Compute the area of a simple polygon, and the direction (clockwise or counterclockwise) in which its vertices are given.InputInput contains up to 25 test cases. Each test case begins with an integer 𝑛 (3≤𝑛≤1000). Then follow the 𝑛 vertices of a simple polygon, one per line, each of the form 𝑥 𝑦. The points may be given in either clockwise or counterclockwise order. Coordinates are integers with absolute value bounded by 10000. The input is terminated by a case beginning with 0.OutputFor each test case, output a line 𝐷 𝐴, where 𝐷 is one of “CW” or “CCW”, indicating whether the polygon was given in clockwise or counterclockwise order, respectively. 𝐴 is the area of the polygon, given with exactly one digit after the decimal point.Sample Input 1 Sample Output 130 010 00 10541 -6-24 -74-51 -673 17-30 -340CCW 50.0CW 3817.5
Question
Compute the area of a simple polygon, and the direction (clockwise or counterclockwise) in which its vertices are given.InputInput contains up to 25 test cases. Each test case begins with an integer 𝑛 (3≤𝑛≤1000). Then follow the 𝑛 vertices of a simple polygon, one per line, each of the form 𝑥 𝑦. The points may be given in either clockwise or counterclockwise order. Coordinates are integers with absolute value bounded by 10000. The input is terminated by a case beginning with 0.OutputFor each test case, output a line 𝐷 𝐴, where 𝐷 is one of “CW” or “CCW”, indicating whether the polygon was given in clockwise or counterclockwise order, respectively. 𝐴 is the area of the polygon, given with exactly one digit after the decimal point.Sample Input 1 Sample Output 130 010 00 10541 -6-24 -74-51 -673 17-30 -340CCW 50.0CW 3817.5
Solution
To solve this problem, we can use the Shoelace formula, which is a simple method to determine the area of a polygon when given the coordinates of its vertices. The formula is:
Area = 1/2 * abs(sum(xiyi+1 - xi+1yi))
where (xi, yi) are the coordinates of the i-th vertex and (xi+1, yi+1) are the coordinates of the next vertex, with the last pair of vertices being (xn, yn) and (x1, y1).
The direction (clockwise or counterclockwise) can be determined by the sign of the sum in the formula. If the sum is positive, the vertices are given in counterclockwise order. If the sum is negative, the vertices are given in clockwise order.
Here is a step-by-step guide to solve the problem:
- Read the number of vertices n.
- Read the coordinates of the vertices and store them in a list.
- Initialize a variable sum to 0.
- For each pair of consecutive vertices (xi, yi) and (xi+1, yi+1), add to sum the value of (xiyi+1 - xi+1yi).
- The area of the polygon is 1/2 * abs(sum).
- If sum is positive, the direction is counterclockwise. If sum is negative, the direction is clockwise.
- Print the direction and the area with one digit after the decimal point.
- Repeat the process for the next test case.
This algorithm works for all simple polygons, regardless of their shape or the order in which their vertices are given.
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