Geometry.hs

{-# LANGUAGE MultiParamTypeClasses, TypeFamilies, FlexibleContexts, FlexibleInstances #-}
{- |
This module provides functionality for working with affine transformations (i.e. in the unit square)
 
-}
module Graphics.Rendering.Plot.Light.Internal.Geometry where

-- import Data.Monoid ((<>))



-- | A `Point` defines a point in R2
data Point a = Point { _px :: a,
                       _py :: a } deriving (Eq)

instance Show a => Show (Point a) where
  show (Point x y) = show x ++ "," ++ show y

mkPoint :: a -> a -> Point a
mkPoint = Point

-- | Overwrite either coordinate of a Point, to e.g. project on an axis
setPointCoord :: Axis -> a -> Point a -> Point a
setPointCoord axis c (Point x y)
  | axis == X = Point c y
  | otherwise = Point x c

setPointX, setPointY :: a -> Point a -> Point a
setPointX = setPointCoord X
setPointY = setPointCoord Y

-- | A `LabeledPoint` carries a "label" (i.e. any additional information such as a text tag, or any other data structure), in addition to position information. Data points on a plot are `LabeledPoint`s.
data LabeledPoint l a =
  LabeledPoint {
   _lp :: Point a,
   _lplabel :: l
   } deriving (Eq, Show)

mkLabeledPoint :: Point a -> l -> LabeledPoint l a
mkLabeledPoint = LabeledPoint

moveLabeledPoint :: (Point a -> Point b) -> LabeledPoint l a -> LabeledPoint l b
moveLabeledPoint f (LabeledPoint p l) = LabeledPoint (f p) l

-- | A frame, i.e. a bounding box for objects
data Frame a = Frame {
   _fpmin :: Point a,
   _fpmax :: Point a
   } deriving (Eq, Show)

mkFrame :: Point a -> Point a -> Frame a
mkFrame = Frame

-- | Build a frame rooted at the origin (0, 0)
mkFrameOrigin :: Num a => a -> a -> Frame a
mkFrameOrigin w h = Frame origin (Point w h)


-- | Create a `Frame` from a container of `Point`s `P`, i.e. construct two points `p1` and `p2` such that :
--
-- p1 := inf(x,y) P
-- p2 := sup(x,y) P
frameFromPoints :: (Ord a, Foldable t, Functor t) =>
                         t (Point a) -> Frame a
frameFromPoints ds = mkFrame (Point mx my) (Point mmx mmy)
  where
    xcoord = _px <$> ds
    ycoord = _py <$> ds
    mmx = maximum xcoord 
    mmy = maximum ycoord 
    mx = minimum xcoord 
    my = minimum ycoord


 

-- | Frame corner coordinates
xmin, xmax, ymin, ymax :: Frame a -> a
xmin = _px . _fpmin
xmax = _px . _fpmax
ymin = _py . _fpmin
ymax = _py . _fpmax

-- | The `width` is the extent in the `x` direction and `height` is the extent in the `y` direction
width, height :: Num a => Frame a -> a
width f = xmax f - xmin f
height f = ymax f - ymin f



-- * Axis

data Axis = X | Y deriving (Eq, Show)

otherAxis :: Axis -> Axis
otherAxis X = Y
otherAxis _ = X






  



-- | V2 is a vector in R^2
data V2 a = V2 a a deriving (Eq, Show)

-- | Vectors form a monoid w.r.t. vector addition
instance Num a => Monoid (V2 a) where
  mempty = V2 0 0
  (V2 a b) `mappend` (V2 c d) = V2 (a + c) (b + d)

-- | Additive group :
-- 
-- > v ^+^ zero == zero ^+^ v == v
--
-- > v ^-^ v == zero
class AdditiveGroup v where
  -- | Identity element
  zero :: v
  -- | Group action ("sum")
  (^+^) :: v -> v -> v
  -- | Inverse group action ("subtraction")
  (^-^) :: v -> v -> v

-- | Vectors form an additive group
instance Num a => AdditiveGroup (V2 a) where
  zero = mempty
  (^+^) = mappend
  (V2 a b) ^-^ (V2 c d) = V2 (a - c) (b - d)

-- | Vector space : multiplication by a scalar quantity
class AdditiveGroup v => VectorSpace v where
  type Scalar v :: *
  -- | Scalar multiplication
  (.*) :: Scalar v -> v -> v
  
instance Num a => VectorSpace (V2 a) where
  type Scalar (V2 a) = a
  n .* (V2 vx vy) = V2 (n*vx) (n*vy)

