{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies, FlexibleInstances #-}
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-- |
-- Module : Data.AffineSpace
-- Copyright : (c) Antony Courtney and Henrik Nilsson, Yale University, 2003
-- License : BSD-style (see the LICENSE file in the distribution)
--
-- Maintainer : ivan.perez@keera.co.uk
-- Stability : provisional
-- Portability : non-portable (GHC extensions)
--
-- Affine space type relation.
--
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module Data.AffineSpace where
import Data.VectorSpace
infix 6 .+^, .-^, .-.
-- Maybe origin should not be a class method, even though an origin
-- can be assocoated with any affine space.
-- Maybe distance should not be a class method, in which case the constraint
-- on the coefficient space (a) could be Fractional (i.e., a Field), which
-- seems closer to the mathematical definition of affine space, provided
-- the constraint on the coefficient space for VectorSpace is also Fractional.
-- | Affine Space type relation.
--
-- An affine space is a set (type) @p@, and an associated vector space @v@ over
-- a field @a@.
class (Floating a, VectorSpace v a) => AffineSpace p v a | p -> v, v -> a where
-- | Origin of the affine space.
origin :: p
-- | Addition of affine point and vector.
(.+^) :: p -> v -> p
-- | Subtraction of affine point and vector.
(.-^) :: p -> v -> p
p .-^ v = p .+^ (negateVector v)
-- | Subtraction of two points in the affine space, giving a vector.
(.-.) :: p -> p -> v
-- | Distance between two points in the affine space, same as the 'norm' of
-- the vector they form (see '(.-.)'.
distance :: p -> p -> a
distance p1 p2 = norm (p1 .-. p2)