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|
{-# LANGUAGE GADTSyntax #-}
{-# LANGUAGE ViewPatterns #-}
{-# OPTIONS_GHC -fno-warn-type-defaults #-}
--------------------------------------------------------------------------------------------
-- |
-- Copyright : (C) 2018 Nathan Waivio
-- License : BSD3
-- Maintainer : Nathan Waivio <nathan.waivio@gmail.com>
-- Stability : Stable
-- Portability : unportable
--
-- Library implementing standard functions for the <https://en.wikipedia.org/wiki/Algebra_of_physical_space Algebra of Physical Space> Cl(3,0)
--
---------------------------------------------------------------------------------------------
module T15304 where
data Cl3 where
R :: !Double -> Cl3 -- Real Scalar Sub-algebra (G0)
V3 :: !Double -> !Double -> !Double -> Cl3 -- Vectors (G1)
BV :: !Double -> !Double -> !Double -> Cl3 -- Bivectors (G2)
I :: !Double -> Cl3 -- Trivector Imaginary Pseudo-Scalar (G3)
PV :: !Double -> !Double -> !Double -> !Double -> Cl3 -- Paravector (G0 + G1)
H :: !Double -> !Double -> !Double -> !Double -> Cl3 -- Quaternion Even Sub-algebra (G0 + G2)
C :: !Double -> !Double -> Cl3 -- Complex Sub-algebra (G0 + G3)
BPV :: !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> Cl3 -- Biparavector (G1 + G2)
ODD :: !Double -> !Double -> !Double -> !Double -> Cl3 -- Odd (G1 + G3)
TPV :: !Double -> !Double -> !Double -> !Double -> Cl3 -- Triparavector (G2 + G3)
APS :: !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> Cl3 -- Algebra of Physical Space (G0 + G1 + G2 + G3)
instance Num Cl3 where
(R a0) + (R b0) = R (a0 + b0)
(R a0) + (V3 b1 b2 b3) = PV a0 b1 b2 b3
(R a0) + (BV b23 b31 b12) = H a0 b23 b31 b12
(R a0) + (I b123) = C a0 b123
(R a0) + (PV b0 b1 b2 b3) = PV (a0 + b0) b1 b2 b3
(R a0) + (H b0 b23 b31 b12) = H (a0 + b0) b23 b31 b12
(R a0) + (C b0 b123) = C (a0 + b0) b123
(R a0) + (BPV b1 b2 b3 b23 b31 b12) = APS a0 b1 b2 b3 b23 b31 b12 0
(R a0) + (ODD b1 b2 b3 b123) = APS a0 b1 b2 b3 0 0 0 b123
(R a0) + (TPV b23 b31 b12 b123) = APS a0 0 0 0 b23 b31 b12 b123
(R a0) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0 + b0) b1 b2 b3 b23 b31 b12 b123
(V3 a1 a2 a3) + (R b0) = PV b0 a1 a2 a3
(BV a23 a31 a12) + (R b0) = H b0 a23 a31 a12
(I a123) + (R b0) = C b0 a123
(PV a0 a1 a2 a3) + (R b0) = PV (a0 + b0) a1 a2 a3
(H a0 a23 a31 a12) + (R b0) = H (a0 + b0) a23 a31 a12
(C a0 a123) + (R b0) = C (a0 + b0) a123
