雑記帳
僕用勉強ノート 「レイトレーシング」の巻

Haskell でレイトレーシングのチュートリアルを追いかける その3 - アンチエイリアス

引き続きこのサイトのチュートリアルに則って、レイトレーシングによる画像の生成に挑戦。
進捗状況としては、ひとまず「section 7」まで完了。
ここで、球面の方程式で球面が描けるなら、2次元トーラスの方程式を与えればトーラスが描けるはずと思って実践してみた。
(後々気付くが、一見ちゃんと描画できているように見えても、実はバグがある。)
コードの実行結果
実行結果
ソースコード
{-# LANGUAGE TypeOperators #-}

module Main where

import Control.Lens
import System.Random
import Linear.Vector
import Linear.Metric
import Linear.V3

-- https://raytracing.github.io/books/RayTracingInOneWeekend.html section 7 + Torus with Haskell!! --

main :: IO ()
main = do
  let
    -- Image
    aspect_ratio = 16.0 / 9
    image_width = 256
    image_height = round $ fromIntegral image_width / aspect_ratio
    samples_per_pixel = 100
    -- World
    world = (([]
      `add` RT_Torus{
        centerOfTorus = V3 0 0 (-1),
        majorRadius = 0.35,
        minorRadius = 0.15,
        orientationOfTorus = normalize $ V3 (-0.2) 1.2 1.9
        })
      `add` RT_Sphere{center = V3 0 (-10000.5) (-1), radius = 10000})
    -- Camera
    camera = Camera {
      viewport_height = 2.0,
      viewport_width = aspect_ratio * viewport_height camera,
      focal_length = 1.0,
      origin = zero,
      horizontal = viewport_width camera *^ unit _x,
      vertical = viewport_height camera *^ unit _y,
      lower_left_corner =
        origin camera - horizontal camera ^/2 - vertical camera ^/2
        - focal_length camera *^ unit _z
      }


  -- Render

  img_data <- return $ "P3\n" ++ show image_width ++ " " ++ show image_height ++ "\n255\n"
  putStr $ img_data

  foldr (>>) (return ()) $ (fmap $ ($) $ \((j, i), seed) ->
    let
      f h currentData gen =
        if h < samples_per_pixel then
          let
            (randNum1, newGen1) = random gen :: (Double, StdGen)
            (randNum2, newGen2) = random newGen1 :: (Double, StdGen)
            u = (fromIntegral i + randNum1) / (fromIntegral image_width - 1.0)
            v = (fromIntegral j + randNum2) / (fromIntegral image_height - 1.0)
            r = get_ray camera (u, v)
            pixcel_color = ray_color r world
          in
            f (succ h) (currentData + pixcel_color) newGen2
        else
          currentData
    in
      write_color (f 0 zero (mkStdGen seed)) samples_per_pixel) $
      zip ((,) <$> [image_height - 1, image_height - 2 .. 0] <*> [0, 1 .. image_width - 1])
      ((randomRs (0, 536870912) (mkStdGen 21)) :: [Int])


---------------------
-- A Ray Data Type --
---------------------

data Ray = Ray {
  orig :: V3 Double,  -- Origin of this ray (As a position in 3D Euclidean space)
  dir :: V3 Double    -- Direction of this ray (As a direction vector in 3D Euclidean space)
} deriving (Show)


at' :: Ray -> Double -> V3 Double
at' r t = (orig r) + t *^ (dir r)


------------------------
-- A Camera Data Type --
------------------------

data Camera = Camera {
  viewport_height :: Double,
  viewport_width :: Double,
  focal_length :: Double,
  origin :: V3 Double,
  horizontal :: V3 Double,
  vertical :: V3 Double,
  lower_left_corner :: V3 Double
} deriving (Show)

get_ray :: Camera -> (Double, Double) -> Ray
get_ray this (u, v) =
  Ray {
    orig = origin this,
    dir = lower_left_corner this + u *^ horizontal this + v *^ vertical this - origin this
    }


