import math
import sys
from collections import namedtuple

Point = namedtuple('Point', ['x', 'y'])

EPSILON = math.sqrt(sys.float_info.epsilon)


def earclip(polygon):
    """
    Simple earclipping algorithm for a given polygon p.
    polygon is expected to be an array of 2-tuples of the cartesian points of the polygon

    For a polygon with n points it will return n-2 triangles.
    The triangles are returned as an array of 3-tuples where each item in the tuple is a 2-tuple of the cartesian point.

    e.g
    >>> polygon = [(0,1), (-1, 0), (0, -1), (1, 0)]
    >>> triangles = earclip(polygon)
    >>> triangles
    [((1, 0), (0, 1), (-1, 0)), ((1, 0), (-1, 0), (0, -1))]

    Implementation Reference:
        - https://www.geometrictools.com/Documentation/TriangulationByEarClipping.pdf
    """
    ear_vertex = []
    triangles = []

    polygon = [Point(*point) for point in polygon]

    if _is_clockwise(polygon):
        polygon.reverse()

    point_count = len(polygon)
    for i in range(point_count):
        prev_index = i - 1
        prev_point = polygon[prev_index]
        point = polygon[i]
        next_index = (i + 1) % point_count
        next_point = polygon[next_index]

        if _is_ear(prev_point, point, next_point, polygon):
            ear_vertex.append(point)

    while ear_vertex and point_count >= 3:
        ear = ear_vertex.pop(0)
        i = polygon.index(ear)
        prev_index = i - 1
        prev_point = polygon[prev_index]
        next_index = (i + 1) % point_count
        next_point = polygon[next_index]

        polygon.remove(ear)
        point_count -= 1
        triangles.append(((prev_point.x, prev_point.y), (ear.x, ear.y), (next_point.x, next_point.y)))
        if point_count > 3:
            prev_prev_point = polygon[prev_index - 1]
            next_next_index = (i + 1) % point_count
            next_next_point = polygon[next_next_index]

            groups = [
                (prev_prev_point, prev_point, next_point, polygon),
                (prev_point, next_point, next_next_point, polygon),
            ]
            for group in groups:
                p = group[1]
                if _is_ear(*group):
                    if p not in ear_vertex:
                        ear_vertex.append(p)
                elif p in ear_vertex:
                    ear_vertex.remove(p)
    return triangles


def _is_clockwise(polygon):
    s = 0
    polygon_count = len(polygon)
    for i in range(polygon_count):
        point = polygon[i]
        point2 = polygon[(i + 1) % polygon_count]
        s += (point2.x - point.x) * (point2.y + point.y)
    return s > 0


def _is_convex(prev, point, next):
    return _triangle_sum(prev.x, prev.y, point.x, point.y, next.x, next.y) < 0


def _is_ear(p1, p2, p3, polygon):
    ear = _contains_no_points(p1, p2, p3, polygon) and \
          _is_convex(p1, p2, p3) and \
          _triangle_area(p1.x, p1.y, p2.x, p2.y, p3.x, p3.y) > 0
    return ear


def _contains_no_points(p1, p2, p3, polygon):
    for pn in polygon:
        if pn in (p1, p2, p3):
            continue
        elif _is_point_inside(pn, p1, p2, p3):
            return False
    return True


def _is_point_inside(p, a, b, c):
    area = _triangle_area(a.x, a.y, b.x, b.y, c.x, c.y)
    area1 = _triangle_area(p.x, p.y, b.x, b.y, c.x, c.y)
    area2 = _triangle_area(p.x, p.y, a.x, a.y, c.x, c.y)
    area3 = _triangle_area(p.x, p.y, a.x, a.y, b.x, b.y)
    areadiff = abs(area - sum([area1, area2, area3])) < EPSILON
    return areadiff


def _triangle_area(x1, y1, x2, y2, x3, y3):
    return abs((x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2)) / 2.0)


def _triangle_sum(x1, y1, x2, y2, x3, y3):
    return x1 * (y3 - y2) + x2 * (y1 - y3) + x3 * (y2 - y1)


def calculate_total_area(triangles):
    result = []
    for triangle in triangles:
        sides = []
        for i in range(3):
            next_index = (i + 1) % 3
            pt = triangle[i]
            pt2 = triangle[next_index]
            # Distance between two points
            side = math.sqrt(math.pow(pt2[0] - pt[0], 2) + math.pow(pt2[1] - pt[1], 2))
            sides.append(side)
        # Heron's numerically stable forumla for area of a triangle:
        # https://en.wikipedia.org/wiki/Heron%27s_formula
        # However, for line-like triangles of zero area this formula can produce an infinitesimally negative value
        # as an input to sqrt() due to the cumulative arithmetic errors inherent to floating point calculations:
        # https://people.eecs.berkeley.edu/~wkahan/Triangle.pdf
        # For this purpose, abs() is used as a reasonable guard against this condition.
        c, b, a = sorted(sides)
        area = .25 * math.sqrt(abs((a + (b + c)) * (c - (a - b)) * (c + (a - b)) * (a + (b - c))))
        result.append((area, a, b, c))
    triangle_area = sum(tri[0] for tri in result)
    return triangle_area