(BPV a1 a2 a3 a23 a31 a12) + (R b0) = APS b0 a1 a2 a3 a23 a31 a12 0
(ODD a1 a2 a3 a123) + (R b0) = APS b0 a1 a2 a3 0 0 0 a123
(TPV a23 a31 a12 a123) + (R b0) = APS b0 0 0 0 a23 a31 a12 a123
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (R b0) = APS (a0 + b0) a1 a2 a3 a23 a31 a12 a123
(V3 a1 a2 a3) + (V3 b1 b2 b3) = V3 (a1 + b1) (a2 + b2) (a3 + b3)
(V3 a1 a2 a3) + (BV b23 b31 b12) = BPV a1 a2 a3 b23 b31 b12
(V3 a1 a2 a3) + (I b123) = ODD a1 a2 a3 b123
(V3 a1 a2 a3) + (PV b0 b1 b2 b3) = PV b0 (a1 + b1) (a2 + b2) (a3 + b3)
(V3 a1 a2 a3) + (H b0 b23 b31 b12) = APS b0 a1 a2 a3 b23 b31 b12 0
(V3 a1 a2 a3) + (C b0 b123) = APS b0 a1 a2 a3 0 0 0 b123
(V3 a1 a2 a3) + (BPV b1 b2 b3 b23 b31 b12) = BPV (a1 + b1) (a2 + b2) (a3 + b3) b23 b31 b12
(V3 a1 a2 a3) + (ODD b1 b2 b3 b123) = ODD (a1 + b1) (a2 + b2) (a3 + b3) b123
(V3 a1 a2 a3) + (TPV b23 b31 b12 b123) = APS 0 a1 a2 a3 b23 b31 b12 b123
(V3 a1 a2 a3) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS b0 (a1 + b1) (a2 + b2) (a3 + b3) b23 b31 b12 b123
(BV a23 a31 a12) + (V3 b1 b2 b3) = BPV b1 b2 b3 a23 a31 a12
(I a123) + (V3 b1 b2 b3) = ODD b1 b2 b3 a123
(PV a0 a1 a2 a3) + (V3 b1 b2 b3) = PV a0 (a1 + b1) (a2 + b2) (a3 + b3)
(H a0 a23 a31 a12) + (V3 b1 b2 b3) = APS a0 b1 b2 b3 a23 a31 a12 0
(C a0 a123) + (V3 b1 b2 b3) = APS a0 b1 b2 b3 0 0 0 a123
(BPV a1 a2 a3 a23 a31 a12) + (V3 b1 b2 b3) = BPV (a1 + b1) (a2 + b2) (a3 + b3) a23 a31 a12
(ODD a1 a2 a3 a123) + (V3 b1 b2 b3) = ODD (a1 + b1) (a2 + b2) (a3 + b3) a123
(TPV a23 a31 a12 a123) + (V3 b1 b2 b3) = APS 0 b1 b2 b3 a23 a31 a12 a123
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (V3 b1 b2 b3) = APS a0 (a1 + b1) (a2 + b2) (a3 + b3) a23 a31 a12 a123
(BV a23 a31 a12) + (BV b23 b31 b12) = BV (a23 + b23) (a31 + b31) (a12 + b12)
(BV a23 a31 a12) + (I b123) = TPV a23 a31 a12 b123
(BV a23 a31 a12) + (PV b0 b1 b2 b3) = APS b0 b1 b2 b3 a23 a31 a12 0
(BV a23 a31 a12) + (H b0 b23 b31 b12) = H b0 (a23 + b23) (a31 + b31) (a12 + b12)
(BV a23 a31 a12) + (C b0 b123) = APS b0 0 0 0 a23 a31 a12 b123
(BV a23 a31 a12) + (BPV b1 b2 b3 b23 b31 b12) = BPV b1 b2 b3 (a23 + b23) (a31 + b31) (a12 + b12)
(BV a23 a31 a12) + (ODD b1 b2 b3 b123) = APS 0 b1 b2 b3 a23 a31 a12 b123
(BV a23 a31 a12) + (TPV b23 b31 b12 b123) = TPV (a23 + b23) (a31 + b31) (a12 + b12) b123
(BV a23 a31 a12) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS b0 b1 b2 b3 (a23 + b23) (a31 + b31) (a12 + b12) b123