----------------------
-- A Hittable Class --
----------------------

type HittableData = (RT_Sphere + RT_Torus) + RT_Sphere -- Third RT_Sphere is just dummies

class Hittable a where
  toSum :: a -> HittableData
  hit :: a -> Ray -> Double -> Double -> Maybe HitRecord

instance (Hittable a, Hittable b) => Hittable (a + b) where
  toSum = coPair(toSum, toSum)
  hit = coPair(hit, hit)

add :: Hittable a => [HittableData] -> a -> [HittableData]
add list obj = (toSum obj) : list

data HitRecord = HitRecord {
  p :: V3 Double,
  normal :: V3 Double,
  t :: Double,
  front_face :: Bool
} deriving (Show)


set_face_normal :: HitRecord -> Ray -> V3 Double -> HitRecord
set_face_normal this r outward_normal =
  HitRecord {
    p = p this,
    normal = if dir r `dot` outward_normal < 0 then outward_normal else -outward_normal,
    t = t this,
    front_face = (dir r `dot` outward_normal < 0)
  }

hitSomething :: [HittableData] -> Ray -> Double -> Double -> Maybe HitRecord
hitSomething list r t_min t_max =
  let
    f (list', r', closest_so_far, currentRecord) =
      case list' of
        x:xs ->
          let
            temp = hit x r' t_min t_max
          in
            case temp of
              Just a ->
                f $ (xs, r', t a, temp)
              Nothing ->
                f $ (xs, r', closest_so_far, currentRecord)
        [] ->
          currentRecord

  in
    f $ (list, r, t_max, Nothing)


----------------------
-- Hittable Objects --
----------------------

-- Sphere
data RT_Sphere = RT_Sphere {
  center :: V3 Double,
  radius :: Double
} deriving (Show)

instance Hittable RT_Sphere where
  toSum = Inj1 -: Inj1
  hit obj r t_min t_max =
    let
      p0 = orig r
      c1 = center obj
      r1 = radius obj
      oc = p0 - c1
      a = quadrance (dir r)
      half_b = oc `dot` dir r
      c = quadrance oc - (radius obj) ^ 2
      discriminant = half_b ^ 2 - a*c in

      if discriminant > 0 then
        let
          root = sqrt discriminant
          f k =
            case k of
              x:xs ->
                if t_min < x && x < t_max then
                  return $ set_face_normal HitRecord {
                    p = at' r x,
                    normal = zero,
                    t = x,
                    front_face = False
                    } r ((at' r x - c1) ^/ r1)
                else
                  f $ xs

              [] ->
                Nothing

        in
          f $ [(-half_b - root) / a, (-half_b + root) / a]
      else
        Nothing

-- Torus
data RT_Torus = RT_Torus {
  centerOfTorus :: V3 Double,
  majorRadius :: Double,
  minorRadius :: Double,
  orientationOfTorus :: V3 Double -- [BEWARB] this pseudo-vector must be normalized
} deriving (Show)

instance Hittable RT_Torus where
  toSum = Inj2 -: Inj1
  hit obj r t_min t_max =
    let
      p0 = orig r
      c1 = centerOfTorus obj
      r1 = majorRadius obj
      r2 = minorRadius obj
      n  = orientationOfTorus obj
      oc = p0 - c1
      a  = quadrance (dir r)
      half_b = oc `dot` dir r
      c = quadrance oc - (r1 + r2 + 0.01) ^ 2
      discriminant = half_b ^ 2 - a*c in

      if discriminant > 0 then
        let
          root = sqrt discriminant
          s = newton's_method
            50
            (max 0 ((-half_b - root) / a), max 0 ((-half_b + root) / a))
            (max 0 ((-half_b - root) / a), 0)
            (findIntersection_forTorus obj r)
            (findIntersection_forTorus' obj r)
        in
          s >>= (\k ->
            let
              x = at' r k - c1
            in
              return $ HitRecord {
                p = c1 + x,
                normal = (x - (r1 *^ (normalize $ x - (n `dot` x) *^ n))) ^/ r2,
                t = k,
                front_face = False
                }
            )
      else
        Nothing