-- | Hermitian space : inner product
class VectorSpace v => Hermitian v where
  type InnerProduct v :: *
  -- | Inner product
  (<.>) :: v -> v -> InnerProduct v

instance Num a => Hermitian (V2 a) where
  type InnerProduct (V2 a) = a
  (V2 a b) <.> (V2 c d) = (a*c) + (b*d)

-- | Euclidean (L^2) norm
norm2 ::
  (Hermitian v, Floating n, n ~ (InnerProduct v)) => v -> n
norm2 v = sqrt $ v <.> v

-- | Normalize a V2 w.r.t. its Euclidean norm
normalize2 :: (InnerProduct v ~ Scalar v, Floating (Scalar v), Hermitian v) =>
     v -> v
normalize2 v = (1/norm2 v) .* v


-- | Create a V2 `v` from two endpoints p1, p2. That is `v` can be seen as pointing from `p1` to `p2`
v2fromEndpoints, (-.) :: Num a => Point a -> Point a -> V2 a
v2fromEndpoints (Point px py) (Point qx qy) = V2 (qx-px) (qy-py)
(-.) = v2fromEndpoints

-- | The origin of the axes, point (0, 0)
origin :: Num a => Point a
origin = Point 0 0


-- | A Mat2 can be seen as a linear operator that acts on points in the plane
data Mat2 a = Mat2 a a a a deriving (Eq, Show)

-- | Linear maps, i.e. linear transformations of vectors
class Hermitian v => LinearMap m v where
  -- | Matrix action, i.e. linear transformation of a vector
  (#>) :: m -> v -> v

-- | Multiplicative matrix semigroup ("multiplying" two matrices together)
class MultiplicativeSemigroup m where
  -- | Matrix product
  (##) :: m -> m -> m

instance Num a => MultiplicativeSemigroup (Mat2 a) where
  Mat2 a00 a01 a10 a11 ## Mat2 b00 b01 b10 b11 = Mat2 (a00*b00+a01*b10) (a00*b01+a01*b11) (a10*b00+a11*b10) (a10*b01+a11*b11)

instance Num a => LinearMap (Mat2 a) (V2 a) where
  (Mat2 a00 a01 a10 a11) #> (V2 vx vy) = V2 (a00 * vx + a01 * vy) (a10 * vx + a11 * vy)



-- | Diagonal matrices in R2 behave as scaling transformations
data DiagMat2 a = DMat2 a a deriving (Eq, Show)

-- | Diagonal matrices form a monoid w.r.t. matrix multiplication and have the identity matrix as neutral element
instance Num a => Monoid (DiagMat2 a) where
  mempty = DMat2 1 1
  mappend = (##)

-- | Matrices form a monoid w.r.t. matrix multiplication and have the identity matrix as neutral element
instance Num a => Monoid (Mat2 a) where
  mempty = Mat2 1 0 0 1
  mappend = (##)

-- | Create a diagonal matrix
diagMat2 :: Num a => a -> a -> DiagMat2 a
diagMat2 = DMat2


-- | Rotation matrix
rotMtx :: Floating a => a -> Mat2 a
rotMtx r = Mat2 (cos r) (- (sin r)) (sin r) (cos r)


-- | The class of invertible linear transformations
class LinearMap m v => MatrixGroup m v where
  -- | Inverse matrix action on a vector
  (<\>) :: m -> v -> v
  
instance Num a => MultiplicativeSemigroup (DiagMat2 a) where
  DMat2 a b ## DMat2 c d = DMat2 (a*c) (b*d)

instance Num a => LinearMap (DiagMat2 a) (V2 a) where
  DMat2 d1 d2 #> V2 vx vy = V2 (d1 * vx) (d2 * vy)

-- | Diagonal matrices can always be inverted
instance Fractional a => MatrixGroup (DiagMat2 a) (V2 a) where
  DMat2 d1 d2 <\> V2 vx vy = V2 (vx / d1) (vy / d2)

-- | Build a `V2` v from a `Point` p (i.e. assuming v points from the origin (0,0) to p)
v2fromPoint :: Num a => Point a -> V2 a
v2fromPoint p = origin -. p

-- | Build a `Point` p from a `V2` v (i.e. assuming v points from the origin (0,0) to p)
pointFromV2 :: V2 a -> Point a
pointFromV2 (V2 x y) = Point x y

-- | Move a point along a vector
movePoint :: Num a => V2 a -> Point a -> Point a
movePoint (V2 vx vy) (Point px py) = Point (px + vx) (py + vy)

