Problem:

TopCoder Problem Statement - ConvexPolygon |

Single Round Match 166 Round 1 - Division II, Level Three |

Overview:

Calculate the area of a convex polygon.

Java Source:

01: public class ConvexPolygon { 02: 03: public double findArea(int[] x, int[] y) { 04: 05: double area = 0.0; 06: 07: /* 08: * Sum the areas of the triangles {0,1,2}, {0,2,3}, ... {0,n-1,n} 09: */ 10: for (int v = 2; v < x.length; v++) { 11: area += areaOfTriangle(x[0], y[0], x[v-1], y[v-1], x[v], y[v]); 12: } 13: 14: return area; 15: } 16: 17: /* 18: * Given the coordinates of the three vertices, the area of a triangle is 19: * given by: 20: * 21: * area = abs((X1(Y2-Y3) + X2(Y3 - Y1) + X3(Y1-Y2)) / 2) 22: * 23: */ 24: private double areaOfTriangle(int x1, int y1, 25: int x2, int y2, 26: int x3, int y3) { 27: 28: double term1 = x1 * (y2 - y3); 29: double term2 = x2 * (y3 - y1); 30: double term3 = x3 * (y1 - y2); 31: 32: double result = (term1 + term2 + term3) / 2.0; 33: 34: return Math.abs(result); 35: 36: } 37: 38: }

Notes:

The simple formula for the area of a triangle (A = (b * h) / 2) won't help us much. Instead we use a formula like the one given here

The area of the polygon can be calculated by dividing it up into triangles. For a polygon with n points, the area is the sum of the triangles formed by the points {0,1,2} + {0,2,3} + ... {0,n-1,n}. This becomes obvious is you draw any convex polygon on a piece of paper.

An alternate approach is to carve off a triangle (and calculate its area), and then recursively call findArea() to get the area of the remaining polygon.

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