Fast Approach for Identifying the Convexity and Concavity of Line-Arc Closed-Curve

The orientation of a closed-curve is determined by first constructing a new convex polygon. By using the orientation of the closed curve the convexity-concavity of vertices and arcs are identified. Finally,the convexity-concavity of the closed curve is detected.The presented algorithm is robust and highly efficient.The complexity of its computing time reaches O(n) and it needs only 15n+33k+25 comparisons,12n-6k+14 multiplications or divisions,10n+16k+21additions and subtractions,2 sine and cosine,k square roots at most,where n is the number of vertices of the closed curve including the centers of arcs and k is the number of arcs.