American Journal of Engineering Research (AJER) 2013 www.ajer.org Page 62 American Journal of Engineering Research (AJER) e-ISSN : 2320-0847 p-ISSN : 2320-0936 Volume-02, Issue-07, pp-62-65 www.ajer.org Research Paper Open Access The fundamental formulas for vertices of convex hull Md. Kazi Salimullah 1 , Md. Khalilur Rahman 2* , Md. Mojahidul Islam 3 1. Department of Computer Science and Engineering, Islamic University, Kushtia, Bangladesh. 2. Associate Professor, Department of Applied Physics, Electronics and Communication Engineering, Islamic University, Kushtia, BangladeshAuthor. 3. Assistant Professor, Department of Computer Science and Engineering, Islamic University, Kushtia, Bangladesh. Abstract: - This paper represents four formulas for solution of convex hull problem. It aims to analyze how many points are vertices out of total input points, how many vertices lie on a horizontal or vertical lines, position of vertices and number of vertices on lower and higher lines(horizontal or vertical). Keywords: - Jarvis’s March method, horizontal line (HL), vertical line(VL), vertices , Convex Hull(CH). I. INTRODUCTION Convex hull is a part of computational geometry. Convex hull of a set S of points is the smallest convex polygon P for which each point in S is either on the boundary of P or in its interior. We denote the convex hull of S by CH(S). Convexity has a number of properties that makes convex polygons easier to work with than arbitrary polygons. For example, every diagonal of a convex polygon is a chord, every vertex of convex polygon is convex (that means its interior angle is less than or equal to 180 degree). There are some methods generated for solving convex hull problem. Among these methods Graham Scan method[1], Jarvis’s March method[1], Divide and Conquer method[2], Incremental method[3] and Prune-Search methods[4] are remarkable. Number of horizontal line indicates the number of different values of y among the input points. Number of vertical line indicates the number of different values of x among the input points. The end points where two segments meet are called it vertices. The vertices of a polygon are classified as convex or reflex. A vertex is convex if the interior angle at the vertex--through the polygon interior —measures less than or equal to 180 degrees. A vertex is reflex otherwise (its interior angle measures greater than180 degrees). II. DESCRIPTION Statement of formulas i) Every vertices of the convex hull must be the starting or ending points (among input points) of any horizontal or vertical line. ii) For top and bottom horizontal line or leftmost and rightmost vertical line the starting and ending points are must be vertices of desired convex hull. iii) The highest number of vertices in one line (horizontal or vertical) is less than or equal to two. iv) If the number of lines(horizontal or vertical) is K at which all the points lie then the total number of vertices is less than or equal to 2K. III. PROOF OF FORMULAS 1st Formula Every vertices of the convex hull must be starting or end points of any HL or VL.