(I a123) + (BV b23 b31 b12) = TPV b23 b31 b12 a123
(PV a0 a1 a2 a3) + (BV b23 b31 b12) = APS a0 a1 a2 a3 b23 b31 b12 0
(H a0 a23 a31 a12) + (BV b23 b31 b12) = H a0 (a23 + b23) (a31 + b31) (a12 + b12)
(C a0 a123) + (BV b23 b31 b12) = APS a0 0 0 0 b23 b31 b12 a123
(BPV a1 a2 a3 a23 a31 a12) + (BV b23 b31 b12) = BPV a1 a2 a3 (a23 + b23) (a31 + b31) (a12 + b12)
(ODD a1 a2 a3 a123) + (BV b23 b31 b12) = APS 0 a1 a2 a3 b23 b31 b12 a123
(TPV a23 a31 a12 a123) + (BV b23 b31 b12) = TPV (a23 + b23) (a31 + b31) (a12 + b12) a123
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (BV b23 b31 b12) = APS a0 a1 a2 a3 (a23 + b23) (a31 + b31) (a12 + b12) a123
(I a123) + (I b123) = I (a123 + b123)
(I a123) + (PV b0 b1 b2 b3) = APS b0 b1 b2 b3 0 0 0 a123
(I a123) + (H b0 b23 b31 b12) = APS b0 0 0 0 b23 b31 b12 a123
(I a123) + (C b0 b123) = C b0 (a123 + b123)
(I a123) + (BPV b1 b2 b3 b23 b31 b12) = APS 0 b1 b2 b3 b23 b31 b12 a123
(I a123) + (ODD b1 b2 b3 b123) = ODD b1 b2 b3 (a123 + b123)
(I a123) + (TPV b23 b31 b12 b123) = TPV b23 b31 b12 (a123 + b123)
(I a123) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS b0 b1 b2 b3 b23 b31 b12 (a123 + b123)
(PV a0 a1 a2 a3) + (I b123) = APS a0 a1 a2 a3 0 0 0 b123
(H a0 a23 a31 a12) + (I b123) = APS a0 0 0 0 a23 a31 a12 b123
(C a0 a123) + (I b123) = C a0 (a123 + b123)
(BPV a1 a2 a3 a23 a31 a12) + (I b123) = APS 0 a1 a2 a3 a23 a31 a12 b123
(ODD a1 a2 a3 a123) + (I b123) = ODD a1 a2 a3 (a123 + b123)
(TPV a23 a31 a12 a123) + (I b123) = TPV a23 a31 a12 (a123 + b123)
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (I b123) = APS a0 a1 a2 a3 a23 a31 a12 (a123 + b123)
(PV a0 a1 a2 a3) + (PV b0 b1 b2 b3) = PV (a0 + b0) (a1 + b1) (a2 + b2) (a3 + b3)
(PV a0 a1 a2 a3) + (H b0 b23 b31 b12) = APS (a0 + b0) a1 a2 a3 b23 b31 b12 0
(PV a0 a1 a2 a3) + (C b0 b123) = APS (a0 + b0) a1 a2 a3 0 0 0 b123
(PV a0 a1 a2 a3) + (BPV b1 b2 b3 b23 b31 b12) = APS a0 (a1 + b1) (a2 + b2) (a3 + b3) b23 b31 b12 0
(PV a0 a1 a2 a3) + (ODD b1 b2 b3 b123) = APS a0 (a1 + b1) (a2 + b2) (a3 + b3) 0 0 0 b123
(PV a0 a1 a2 a3) + (TPV b23 b31 b12 b123) = APS a0 a1 a2 a3 b23 b31 b12 b123
(PV a0 a1 a2 a3) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0 + b0) (a1 + b1) (a2 + b2) (a3 + b3) b23 b31 b12 b123
(H a0 a23 a31 a12) + (PV b0 b1 b2 b3) = APS (a0 + b0) b1 b2 b3 a23 a31 a12 0