-- | Move a `LabeledPoint` along a vector
moveLabeledPointV2 :: Num a => V2 a -> LabeledPoint l a -> LabeledPoint l a
moveLabeledPointV2 = moveLabeledPoint . movePoint


-- | `pointRange n p q` returns a list of `n+1` equi-spaced `Point`s between `p` and `q` (i.e. the input points are included as the first and last points in the list)
pointRange :: (Fractional a, Integral n) =>
     n -> Point a -> Point a -> [Point a]
pointRange n p q = [ movePoint (fromIntegral x .* vnth) p | x <- [0 .. n]]
  where
    v = p -. q
    vnth = (1/fromIntegral n) .* v





fromFrame :: Fractional a => Frame a -> V2 a -> V2 a
fromFrame from v = mfrom <\> (v ^-^ vfrom) where
  vfrom = v2fromPoint (_fpmin from) -- min.point vector of `from`
  mfrom = diagMat2 (width from) (height from) -- rescaling matrix of `from`

toFrame :: Num a => Frame a -> V2 a -> V2 a
toFrame to v01 = (mto #> v01) ^+^ vto where
  vto = v2fromPoint (_fpmin to)     -- min.point vector of `to`
  mto = diagMat2 (width to) (height to)       -- rescaling matrix of `to`




-- | Given two frames `F1` and `F2`, returns a function `f` that maps an arbitrary vector `v` contained within `F1` onto one contained within `F2`.
--
-- This function is composed of three affine maps :
--
-- 1. map `v` into a vector `v01` that points within the unit square,
--
-- 2. map `v01` onto `v01'`. This transformation serves to e.g. flip the dataset along the y axis (since the origin of the SVG canvas is the top-left corner of the screen). If this is not needed one can just supply the identity matrix and the zero vector,
--
-- 3. map `v01'` onto the target frame `F2`. 
--
-- NB: we do not check that `v` is actually contained within the `F1`, nor that `v01'` is still contained within [0,1] x [0, 1]. This has to be supplied correctly by the user.
frameToFrame :: Fractional a =>
                      Frame a  -- ^ Initial frame
                   -> Frame a  -- ^ Final frame
                   -> Bool        -- ^ Flip L-R in [0,1] x [0,1]
                   -> Bool     -- ^ Flip U-D in [0,1] x [0,1]
                   -> V2 a     -- ^ Initial vector
                   -> V2 a
frameToFrame from to fliplr flipud v = toFrame to v01'
  where
    v01 = fromFrame from v
    v01' | fliplr && flipud = flipLR01 (flipUD01 v01)
         | fliplr = flipLR01 v01
         | flipud = flipUD01 v01
         | otherwise = v01


flipLR01, flipUD01 :: Num a => V2 a -> V2 a
flipLR01 (V2 a b) = V2 (1 - a) b
flipUD01 (V2 a b) = V2 a (1 - b)




moveLabeledPointV2Frames ::
  Fractional a =>
     Frame a          -- ^ Initial frame
  -> Frame a          -- ^ Final frame
  -> Bool             -- ^ Flip L-R in [0,1] x [0,1]
  -> Bool             -- ^ Flip U-D in [0,1] x [0,1]
  -> LabeledPoint l a -- ^ Initial `LabeledPoint`
  -> LabeledPoint l a
moveLabeledPointV2Frames from to fliplr flipud lp = LabeledPoint p' (_lplabel lp)
  where
    vlp = v2fromPoint $ _lp lp -- vector associated with starting point
    vlp' = frameToFrame from to fliplr flipud vlp -- vector associated w new point
    p' = pointFromV2 vlp'



-- -- * HasFrame : things which have a bounding box 
-- class HasFrame v where
--   type UnitInterval v :: *
--   type FrameType v :: *
--   fromFrame :: v -> UnitInterval v
--   toFrame :: UnitInterval v -> v



-- | X-aligned unit vector
e1 :: Num a => V2 a
e1 = V2 1 0
-- | Y-aligned unit vector
e2 :: Num a => V2 a
e2 = V2 0 1



-- | Numerical equality 
class Eps a where
  -- | Comparison within numerical precision
  (~=) :: a -> a -> Bool

instance Eps Double where
  a ~= b = abs (a - b) <= 1e-12

instance Eps Float where
  a ~= b = abs (a - b) <= 1e-6

instance Eps (V2 Double) where
  v1 ~= v2 = norm2 (v1 ^-^ v2) <= 1e-8
  
instance Eps (V2 Float) where
  v1 ~= v2 = norm2 (v1 ^-^ v2) <= 1e-3