solution4.py

# Q4
from math import pi, cos, sin, acos, asin, sqrt
from random import random
from functools import partial

def output(case_num, result):
    print("Case #{}".format(case_num))
    for x, z, y in result:
        print("{} {} {}".format(y, z, x))


t = int(input())

A_min = 1.0
A_max = sqrt(3)

alpha_min = 0.0
alpha_max = pi / 4

initial_points = [(0.5, 0, 0), (0, 0.5, 0), (0, 0, 0.5)]

def calc_beta(A, alpha):
    return pi / 4 + acos((A - sin(alpha)) / ( sqrt(2) * cos(alpha)))

def rotate_point(alpha, beta, point):
    x, y, z = point
    ## rotate about x
    x = x
    y = y*cos(alpha) - y*sin(alpha)
    z = y*sin(alpha) + z*cos(alpha)

    ## rotation about z
    x = x*cos(beta) + y*sin(beta)
    y = y
    z = -x*sin(beta) + z*cos(beta)
    return (x, y, z)

def area(alpha, beta):
    return sin(alpha) + sqrt(2)*cos(alpha)*cos(beta - pi /4)


for case_num in range(1, t + 1):
    A = float(input())

    beta = None
    alpha = random() * pi / 4
    while beta is None:
        try:
            beta = calc_beta(A, alpha)
            break
        except ValueError:
            alpha = random() * pi / 4

    # now that we have alpha and beta, calculate new points
    rotate = partial(rotate_point, alpha, beta)
    output(case_num, map(rotate, initial_points))