(C a0 a123) + (PV b0 b1 b2 b3) = APS (a0 + b0) b1 b2 b3 0 0 0 a123
(BPV a1 a2 a3 a23 a31 a12) + (PV b0 b1 b2 b3) = APS b0 (a1 + b1) (a2 + b2) (a3 + b3) a23 a31 a12 0
(ODD a1 a2 a3 a123) + (PV b0 b1 b2 b3) = APS b0 (a1 + b1) (a2 + b2) (a3 + b3) 0 0 0 a123
(TPV a23 a31 a12 a123) + (PV b0 b1 b2 b3) = APS b0 b1 b2 b3 a23 a31 a12 a123
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (PV b0 b1 b2 b3) = APS (a0 + b0) (a1 + b1) (a2 + b2) (a3 + b3) a23 a31 a12 a123
(H a0 a23 a31 a12) + (H b0 b23 b31 b12) = H (a0 + b0) (a23 + b23) (a31 + b31) (a12 + b12)
(H a0 a23 a31 a12) + (C b0 b123) = APS (a0 + b0) 0 0 0 a23 a31 a12 b123
(H a0 a23 a31 a12) + (BPV b1 b2 b3 b23 b31 b12) = APS a0 b1 b2 b3 (a23 + b23) (a31 + b31) (a12 + b12) 0
(H a0 a23 a31 a12) + (ODD b1 b2 b3 b123) = APS a0 b1 b2 b3 a23 a31 a12 b123
(H a0 a23 a31 a12) + (TPV b23 b31 b12 b123) = APS a0 0 0 0 (a23 + b23) (a31 + b31) (a12 + b12) b123
(H a0 a23 a31 a12) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0 + b0) b1 b2 b3 (a23 + b23) (a31 + b31) (a12 + b12) b123
(C a0 a123) + (H b0 b23 b31 b12) = APS (a0 + b0) 0 0 0 b23 b31 b12 a123
(BPV a1 a2 a3 a23 a31 a12) + (H b0 b23 b31 b12) = APS b0 a1 a2 a3 (a23 + b23) (a31 + b31) (a12 + b12) 0
(ODD a1 a2 a3 a123) + (H b0 b23 b31 b12) = APS b0 a1 a2 a3 b23 b31 b12 a123
(TPV a23 a31 a12 a123) + (H b0 b23 b31 b12) = APS b0 0 0 0 (a23 + b23) (a31 + b31) (a12 + b12) a123
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (H b0 b23 b31 b12) = APS (a0 + b0) a1 a2 a3 (a23 + b23) (a31 + b31) (a12 + b12) a123
(C a0 a123) + (C b0 b123) = C (a0 + b0) (a123 + b123)
(C a0 a123) + (BPV b1 b2 b3 b23 b31 b12) = APS a0 b1 b2 b3 b23 b31 b12 a123
(C a0 a123) + (ODD b1 b2 b3 b123) = APS a0 b1 b2 b3 0 0 0 (a123 + b123)
(C a0 a123) + (TPV b23 b31 b12 b123) = APS a0 0 0 0 b23 b31 b12 (a123 + b123)
(C a0 a123) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0 + b0) b1 b2 b3 b23 b31 b12 (a123 + b123)
(BPV a1 a2 a3 a23 a31 a12) + (C b0 b123) = APS b0 a1 a2 a3 a23 a31 a12 b123
(ODD a1 a2 a3 a123) + (C b0 b123) = APS b0 a1 a2 a3 0 0 0 (a123 + b123)
(TPV a23 a31 a12 a123) + (C b0 b123) = APS b0 0 0 0 a23 a31 a12 (a123 + b123)
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (C b0 b123) = APS (a0 + b0) a1 a2 a3 a23 a31 a12 (a123 + b123)
(BPV a1 a2 a3 a23 a31 a12) + (BPV b1 b2 b3 b23 b31 b12) = BPV (a1 + b1) (a2 + b2) (a3 + b3) (a23 + b23) (a31 + b31) (a12 + b12)
