csci5607/exam-2/exam2.py

import itertools
import numpy as np
from sympy import *
import math

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

def perspective_matrix(vfov, width, height, left, right, bottom, top, near, far):
  aspect = width / height
  return np.array([
    [1.0 / math.tan(vfov / 2.0) / aspect, 0, 0, 0],
    [0, 1.0 / math.tan(vfov / 2.0), 0, 0],
    [0, 0, -(far + near) / (far - near), -2.0 * far * near / (far - near)],
    [0, 0, -1, 0]
  ])
  # 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],
  # ])

def view_matrix(camera_pos, view_dir, up_dir):
  n = unit(-view_dir)
  u = unit(np.cross(up_dir, n))
  v = np.cross(n, u)
  return np.array([
    [u[0], u[1], u[2], -np.dot(camera_pos, u)],
    [v[0], v[1], v[2], -np.dot(camera_pos, v)],
    [n[0], n[1], n[2], -np.dot(camera_pos, n)],
    [0, 0, 0, 1],
  ])

def print_trans(before, after):
  def style_vec(v):
    start = "$[\\begin{matrix}"
    mid = str(v[0]) + " & " + str(v[1]) + " & " + str(v[2])
    end = "\\end{matrix}]$"
    return f"{start}{mid}{end}"
  return style_vec(before) + " $\\rightarrow$ " + style_vec(after)

def compute_view(near, vfov, hfov):
  width = 2.0 * near * math.tan(hfov / 2.0)
  height = 2.0 * near * math.tan(hfov / 2.0)

  left = -width / 2.0
  right = width / 2.0
  bottom = -height / 2.0
  top = height / 2.0
  return width, height, left, right, bottom, top

def print_bmatrix(arr):
  for row in arr:
    for j, col in enumerate(row):
      end = " " if j == len(row) - 1 else " & "
      print(col, end=end)
    print("\\\\")

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)

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

  uin = unit(n_norm)
  print("unit inv normal", uin)

  uperp = unit(v2)
  print("uperp", uperp)

  answer_1e = uin + math.tan(theta_t) * uperp - p
  print("1e answer", unit(answer_1e))

def problem_4():
  camera_pos = np.array([2, 3, 5])
  view_dir = np.array([1, -1, -1])
  up_dir = np.array([0, 1, 0])
  V = view_matrix(camera_pos, view_dir, up_dir)
  print(V)

  f = np.vectorize(lambda c: (c * c))
  print(f(V))


def build_translation_matrix(vec):
  return np.array([
    [1, 0, 0, vec[0]],
    [0, 1, 0, vec[1]],
    [0, 0, 1, vec[2]],
    [0, 0, 0, 1],
  ])

def problem_5():
  b1 = build_translation_matrix(np.array([0, 0, 5]))
  theta = math.radians(-90)
  sin_theta = round( math.sin(theta), 5)
  cos_theta = round(math.cos(theta), 5)
  b2 = np.array([
    [cos_theta, 0, sin_theta, 0],
    [0, 1, 0, 0],
    [-sin_theta, 0, cos_theta, 0],
    [0, 0, 0, 1],
  ])
  b3 = build_translation_matrix(np.array([0, 0, -5]))

  print("b1", b1)
  print("b2", b2)
  print("b3", b3)
  M = b3 @ b2 @ b1
  print("M", M)

  ex1 = np.array([1, 1, -4, 1])
  print("ex1", ex1, M @ ex1)

  up = np.array([0, 1, 0])
  n = np.array([1, 0, 0])
  u = unit(np.cross(up, n))
  v = np.cross(n, u)

  print(f"{up = }, {n = }, {u = }, {v = }")

  eye = np.array([5, 0, -5])
  dx = -(np.dot(eye, u))
  dy = -(np.dot(eye, v))
  dz = -(np.dot(eye, n))
  print(f"{dx = }, {dy = }, {dz = }")

def problem_8():
  near = 0.5
  far = 20



  print("part 8a")
  vfov = hfov = math.radians(60)
  width, height, left, right, bottom, top = compute_view(near, vfov, hfov)
  print(perspective_matrix(vfov, width, height, left, right, bottom, top, near, far))
  print()

def problem_7():

