    {-# LANGUAGE FlexibleContexts #-}
    {-# LANGUAGE TypeFamilies     #-}
    {-# OPTIONS_GHC -fno-warn-incomplete-patterns #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Diagrams.CubicSpline
    -- Copyright   :  (c) 2011 diagrams-lib team (see LICENSE)
    -- License     :  BSD-style (see LICENSE)
    -- Maintainer  :  diagrams-discuss@googlegroups.com
    --
    -- A /cubic spline/ is a smooth, connected sequence of cubic curves.
    -- This module provides two methods for constructing splines.
    --
    -- The 'cubicSpline' method can be used to create closed or open cubic
    -- splines from a list of points. The resulting splines /pass through/
    -- all the control points, but depend on the control points in a
    -- "global" way (that is, changing one control point may alter the
    -- entire curve).  For access to the internals of the spline
    -- generation algorithm, see "Diagrams.CubicSpline.Internal".
    --
    -- 'bspline' creates a cubic B-spline, which starts and ends at the
    -- first and last control points, but does not necessarily pass
    -- through any of the other control points.  It depends on the control
    -- points in a "local" way, that is, changing one control point will
    -- only affect a local portion of the curve near that control point.
    --
    -----------------------------------------------------------------------------
    module Diagrams.CubicSpline
           (
             -- * Constructing paths from cubic splines
             cubicSpline
           , BSpline
           , bspline
           ) where

    import           Control.Lens                  (view)

    import           Diagrams.Core
    import           Diagrams.CubicSpline.Boehm
    import           Diagrams.CubicSpline.Internal
    import           Diagrams.Located              (Located, at, mapLoc)
    import           Diagrams.Segment
    import           Diagrams.Trail
    import           Diagrams.TrailLike            (TrailLike (..))

    import           Linear.Affine
    import           Linear.Metric

    -- | Construct a spline path-like thing of cubic segments from a list of
    --   vertices, with the first vertex as the starting point.  The first
    --   argument specifies whether the path should be closed.
    --
    --   <<diagrams/src_Diagrams_CubicSpline_cubicSplineEx.svg#diagram=cubicSplineEx&width=600>>
    --
    --   > pts = map p2 [(0,0), (2,3), (5,-2), (-4,1), (0,3)]
    --   > spot = circle 0.2 # fc blue # lw none
    --   > mkPath closed = position (zip pts (repeat spot))
    --   >              <> cubicSpline closed pts
    --   > cubicSplineEx = (mkPath False ||| strutX 2 ||| mkPath True)
    --   >               # centerXY # pad 1.1
    --
    --   For more information, see <http://mathworld.wolfram.com/CubicSpline.html>.
    cubicSpline :: (V t ~ v, N t ~ n, TrailLike t, Fractional (v n)) => Bool -> [Point v n] -> t
    cubicSpline closed []  = trailLike . closeIf closed $ emptyLine `at` origin
    cubicSpline closed [p] = trailLike . closeIf closed $ emptyLine `at` p
    cubicSpline closed ps  = flattenBeziers . map f . solveCubicSplineCoefficients closed . map (view lensP) $ ps
      where
        f [a,b,c,d] = [a, (3*a+b)/3, (3*a+2*b+c)/3, a+b+c+d]
        flattenBeziers bs@((b:_):_)
          = trailLike . closeIf closed $ lineFromSegments (map bez bs) `at` P b
        bez [a,b,c,d] = bezier3 (b - a) (c - a) (d - a)

    closeIf :: (Metric v, OrderedField n)
            => Bool -> Located (Trail' Line v n) -> Located (Trail v n)
    closeIf c = mapLoc (if c then wrapLoop . glueLine else wrapLine)