Abdulfattah A. Aboaba, M. S. H. Chowdhury, Shihab A. Hameed, Othman O. Khalifa, & Aisha H. Abdalla, Department of Science in Engineering, Kulliyyah of Engineering, International Islamic University Malaysia …… Poster ID:1243 Iterative Spatial Sectoring: A Innovative Extension of Simpson’s Rule for Determining Area of Irregular Close Shape IIUM Research, Invention and Innovation Exhibition 2012 ABSTRACT Much as the method for determining the properties of shapes like triangles, parallelogram, cuboids, and geometric figures like circle, and spheres have standard formulae, the determination of a function F(x) in many dimensions have formulae or standard methods. Complex 2D and 3D figures consisting of intricate merger of regular shapes have had their parameters determined through careful separation and determination of the properties of individual components that make up the figure. Moreover, the case of irregular close shapes mostly but not limited to 2D and 3D which abound in science and engineering have equally received attention resulting in lots of approximation method evolved over time. The Iterative spatial sectoring is a novel way of determining the area of 2D irregular close shape by expanding on the famours Simpson’s method for finding the area of a function F(x) whose definite integral is either tedious or impossible by standard approaches. INTRODUCTION The iterative partial sectoring (IPS) is an idea conceived to evolve yet another approximation method of finding the area of irregular close shapes of the type in figure 1 neither from the computer programming perspective nor by integrating an approximate function that describes the curve, but from the classical mathematics standpoint. Hence, the IPS methodology follows from approximate integral and integral application methods of solving area bounded by functions difficult to express mathematically or shapes whose function is either non-existent. We therefore reasoned that since the areas whose complex curves could be evaluated using those approaches, the area of irregular close shape could also be found by careful extension of those methods especially the Simpson’s rule. ALGORITHM FOR SOLVING REAL PROBLEM ( a)Construct a circle of known radius round the shape (b)Construct another circle within the shape similar to (c)Find the ratio of the two circles (d) Divide the irregular shape into sectors using ISS sector angle formula (e)Mark the points of intersection of the radii with the irregular shape and measure its length (r) (f)Determine the area of each sector using A e (g)Calculate the approximate area of the irregular shape using R Tx (h)Reduce the size of the sectors (by halving if possible), and recalculate area of irregular shape using R Tx+1 (i)Check for convergence by comparing step (g) and (h). If the modulus of the difference is zero, then convergence is reached otherwise iteration continues. NOVELTY Area of the irregular shape using ISS in two iterations is 7.7cm 2 Area of the irregular shape using Measuration is 8 cm 2 PUBLICATIONS [1] ABDULFATTAH A. ABOABA, SHIHAB A.HAMEED, OTHMAN O. KHALIFA, and AISHA H. ABDALLA (2011). Iterative Spatial Sectoring: An Extension of Simpson’s Rule for Determining Area of Irregular Close Shape, IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering, pp [2] Abdulfattah A. Aboaba, Shihab A. Hameed, Othman O. KhaliAisha H. Abdalla, Rahmat B. Harun, & Nurzaini Rose Mohd Zain (2011). Iterative Spatial Sectoring Algorithm: A Computer Based Approach to Determining Area of Irregular Closed Space and Tumor Volume, Proceedings of National Postgraduate Conference NPC 2011, Universiti Petronas Malaysia (UTP). IEEE 2011.fa, SymbolsMeaning Radius of a Sector of a Circle First Radius of a Sector of irregular Shape Second Radius of a Sector of irregular Shape Last Sector Second Radius of the Last Sector of irregular Shape PROBLEM Lack of classical approach to determine the area bounded by irregular close shape OBJECTIVE Find a classical method of determining the area bounded by irregular close shape METHODOLOGY Combine approximate integral, classical measuration, and integral application methods of solving area bounded by functions difficult to express mathematically or shapes whose function is either non-existent. SCIENTIFIC THOUGHT Systematic derivation of New ISS Formulae that are capable of converging (solving problems) with fewer iterations NOVELTY Mathematical combination of approximate integral (Simpson’s Rule), integral application (method of solving arbitrary curves), and classical measuration to form Iterative Spatial Sectoring (ISS) NUMERICAL EXAMPLE SCIENTIFIC THOUGHT