(BPV a1 a2 a3 a23 a31 a12) + (ODD b1 b2 b3 b123) = APS 0 (a1 + b1) (a2 + b2) (a3 + b3) a23 a31 a12 b123
(BPV a1 a2 a3 a23 a31 a12) + (TPV b23 b31 b12 b123) = APS 0 a1 a2 a3 (a23 + b23) (a31 + b31) (a12 + b12) b123
(BPV a1 a2 a3 a23 a31 a12) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS b0 (a1 + b1) (a2 + b2) (a3 + b3) (a23 + b23) (a31 + b31) (a12 + b12) b123
(ODD a1 a2 a3 a123) + (BPV b1 b2 b3 b23 b31 b12) = APS 0 (a1 + b1) (a2 + b2) (a3 + b3) b23 b31 b12 a123
(TPV a23 a31 a12 a123) + (BPV b1 b2 b3 b23 b31 b12) = APS 0 b1 b2 b3 (a23 + b23) (a31 + b31) (a12 + b12) a123
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (BPV b1 b2 b3 b23 b31 b12) = APS a0 (a1 + b1) (a2 + b2) (a3 + b3) (a23 + b23) (a31 + b31) (a12 + b12) a123
(ODD a1 a2 a3 a123) + (ODD b1 b2 b3 b123) = ODD (a1 + b1) (a2 + b2) (a3 + b3) (a123 + b123)
(ODD a1 a2 a3 a123) + (TPV b23 b31 b12 b123) = APS 0 a1 a2 a3 b23 b31 b12 (a123 + b123)
(ODD a1 a2 a3 a123) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS b0 (a1 + b1) (a2 + b2) (a3 + b3) b23 b31 b12 (a123 + b123)
(TPV a23 a31 a12 a123) + (ODD b1 b2 b3 b123) = APS 0 b1 b2 b3 a23 a31 a12 (a123 + b123)
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (ODD b1 b2 b3 b123) = APS a0 (a1 + b1) (a2 + b2) (a3 + b3) a23 a31 a12 (a123 + b123)
(TPV a23 a31 a12 a123) + (TPV b23 b31 b12 b123) = TPV (a23 + b23) (a31 + b31) (a12 + b12) (a123 + b123)
(TPV a23 a31 a12 a123) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS b0 b1 b2 b3 (a23 + b23) (a31 + b31) (a12 + b12) (a123 + b123)
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (TPV b23 b31 b12 b123) = APS a0 a1 a2 a3 (a23 + b23) (a31 + b31) (a12 + b12) (a123 + b123)
(APS a0 a1 a2 a3 a23 a31 a12 a123) + (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0 + b0)
(a1 + b1) (a2 + b2) (a3 + b3)
(a23 + b23) (a31 + b31) (a12 + b12)
(a123 + b123)
-- | Multiplication Instance implementing a Geometric Product
(R a0) * (R b0) = R (a0*b0)
(R a0) * (V3 b1 b2 b3) = V3 (a0*b1) (a0*b2) (a0*b3)
(R a0) * (BV b23 b31 b12) = BV (a0*b23) (a0*b31) (a0*b12)
(R a0) * (I b123) = I (a0*b123)
(R a0) * (PV b0 b1 b2 b3) = PV (a0*b0)
(a0*b1) (a0*b2) (a0*b3)
(R a0) * (H b0 b23 b31 b12) = H (a0*b0)
(a0*b23) (a0*b31) (a0*b12)
(R a0) * (C b0 b123) = C (a0*b0)
(a0*b123)
(R a0) * (BPV b1 b2 b3 b23 b31 b12) = BPV (a0*b1) (a0*b2) (a0*b3)
(a0*b23) (a0*b31) (a0*b12)
(R a0) * (ODD b1 b2 b3 b123) = ODD (a0*b1) (a0*b2) (a0*b3)
