competition ration by means of this method.