  def solve(camera_pos, angle):
    angle_radians = math.radians(angle)
    near = 1
    far = 10
    view_dir = np.array([0, 0, -1])
    up_dir = np.array([0, 1, 0])
    width, height, left, right, bottom, top = compute_view(near, angle_radians, angle_radians)
    print("faces of the viewing frustum", left, right, bottom, top)
    P = perspective_matrix(angle_radians, width, height, left, right, bottom, top, near, far)
    V = view_matrix(camera_pos, view_dir, up_dir)
    print("P")
    print_bmatrix(np.around(P, 4))
    print("V")
    print_bmatrix(np.around(V, 4))
    m = P @ V

    points = [
      np.array([0.5, 0.5, -4]),
      np.array([0.5, -0.5, -4]),
      np.array([-0.5, -0.5, -4]),
      np.array([-0.5, 0.5, -4]),

      np.array([0.5, 0.5, -5]),
      np.array([0.5, -0.5, -5]),
      np.array([-0.5, -0.5, -5]),
      np.array([-0.5, 0.5, -5]),

      np.array([1, 1.5, -7]),
      np.array([1, -0.5, -7]),
      np.array([-1, -0.5, -7]),
      np.array([-1, 1.5, -7]),

      np.array([1, 1.5, -9]),
      np.array([1, -0.5, -9]),
      np.array([-1, -0.5, -9]),
      np.array([-1, 1.5, -9]),
    ]

    for point in points:
      point_ = np.r_[point, [1]]
      trans = m @ point_
      def style_vec(v):
        start = "$[\\begin{matrix}"
        mid = str(v[0]) + " & " + str(v[1]) + " & " + str(v[2])
        end = "\\end{matrix}]$"
        return f"{start}{mid}{end}"
      print("-", style_vec(point), "$\\rightarrow$", style_vec(np.around(trans[:3], 4)))

  print("Part A")
  camera_pos = np.array([0, 0, 0])
  angle = 90
  solve(camera_pos, angle)

  print("Part B")
  camera_pos = np.array([0, 0, -2])
  angle = 90
  solve(camera_pos, angle)

  print("Part C")
  camera_pos = np.array([0, 0, 0])
  angle = 45
  solve(camera_pos, angle)

def problem_6():
  def calculate(points):
    left = min(map(lambda p: p[0], points))
    right = max(map(lambda p: p[0], points))
    bottom = min(map(lambda p: p[1], points))
    top = max(map(lambda p: p[1], points))
    near = min(map(lambda p: p[2], points))
    far = max(map(lambda p: p[2], points))

    step_1 = np.array([
      [1, 0, 0, 0],
      [0, 1, 0, 0],
      [0, 0, -1, 0],
      [0, 0, 0, 1],
    ])

    step_2 = np.array([
      [1, 0, 0, -(left + right) / 2.0],
      [0, 1, 0, -(bottom + top) / 2.0],
      [0, 0, 1, -(near + far) / 2.0],
      [0, 0, 0, 1],
    ])

    step_3 = np.array([
      [2.0 / (right - left), 0, 0, 0],
      [0, 2.0 / (top - bottom), 0, 0],
      [0, 0, 2.0 / (far - near), 0],
      [0, 0, 0, 1],
    ])

    M_ortho = step_3 @ step_2 @ step_1

    for point in points:
      point_ = np.r_[point, [1]]
      trans_ = M_ortho @ point_
      trans = trans_[:3]
      print("-", print_trans(np.around(point, 3), np.around(trans, 3)))

  sqrt3 = math.sqrt(3)
  width = 2.0 * sqrt3
  cube_center = np.array([0, 0, -3.0 * sqrt3])
  print("HELLOSU")

  points = []
  for (dx, dy, dz) in itertools.product([-1, 1], [-1, 1], [-1, 1]):
    point = cube_center + np.array([dx * width / 2, dy * width / 2, dz * width / 2])
    points.append(point)

  calculate(points)

def problem_9():
  pass

print("\nPROBLEM 8 -------------------------"); problem_8()
print("\nPROBLEM 1 -------------------------"); problem_1()
print("\nPROBLEM 5 -------------------------"); problem_5()
print("\nPROBLEM 9 -------------------------"); problem_9()
print("\nPROBLEM 7 -------------------------"); problem_7()
print("\nPROBLEM 6 -------------------------"); problem_6()
print("\nPROBLEM 4 -------------------------"); problem_4()