(a0*b123)
(R a0) * (TPV b23 b31 b12 b123) = TPV (a0*b23) (a0*b31) (a0*b12)
(a0*b123)
(R a0) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0*b0)
(a0*b1) (a0*b2) (a0*b3)
(a0*b23) (a0*b31) (a0*b12)
(a0*b123)
(V3 a1 a2 a3) * (R b0) = V3 (a1*b0) (a2*b0) (a3*b0)
(BV a23 a31 a12) * (R b0) = BV (a23*b0) (a31*b0) (a12*b0)
(I a123) * (R b0) = I (a123*b0)
(PV a0 a1 a2 a3) * (R b0) = PV (a0*b0)
(a1*b0) (a2*b0) (a3*b0)
(H a0 a23 a31 a12) * (R b0) = H (a0*b0)
(a23*b0) (a31*b0) (a12*b0)
(C a0 a123) * (R b0) = C (a0*b0)
(a123*b0)
(BPV a1 a2 a3 a23 a31 a12) * (R b0) = BPV (a1*b0) (a2*b0) (a3*b0)
(a23*b0) (a31*b0) (a12*b0)
(ODD a1 a2 a3 a123) * (R b0) = ODD (a1*b0) (a2*b0) (a3*b0)
(a123*b0)
(TPV a23 a31 a12 a123) * (R b0) = TPV (a23*b0) (a31*b0) (a12*b0)
(a123*b0)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (R b0) = APS (a0*b0)
(a1*b0) (a2*b0) (a3*b0)
(a23*b0) (a31*b0) (a12*b0)
(a123*b0)
(V3 a1 a2 a3) * (V3 b1 b2 b3) = H (a1*b1 + a2*b2 + a3*b3)
(a2*b3 - a3*b2) (a3*b1 - a1*b3) (a1*b2 - a2*b1)
(V3 a1 a2 a3) * (BV b23 b31 b12) = ODD (a3*b31 - a2*b12) (a1*b12 - a3*b23) (a2*b23 - a1*b31)
(a1*b23 + a2*b31 + a3*b12)
(V3 a1 a2 a3) * (I b123) = BV (a1*b123) (a2*b123) (a3*b123)
(V3 a1 a2 a3) * (PV b0 b1 b2 b3) = APS (a1*b1 + a2*b2 + a3*b3)
(a1*b0) (a2*b0) (a3*b0)
(a2*b3 - a3*b2) (a3*b1 - a1*b3) (a1*b2 - a2*b1)
0
(V3 a1 a2 a3) * (H b0 b23 b31 b12) = ODD (a1*b0 - a2*b12 + a3*b31) (a2*b0 + a1*b12 - a3*b23) (a3*b0 - a1*b31 + a2*b23)
(a1*b23 + a2*b31 + a3*b12)
(V3 a1 a2 a3) * (C b0 b123) = BPV (a1*b0) (a2*b0) (a3*b0)
(a1*b123) (a2*b123) (a3*b123)
(V3 a1 a2 a3) * (BPV b1 b2 b3 b23 b31 b12) = APS (a1*b1 + a2*b2 + a3*b3)
(a3*b31 - a2*b12) (a1*b12 - a3*b23) (a2*b23 - a1*b31)
(a2*b3 - a3*b2) (a3*b1 - a1*b3) (a1*b2 - a2*b1)
(a1*b23 + a2*b31 + a3*b12)
(V3 a1 a2 a3) * (ODD b1 b2 b3 b123) = H (a1*b1 + a2*b2 + a3*b3)
(a1*b123 + a2*b3 - a3*b2) (a2*b123 - a1*b3 + a3*b1) (a3*b123 + a1*b2 - a2*b1)
(V3 a1 a2 a3) * (TPV b23 b31 b12 b123) = APS 0
(a3*b31 - a2*b12) (a1*b12 - a3*b23) (a2*b23 - a1*b31)
(a1*b123) (a2*b123) (a3*b123)
(a1*b23 + a2*b31 + a3*b12)
(V3 a1 a2 a3) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a1*b1 + a2*b2 + a3*b3)
(a1*b0 - a2*b12 + a3*b31) (a2*b0 + a1*b12 - a3*b23) (a3*b0 - a1*b31 + a2*b23)
(a1*b123 + a2*b3 - a3*b2) (a3*b1 - a1*b3 + a2*b123) (a1*b2 - a2*b1 + a3*b123)
(a1*b23 + a2*b31 + a3*b12)
