Step by Step solution about How to resolve the algorithm Find the intersection of a line with a plane step by step in the Haskell programming language

The Haskell code you provided defines a 3D vector type V3 and some operations on it. Let's break down the code step by step:

1. Data Definition:

   data V3 a = V3 a a a
      deriving Show

   This line defines a new data type V3 that represents a 3D vector with three components a, b, and c. The deriving Show part generates an instance of the Show typeclass, allowing you to print V3 vectors easily.

2. Functor Instance:

   instance Functor V3 where
      fmap f (V3 a b c) = V3 (f a) (f b) (f c)

   This instance makes V3 a functor, which means you can apply a function to each component of the vector. The fmap function simply applies the function f to each component of the vector.

3. Applicative Instance:

   instance Applicative V3 where
      pure a = V3 a a a
      V3 a b c <*> V3 d e f = V3 (a d) (b e) (c f)

   This instance makes V3 an applicative functor, which means you can lift functions to operate on V3 vectors. The pure function creates a pure value wrapped in a V3 vector, and the <*> function applies a function lifted inside a V3 vector to another V3 vector.

4. Num Instance:

   instance Num a => Num (V3 a) where
      (+) = liftA2 (+)
      (-) = liftA2 (-)
      (*) = liftA2 (*)
      negate = fmap negate
      abs = fmap abs
      signum = fmap signum
      fromInteger = pure . fromInteger

   This instance makes V3 a numeric type, which means you can perform numeric operations on V3 vectors. The operations are lifted using liftA2 so that they operate on each component of the vector.

5. Dot Product:

   dot ::Num a => V3 a -> V3 a -> a
   dot a b = x + y + z
    where
      V3 x y z = a * b

   This function calculates the dot product of two V3 vectors. It multiplies the corresponding components of the vectors and sums them up.

6. Ray-Plane Intersection:

   intersect :: Fractional a => V3 a -> V3 a -> V3 a -> V3 a -> V3 a
   intersect rayVector rayPoint planeNormal planePoint =
      rayPoint - rayVector * pure prod3
   where
      diff = rayPoint - planePoint
      prod1 = diff `dot` planeNormal
      prod2 = rayVector `dot` planeNormal
      prod3 = prod1 / prod2

   This function calculates the intersection point between a ray and a plane in 3D space. It uses the following parameters:

   • rayVector: The direction vector of the ray.
   • rayPoint: A point on the ray.
   • planeNormal: The normal vector of the plane.
   • planePoint: A point on the plane.

   The function calculates the intersection point by first finding the distance along the ray where it intersects the plane (prod3). Then, it multiplies the ray vector by this distance and subtracts the result from the ray point to get the intersection point.

7. Main Function:

   main = printf "The ray intersects the plane at (%f, %f, %f)\n" x y z
    where
      V3 x y z = intersect rv rp pn pp :: V3 Double
      rv = V3 0 (-1) (-1)
      rp = V3 0 0 10
      pn = V3 0 0 1
      pp = V3 0 0 5

   The main function is the entry point of the program. It calls the intersect function to calculate the intersection point between a ray and a plane. Then, it prints the coordinates of the intersection point using printf. The specific values of the ray and plane parameters are defined in the code.

import Control.Applicative (liftA2)
import Text.Printf (printf)

data V3 a = V3 a a a
    deriving Show

instance Functor V3 where
    fmap f (V3 a b c) = V3 (f a) (f b) (f c)

instance Applicative V3 where
    pure a = V3 a a a
    V3 a b c <*> V3 d e f = V3 (a d) (b e) (c f)

instance Num a => Num (V3 a) where
    (+) = liftA2 (+)
    (-) = liftA2 (-)
    (*) = liftA2 (*)
    negate = fmap negate
    abs = fmap abs
    signum = fmap signum
    fromInteger = pure . fromInteger

dot ::Num a => V3 a -> V3 a -> a
dot a b = x + y + z
  where
    V3 x y z = a * b

intersect :: Fractional a => V3 a -> V3 a -> V3 a -> V3 a -> V3 a
intersect rayVector rayPoint planeNormal planePoint =
    rayPoint - rayVector * pure prod3
  where
    diff = rayPoint - planePoint
    prod1 = diff `dot` planeNormal
    prod2 = rayVector `dot` planeNormal
    prod3 = prod1 / prod2

main = printf "The ray intersects the plane at (%f, %f, %f)\n" x y z
  where
    V3 x y z = intersect rv rp pn pp :: V3 Double
    rv = V3 0 (-1) (-1)
    rp = V3 0 0 10
    pn = V3 0 0 1
    pp = V3 0 0 5