import numpy as np

unit = lambda v: v/np.linalg.norm(v)

def problem_1():
    p = np.array([1, 4, 8])
    e = np.array([0, 0, 0])
    s = np.array([2, 2, 10])

    i = p - e
    print("incoming", i)
    print("|I| =", np.linalg.norm(i))
    n = s - p
    print("normal", n)

    n_norm = unit(n)
    print("normal_norm", n_norm)
    cos_theta_i = np.dot(-i, n) / (np.linalg.norm(i) * np.linalg.norm(n))
    print("part a = cos^{-1} of ", cos_theta_i)
    print(np.arccos(cos_theta_i))

    proj = n_norm * np.linalg.norm(i) * cos_theta_i
    print("proj", proj)

    p_ = p + proj
    print("proj point", p_)

    v2 = p_ - e
    print("v2", v2)

    sin_theta_i = np.sin(np.arccos(cos_theta_i))
    print("sin theta_i =", sin_theta_i)

    print("approx answer for part d", np.arcsin(1.0 / 1.5 * sin_theta_i))

def problem_8():
    def P(left, right, bottom, top, near, far):
        return np.array([
            [2.0 * near / (right - left), 0, (right + left) / (right - left), 0],
            [0, 2.0 * near / (top - bottom), (top + bottom) / (top - bottom), 0],
            [0, 0, -(far + near) / (far - near), -(2.0 * far * near) / (far - near)],
            [0, 0, -1, 0],
        ])

    near, far = left, right = bottom, top = -1, 1
    print("part 8a", P(left, right, bottom, top, near, far))
    print()

problem_8()
problem_1()