(BV a23 a31 a12) * (V3 b1 b2 b3) = ODD (a12*b2 - a31*b3) (a23*b3 - a12*b1) (a31*b1 - a23*b2)
(a23*b1 + a31*b2 + a12*b3)
(I a123) * (V3 b1 b2 b3) = BV (a123*b1) (a123*b2) (a123*b3)
(PV a0 a1 a2 a3) * (V3 b1 b2 b3) = APS (a1*b1 + a2*b2 + a3*b3)
(a0*b1) (a0*b2) (a0*b3)
(a2*b3 - a3*b2) (a3*b1 - a1*b3) (a1*b2 - a2*b1)
0
(H a0 a23 a31 a12) * (V3 b1 b2 b3) = ODD (a0*b1 + a12*b2 - a31*b3) (a0*b2 - a12*b1 + a23*b3) (a0*b3 + a31*b1 - a23*b2)
(a23*b1 + a31*b2 + a12*b3)
(C a0 a123) * (V3 b1 b2 b3) = BPV (a0*b1) (a0*b2) (a0*b3)
(a123*b1) (a123*b2) (a123*b3)
(BPV a1 a2 a3 a23 a31 a12) * (V3 b1 b2 b3) = APS (a1*b1 + a2*b2 + a3*b3)
(a12*b2 - a31*b3) (a23*b3 - a12*b1) (a31*b1 - a23*b2)
(a2*b3 - a3*b2) (a3*b1 - a1*b3) (a1*b2 - a2*b1)
(a23*b1 + a31*b2 + a12*b3)
(ODD a1 a2 a3 a123) * (V3 b1 b2 b3) = H (a1*b1 + a2*b2 + a3*b3)
(a123*b1 + a2*b3 - a3*b2) (a123*b2 - a1*b3 + a3*b1) (a123*b3 + a1*b2 - a2*b1)
(TPV a23 a31 a12 a123) * (V3 b1 b2 b3) = APS 0
(a12*b2 - a31*b3) (a23*b3 - a12*b1) (a31*b1 - a23*b2)
(a123*b1) (a123*b2) (a123*b3)
(a23*b1 + a31*b2 + a12*b3)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (V3 b1 b2 b3) = APS (a1*b1 + a2*b2 + a3*b3)
(a0*b1 + a12*b2 - a31*b3) (a0*b2 - a12*b1 + a23*b3) (a0*b3 + a31*b1 - a23*b2)
(a123*b1 + a2*b3 - a3*b2) (a3*b1 - a1*b3 + a123*b2) (a1*b2 - a2*b1 + a123*b3)
(a23*b1 + a31*b2 + a12*b3)
(BV a23 a31 a12) * (BV b23 b31 b12) = H (negate $ a23*b23 + a31*b31 + a12*b12)
(a12*b31 - a31*b12) (a23*b12 - a12*b23) (a31*b23 - a23*b31)
(BV a23 a31 a12) * (I b123) = V3 (negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(BV a23 a31 a12) * (PV b0 b1 b2 b3) = APS 0
(a12*b2 - a31*b3) (a23*b3 - a12*b1) (a31*b1 - a23*b2)
(a23*b0) (a31*b0) (a12*b0)
(a23*b1 + a31*b2 + a12*b3)
(BV a23 a31 a12) * (H b0 b23 b31 b12) = H (negate $ a23*b23 + a31*b31 + a12*b12)
(a23*b0 - a31*b12 + a12*b31) (a31*b0 + a23*b12 - a12*b23) (a12*b0 - a23*b31 + a31*b23)
(BV a23 a31 a12) * (C b0 b123) = BPV (negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a23*b0) (a31*b0) (a12*b0)
(BV a23 a31 a12) * (BPV b1 b2 b3 b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a12*b2 - a31*b3) (a23*b3 - a12*b1) (a31*b1 - a23*b2)
(a12*b31 - a31*b12) (a23*b12 - a12*b23) (a31*b23 - a23*b31)
(a23*b1 + a31*b2 + a12*b3)
(BV a23 a31 a12) * (ODD b1 b2 b3 b123) = ODD (a12*b2 - a31*b3 - a23*b123) (a23*b3 - a12*b1 - a31*b123) (a31*b1 - a23*b2 - a12*b123)
(a23*b1 + a31*b2 + a12*b3)
