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-- The magic of Lie Algebra
{-# LANGUAGE FlexibleContexts #-}
import Numeric.LinearAlgebra
rot1 :: Double -> Matrix Double
rot1 a = (3><3)
[ 1, 0, 0
, 0, c, s
, 0,-s, c ]
where c = cos a
s = sin a
g1,g2,g3 :: Matrix Double
g1 = (3><3) [0, 0,0
,0, 0,1
,0,-1,0]
rot2 :: Double -> Matrix Double
rot2 a = (3><3)
[ c, 0, s
, 0, 1, 0
,-s, 0, c ]
where c = cos a
s = sin a
g2 = (3><3) [ 0,0,1
, 0,0,0
,-1,0,0]
rot3 :: Double -> Matrix Double
rot3 a = (3><3)
[ c, s, 0
,-s, c, 0
, 0, 0, 1 ]
where c = cos a
s = sin a
g3 = (3><3) [ 0,1,0
,-1,0,0
, 0,0,0]
deg=pi/180
-- commutator
infix 8 &
a & b = a <> b - b <> a
infixl 6 |+|
a |+| b = a + b + a&b /2 + (a-b)&(a & b) /12
main = do
let a = 45*deg
b = 50*deg
c = -30*deg
exact = rot3 a <> rot1 b <> rot2 c
lie = scalar a * g3 |+| scalar b * g1 |+| scalar c * g2
putStrLn "position in the tangent space:"
disp 5 lie
putStrLn "exponential map back to the group (2 terms):"
disp 5 (expm lie)
putStrLn "exact position:"
disp 5 exact
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