----------------------------------------------------------------------------- -- | -- 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 -- passing through a given sequence of points. This module implements -- a straightforward spline generation algorithm based on solving -- tridiagonal systems of linear equations. -- ----------------------------------------------------------------------------- module Diagrams.CubicSpline.Internal ( -- * Solving for spline coefficents solveCubicSplineDerivatives , solveCubicSplineDerivativesClosed , solveCubicSplineCoefficients ) where import Diagrams.Solve.Tridiagonal import Data.List -- | Use the tri-diagonal solver with the appropriate parameters for an open cubic spline. solveCubicSplineDerivatives :: Fractional a => [a] -> [a] solveCubicSplineDerivatives (x:xs) = solveTriDiagonal as bs as ds where as = replicate (l - 1) 1 bs = 2 : replicate (l - 2) 4 ++ [2] l = length ds ds = zipWith f (xs ++ [last xs]) (x:x:xs) f a b = 3*(a - b) solveCubicSplineDerivatives _ = error "argument to solveCubicSplineDerivatives must be nonempty" -- | Use the cyclic-tri-diagonal solver with the appropriate parameters for a closed cubic spline. solveCubicSplineDerivativesClosed :: Fractional a => [a] -> [a] solveCubicSplineDerivativesClosed xs = solveCyclicTriDiagonal as bs as ds 1 1 where as = replicate (l - 1) 1 bs = replicate l 4 l = length xs xs' = cycle xs ds = take l \$ zipWith f (drop 1 xs') (drop (l - 1) xs') f a b = 3*(a - b) -- | Use the cyclic-tri-diagonal solver with the appropriate parameters for a closed cubic spline. solveCubicSplineCoefficients :: Fractional a => Bool -> [a] -> [[a]] solveCubicSplineCoefficients closed xs = [ [x,d,3*(x1-x)-2*d-d1,2*(x-x1)+d+d1] | (x,x1,d,d1) <- zip4 xs' (tail xs') ds' (tail ds') ] where ds | closed = solveCubicSplineDerivativesClosed xs | otherwise = solveCubicSplineDerivatives xs close as | closed = as ++ [head as] | otherwise = as xs' = close xs ds' = close ds