from math import sqrt, sin, cos, atan2

# def add(v1,v2):
#     return (v1[0] + v2[0], v1[1] + v2[1])

def subtract(v1,v2):
    return (v1[0] - v2[0], v1[1] - v2[1])

def add(*vectors):
    return (sum([v[0] for v in vectors]), sum([v[1] for v in vectors]))

def length(v):
    return sqrt(v[0]**2 + v[1]**2)

def distance(v1,v2):
    return length(subtract(v1,v2))

def perimeter(vectors):
    distances = [distance(vectors[i], vectors[(i+1)%len(vectors)])
                    for i in range(0,len(vectors))]
    return sum(distances)

def scale(scalar,v):
    return (scalar * v[0], scalar * v[1])

def to_cartesian(polar_vector):
    length, angle = polar_vector[0], polar_vector[1]
    return (length*cos(angle), length*sin(angle))

def rotate(angle, vectors):
    polars = [to_polar(v) for v in vectors]
    return [to_cartesian((l, a+angle)) for l,a in polars]

def translate(translation, vectors):
    return [add(translation, v) for v in vectors]

def to_polar(vector):
    x, y = vector[0], vector[1]
    angle = atan2(y,x)
    return (length(vector), angle)