Problem p01302 - Generation 3

Orig Description

Problem J: Cave Explorer
Mike Smith is a man exploring caves all over the world.
One day, he faced a scaring creature blocking his way. He got scared, but in short time he took
his knife and then slashed it to attempt to kill it. Then they were split into parts, which soon
died out but the largest one. He slashed the creature a couple of times more to make it small
enough, and finally became able to go forward.
Now let us think of his situation in a mathematical way. The creature is considered to be a
polygon, convex or concave. Mike slashes this creature straight with his knife in some direction.
We suppose here the direction is given for simpler settings, while the position is arbitrary. Then
all split parts but the largest one disappears.
Your task is to write a program that calculates the area of the remaining part when the creature
is slashed in such a way that the area is minimized.
Input
The input is a sequence of datasets. Each dataset is given in the following format:
n
vx vy
x1 y1
       ...
xn yn
The first line contains an integer n, the number of vertices of the polygon that represents the
shape of the creature (3 ≤ n ≤ 100). The next line contains two integers vx and vy, where
(vx, vy) denote a vector that represents the direction of the knife (-10000 ≤ vx, vy ≤ 10000,
vx2 + vy2 > 0). Then n lines follow. The i-th line contains two integers xi and yi, where (xi, yi)
denote the coordinates of the i-th vertex of the polygon (0 ≤ xi, yi ≤ 10000).
The vertices are given in the counterclockwise order. You may assume the polygon is always
simple, that is, the edges do not touch or cross each other except for end points.
The input is terminated by a line with a zero. This should not be processed.
Output
For each dataset, print the minimum possible area in a line. The area may be printed with
an arbitrary number of digits after the decimal point, but should not contain an absolute error
greater than 10-2.
Sample Input
5
0 1
0 0
5 0
1 1
5 2
0 2
7
9999 9998
0 0
2 0
3 1
1 1
10000 9999
2 2
0 2
0
Output for the Sample Input
2.00
2.2500000

Extracted Specification

An integer \( n \) (3 ≤ \( n \) ≤ 100), representing some quantity.
Two integers \( vx \) and \( vy \) (-10000 ≤ \( vx, vy \) ≤ 10000, \( vx^2 + vy^2 > 0 \)), representing some vector.
A list of \( n \) pairs of integers, where each pair \( (x_i, y_i) \) (0 ≤ \( x_i, y_i \) ≤ 10000), representing some coordinates.

### Example Input:

```
5
0 1
0 0
5 0
1 1
5 2
0 2
```

### Function Signature:
Write a function f(n, vx, vy, coordinates) that takes in the input.
```python
def f(n: int, vx: int, vy: int, coordinates: List[Tuple[int, int]]):
    '''
    n: an integer
    vx: an integer
    vy: an integer
    coordinates: a list of tuples, where each tuple contains two integers
    '''
```

Test Case Generator

Sure, here's a test case generator for the provided problem "Cave Explorer":

```python
import random

def generate_polygon(n):
    """ Generate a simple polygon with n vertices """
    points = []
    for _ in range(n):
        x = random.randint(0, 10000)
        y = random.randint(0, 10000)
        points.append((x, y))
    return points

def is_simple_polygon(polygon):
    """ Check if a given polygon is simple (no self-intersecting) """
    def orientation(p, q, r):
        """ Return the orientation of the triplet (p, q, r).
        0 -> p, q and r are collinear
        1 -> Clockwise
        2 -> Counterclockwise
        """
        val = (q[1] - p[1]) * (r[0] - q[0]) - (q[0] - p[0]) * (r[1] - q[1])
        if val == 0:
            return 0
        elif val > 0:
            return 1
        else:
            return 2

    def do_intersect(p1, q1, p2, q2):
        """ Return true if line segment 'p1q1' and 'p2q2' intersect. """
        o1 = orientation(p1, q1, p2)
        o2 = orientation(p1, q1, q2)
        o3 = orientation(p2, q2, p1)
        o4 = orientation(p2, q2, q1)

        if o1 != o2 and o3 != o4:
            return True

        if o1 == 0 and on_segment(p1, p2, q1):
            return True

        if o2 == 0 and on_segment(p1, q2, q1):
            return True

        if o3 == 0 and on_segment(p2, p1, q2):
            return True

        if o4 == 0 and on_segment(p2, q1, q2):
            return True

        return False

    def on_segment(p, q, r):
        """ Check if point q lies on line segment 'pr' """
        if (q[0] <= max(p[0], r[0]) and q[0] >= min(p[0], r[0]) and
            q[1] <= max(p[1], r[1]) and q[1] >= min(p[1], r[1])):
            return True
        return False

    n = len(polygon)
    for i in range(n):
        for j in range(i + 1, n):
            if do_intersect(polygon[i - 1], polygon[i], polygon[j - 1], polygon[j]):
                return False
    return True

def tcgen():
    datasets = []
    while True:
        n = random.randint(3, 100)
        vx = random.randint(-10000, 10000)
        vy = random.randint(-10000, 10000)
        while vx == 0 and vy == 0:
            vx = random.randint(-10000, 10000)
            vy = random.randint(-10000, 10000)
            
        polygon = generate_polygon(n)
        # Ensure the polygon is simple
        while not is_simple_polygon(polygon):
            polygon = generate_polygon(n)
            
        dataset = [str(n)]
        dataset.append(f"{vx} {vy}")
        for x, y in polygon:
            dataset.append(f"{x} {y}")
        
        datasets.append("\n".join(dataset))
        
        # Randomly decide if this is the last dataset
        if random.random() < 0.1:
            break
    
    datasets.append("0")
    return "\n".join(datasets)

# Generate a test case
print(tcgen())
```

### Explanation:
1. **Polygon Generation**: The `generate_polygon` function generates random points within the given constraints, ensuring that the polygon is simple (doesn't self-intersect) using the `is_simple_polygon` function.
2. **Vector Generation**: Generates a random vector \((vx, vy)\) ensuring that it is not \((0, 0)\).
3. **Dataset Construction**: Constructs a dataset with the required format.
4. **Termination**: The generation loop randomly decides when to stop adding datasets to simulate the input termination with a line containing zero.

This generator can produce multiple datasets in one run, adhering to the problem's constraints and ensuring the polygons are simple.

Extract Arguments

def extract_arguments(fh):
    datasets = []
    while True:
        n = int(fh.readline().strip())
        if n == 0:
            break
        vx, vy = map(int, fh.readline().strip().split())
        vertices = []
        for _ in range(n):
            x, y = map(int, fh.readline().strip().split())
            vertices.append((x, y))
        datasets.append((n, (vx, vy), vertices))
    return datasets