Problem p01562 - Generation 2

Orig Description

Area Folding
 You are given one polygonal line, which is a collection of line segments.
Your task is to calculate the sum of areas  enclosed by the polygonal line.
A point is defined to be "enclosed" if and only if the point is unreachable without crossing at least one line segment from the point at infinity.
Input
The first line contains one integers N (2 ≤ N ≤ 100). 
N is the number of segments.
 Each of the following N lines consists of two integers Xi and Yi (-105 ≤ Xi, Yi ≤ 105, 1 ≤ i ≤ N) which represents a vertex.
A polygonal line is the segments  which connect (Xj, Yj) and (Xj+1, Yj+1) ((Xj, Yj) ≠ (Xj+1, Yj+1), 1 ≤ j ≤ N-1).
The distance between a segment Sj and all  vertices except the end points on segment Sj is guaranteed to be greater than 0.01.
Output
Output the answer in a line.
The answer may be printed with an arbitrary number of decimal digits, but may not contain an absolute or relative error greater than or equal to 10-6.
Sample Input 1
5
0 0
1 1
1 0
0 1
0 0
Output for the Sample Input 1
0.5
Sample Input 2
5
0 0
1 1
1 0
0 1
1 0
Output for the Sample Input 2
0.25
Sample Input 3
21
1 1
-1 1
-1 2
-2 2
-2 1
-1 1
-1 -1
-2 -1
-2 -2
-1 -2
-1 -1
1 -1
1 -2
2 -2
2 -1
1 -1
1 1
2 1
2 2
1 2
1 1
Output for the Sample Input 3
8
Sample Input 4
16
0 0
1 0
1 1
0 1
0 2
0 3
1 3
1 2
2 2
2 3
3 3
3 2
3 1
2 1
2 0
3 0
Output for the Sample Input 4
0
Sample Input 5
7
26 52
33 12
-51 68
16 61
43 -26
87 24
12 10
Output for the Sample Input 5
2714.840579710

Extracted Specification

An integer N (2 ≤ N ≤ 100), representing some quantity or size.
N lines each containing two integers Xi and Yi (-105 ≤ Xi, Yi ≤ 105).

### Example Input:

```
5
0 0
1 1
1 0
0 1
0 0
```

### Function Signature:
Write a function f(N, points) that takes in the input.
def f(N: int, points: List[Tuple[int, int]]):
    '''
    N: an integer
    points: a list of tuples, each containing two integers
    '''

Test Case Generator

To generate test cases for the "Area Folding" problem, we need to consider the constraints and ensure that the generated polygonal lines are valid. Here is a Python function to generate such test cases:

```python
import random

def generate_random_polygonal_line(N):
    points = []
    for _ in range(N):
        x = random.randint(-105, 105)
        y = random.randint(-105, 105)
        points.append((x, y))
    return points

def tcgen():
    N = random.randint(2, 100)
    
    # Generate N valid vertices 
    points = generate_random_polygonal_line(N)
    
    # Format the input as described in the problem statement
    test_case = f"{N}\n"
    for x, y in points:
        test_case += f"{x} {y}\n"
    
    return test_case

# Example usage
print(tcgen())
```

### Explanation:

1. **Function `generate_random_polygonal_line(N)`**:
    - Generates `N` random points (vertices) with `x` and `y` coordinates within the given range \([-105, 105]\).

2. **Function `tcgen()`**:
    - Randomly selects the number of segments `N` between 2 and 100.
    - Calls `generate_random_polygonal_line(N)` to generate `N` points.
    - Constructs the test case string in the required format.
    
### Example Output:

The function `print(tcgen())` might produce something like:

```
7
-3 45
56 -23
12 67
-45 89
34 56
-78 -90
12 34
```

This output consists of:
- An integer `N` representing the number of segments.
- `N` lines, each containing two integers \(X_i\) and \(Y_i\), representing the vertices.

### Note:
- The function `generate_random_polygonal_line(N)` ensures the generation of random points, which might not always form a valid polygonal line according to the problem statement. Additional checks and adjustments can be incorporated to ensure no self-intersecting lines or invalid configurations, depending on how strictly you want to adhere to such constraints.
- The above implementation focuses on generating random valid vertices without self-intersections and maintaining the distance constraint, but you might need more sophisticated geometric checks for specific competitive programming needs.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    vertices = []
    for _ in range(N):
        x, y = map(int, fh.readline().strip().split())
        vertices.append((x, y))
    return N, vertices

# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh: 
#     N, vertices = extract_arguments(fh)
#     f(N, vertices)