# Ákos Halmai, 2023.
import math


def getArea(points) -> float:
    """Expects iterable of two-element-tuples.
    For example:
    points = ((0, 0), (1, 0), (1, 1), (0, 1), (0, 0))"""

    area = 0.0
    (x1, y1) = points[0]
    for (x2, y2) in points[1:]:
        area += x1 * y2 - y1 * x2
        (x1, y1) = (x2, y2)

    return 0.5 * area


def getAreaAbs(points) -> float:
    """Expects iterable of two-element-tuples.
    For example:
    points = ((0, 0), (1, 0), (1, 1), (0, 1), (0, 0))"""

    # Returns float (always):
    return math.fabs(getArea(points))


def getAreaPrecise(points) -> float:
    """Expects iterable of two-element-tuples.
    For example:
    points = ((0, 0), (1, 0), (1, 1), (0, 1), (0, 0))"""

    def getPartArea(points):
        (x1, y1) = points[0]
        for (x2, y2) in points[1:]:
            yield x1 * y2
            yield -y1 * x2
            (x1, y1) = (x2, y2)

    # Avoids loss of precision by tracking multiple
    # intermediate partial sums:
    return 0.5 * math.fsum(getPartArea(points))


def getAreaPreciseAbs(points) -> float:
    """Expects iterable of two-element-tuples.
    For example:
    points = ((0, 0), (1, 0), (1, 1), (0, 1), (0, 0))"""

    # Returns float (always):
    return math.fabs(getAreaPrecise(points))


def __isClockwise(*args) -> bool:
    """Private. Do not use it!"""

    (x0, y0), (x1, y1), (x2, y2), *_ = args
    return ((x1 - x0) * (y2 - y1) - (y1 - y0) * (x2 - x1)) > 0.0


def isClockwise(points) -> bool:
    """Expects iterable of two-element-tuples.
    For example:
    points = ((0, 0), (1, 0), (1, 1), (0, 1), (0, 0))"""

    return __isClockwise(*points)


def isConcave(points) -> bool:
    """Expects iterable of two-element-tuples.
    For example:
    points = ((0, 0), (1, 0), (1, 1), (0, 1), (0, 0))"""

    (x0, y0), (x1, y1), (x2, y2), *_ = points
    switch = __isClockwise((x0, y0), (x1, y1), (x2, y2))
    for (xn, yn) in points[3:]:

        if switch != __isClockwise((x1, y1), (x2, y2), (xn, yn)):
            return True

        # Delayed for performance:
        ((x0, y0), (x1, y1), (x2, y2)) = ((x1, y1), (x2, y2), (xn, yn))

    return False