findIntersection_forTorus :: RT_Torus -> Ray -> Double -> Double
findIntersection_forTorus obj r t =
  (quadrance u + r1 ^2 - r2 ^ 2) ^ 2 - 4 * (r1 ^ 2) * (u `dot` (u - (n `dot` u) *^ n))
  -- quadrance u - 2 * r1 * (sqrt $ u `dot` (u - (n `dot` u) *^ n)) + r1 ^2 - r2 ^ 2
  where
    a = dir r
    u = at' r t - centerOfTorus obj
    n = orientationOfTorus obj
    r1 = majorRadius obj
    r2 = minorRadius obj

findIntersection_forTorus' :: RT_Torus -> Ray -> Double -> Double
findIntersection_forTorus' obj r t =
  4 * (a `dot` u) * (quadrance u + r1 ^ 2 - r2 ^ 2)
  - 8 * (r1^2) * (u `dot` (a - (n `dot` a) *^ n))
  -- 2 * (a `dot` u)
  -- - 2 * r1 * (u `dot` (a - (n `dot` a) *^ n)) / (sqrt $ u `dot` (u - (n `dot` u) *^ n))
  where
    a = dir r
    u = at' r t - centerOfTorus obj
    n = orientationOfTorus obj
    r1 = majorRadius obj
    r2 = minorRadius obj


----------------------------------------
-- Computing the color of a given ray --
----------------------------------------

write_color :: RealFrac a => V3 a -> Int -> IO ()
write_color v samples_per_pixel =
  let
    v' = v ^/ fromIntegral samples_per_pixel
    f = show.floor.(255.999*)
  in
    do
      tmp <- return $ f(v' ^._x) ++ " " ++ f(v' ^._y) ++ " " ++ f(v' ^._z) ++ "\n"
      putStr $ tmp


ray_color :: Ray -> [HittableData] -> V3 Double
ray_color r objects =
  let
    record = hitSomething objects r 0 infinity
  in
    case record of
      Just a ->
        0.5 *^ ((normal a) + (V3 1 1 1))

      Nothing ->
        let
          unit_direction = normalize $ (dir r)
          s = 0.5 * (unit_direction ^._y + 1.0)
        in
          lerp s (V3 0.5 0.7 1.0) (V3 1.0 1.0 1.0)


---------------
-- Utilities --
---------------

infinity :: RealFloat a => a
infinity = encodeFloat (floatRadix 0 - 1) (snd $ floatRange 0)

deg2rad :: Floating a => a -> a
deg2rad degrees = degrees * pi / 180

clamp :: (Ord a, Num a) => a -> a -> a -> a
clamp x y val = (max x).(min y) $ val

isInClosedInterval:: (Ord a, Fractional a) => (a, a) -> a -> Bool
isInClosedInterval (a, b) val = (a <= val && val <= b)

isInOpenInterval:: (Ord a, Fractional a) => (a, a) -> a -> Bool
isInOpenInterval (a, b) val = (a < val && val < b)

newton's_method :: (Ord a, Fractional a) => Int -> (a, a) -> (a, a) -> (a -> a) -> (a -> a) -> Maybe a
newton's_method depth interval (current, prev) f f' =
  if (uncurry (/=)) interval && isInClosedInterval interval current && depth > 0 then
    if abs(current - prev) < 1.0E-10 then
      Just current
    else
      newton's_method (pred depth) interval (current - (f(current) / f'(current)), current) f f'
  else
    Nothing

-- Joke
derivativeOf :: (Floating a) => (a -> a) -> Int -> a -> a
derivativeOf f precision =
  \x -> (f(x + dx) - f(x)) / dx
  where dx = 0.1^precision


-- Diagrammatic-order composition
(-:) = flip (.)

-- Sum objects and injections
data (+) a b = Inj1 a | Inj2 b

instance (Show a, Show b) => Show (a + b) where
  show = coPair(show -: (++ ";inj1"), show -: (++ ";inj2"))

coPair :: (a1 -> b, a2 -> b) -> (a1 + a2 -> b)
coPair (f, g) x = case x of
  Inj1 x -> f x
  Inj2 x -> g x