Space.hs

{-# LANGUAGE CPP #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# OPTIONS_GHC -Wall #-}
#if ( __GLASGOW_HASKELL__ < 820 )
{-# OPTIONS_GHC -fno-warn-incomplete-patterns #-}
{-# OPTIONS_GHC -fno-warn-unrecognised-pragmas #-}
#endif

-- | A 'Space' represents an abstract continuous range class for a type. The <https://hackage.haskell.org/package/intervals interval> package is an alternative approach.
module NumHask.Space
  ( Space(..)
  , Pos(..)
  ) where

import NumHask.Prelude hiding (singleton)

-- | space laws
--
-- > a `union` nul == a
-- > a `union` a == a
-- > project o n (lower o) == lower n
-- > project o n (upper o) == upper n
-- > project a a == id
class (Eq (Element s), Ord (Element s), Field (Element s)) => Space s where
    type Element s :: *
    -- | lower boundary of space
    lower :: s -> Element s
    -- | upper boundary of space
    upper :: s -> Element s
    -- | mid-point of the space
    mid :: s -> Element s
    mid s = (lower s + upper s)/(one+one)
    -- | distance between boundaries
    width :: s -> Element s
    width s = upper s - lower s
    -- | singleton space
    singleton :: Element s -> s
    -- | zero-width test
    singular :: s -> Bool
    singular s = lower s == upper s
    -- | determine whether an a is in the space
    element :: Element s -> s -> Bool
    element a s = a >= lower s && a <= upper s
    -- | is a space contained within another?
    contains :: s -> s -> Bool
    contains s0 s1 = lower s0 <= lower s1 && upper s0 >= upper s1
    -- | convex hull
    union :: s -> s -> s
    -- | null space, which can be interpreted as mempty
    nul :: s
    -- | the containing space of a Foldable
    space :: (Foldable f) => f (Element s) -> s
    space = foldr (\a x -> x `union` singleton a) nul
    -- | project a data point from an old range to a new range
    project :: s -> s -> Element s -> Element s
    project s0 s1 p =
        ((p-lower s0)/(upper s0-lower s0)) * (upper s1-lower s1) + lower s1
    type Grid s :: *
    -- | create equally-spaced `a`s from a space
    grid :: Pos -> s -> Grid s -> [Element s]
    -- | create equally-spaced `Space a`s from a space
    gridSpace :: s -> Grid s -> [s]

-- | Pos suggests where data points are placed on a grid across a range. Pos can also be thought about as whether the lower and upper points on the range are open or closed (plus the mid-point as an extra option).
data Pos = OuterPos | InnerPos | LowerPos | UpperPos | MidPos deriving (Eq)