Abstract: Using the y-axis to sweep across the point-line set from right to left, when it reaches a certain given point or endpoint of the given line segment, connect that point with the next endpoint of neighboring line located above or below it and hitting the moving y-axis first. The newly constructed line shouldn't intercept the existing edges of finished triangulation other than at their endpoint. Time complexity of the algorithm is O(N log N), where N is the sum of given points and endpoints of given lines.