(BV a23 a31 a12) * (TPV b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a12*b31 - a31*b12) (a23*b12 - a12*b23) (a31*b23 - a23*b31)
0
(BV a23 a31 a12) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a12*b2 - a31*b3 - a23*b123) (a23*b3 - a31*b123 - a12*b1) (a31*b1 - a23*b2 - a12*b123)
(a23*b0 - a31*b12 + a12*b31) (a31*b0 + a23*b12 - a12*b23) (a12*b0 - a23*b31 + a31*b23)
(a23*b1 + a31*b2 + a12*b3)
(I a123) * (BV b23 b31 b12) = V3 (negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(PV a0 a1 a2 a3) * (BV b23 b31 b12) = APS 0
(a3*b31 - a2*b12) (a1*b12 - a3*b23) (a2*b23 - a1*b31)
(a0*b23) (a0*b31) (a0*b12)
(a1*b23 + a2*b31 + a3*b12)
(H a0 a23 a31 a12) * (BV b23 b31 b12) = H (negate $ a23*b23 + a31*b31 + a12*b12)
(a0*b23 - a31*b12 + a12*b31) (a0*b31 + a23*b12 - a12*b23) (a0*b12 - a23*b31 + a31*b23)
(C a0 a123) * (BV b23 b31 b12) = BPV (negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a0*b23) (a0*b31) (a0*b12)
(BPV a1 a2 a3 a23 a31 a12) * (BV b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a3*b31 - a2*b12) (a1*b12 - a3*b23) (a2*b23 - a1*b31)
(a12*b31 - a31*b12) (a23*b12 - a12*b23) (a31*b23 - a23*b31)
(a1*b23 + a2*b31 + a3*b12)
(ODD a1 a2 a3 a123) * (BV b23 b31 b12) = ODD (negate $ a123*b23 + a2*b12 - a3*b31)
(negate $ a123*b31 - a1*b12 + a3*b23)
(negate $ a123*b12 + a1*b31 - a2*b23)
(a1*b23 + a2*b31 + a3*b12)
(TPV a23 a31 a12 a123) * (BV b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(negate $ a31*b12 - a12*b31) (negate $ a12*b23 - a23*b12) (negate $ a23*b31 - a31*b23)
0
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (BV b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a3*b31 - a123*b23 - a2*b12) (a1*b12 - a3*b23 - a123*b31) (a2*b23 - a123*b12 - a1*b31)
(a0*b23 - a31*b12 + a12*b31) (a0*b31 + a23*b12 - a12*b23) (a0*b12 - a23*b31 + a31*b23)
(a1*b23 + a2*b31 + a3*b12)
(I a123) * (I b123) = R (negate $ a123*b123)
(I a123) * (PV b0 b1 b2 b3) = TPV (a123*b1) (a123*b2) (a123*b3)
(a123*b0)
(I a123) * (H b0 b23 b31 b12) = ODD (negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a123*b0)
(I a123) * (C b0 b123) = C (negate $ a123*b123)
(a123*b0)
(I a123) * (BPV b1 b2 b3 b23 b31 b12) = BPV (negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a123*b1) (a123*b2) (a123*b3)
(I a123) * (ODD b1 b2 b3 b123) = H (negate $ a123*b123)
(a123*b1) (a123*b2) (a123*b3)
(I a123) * (TPV b23 b31 b12 b123) = PV (negate $ a123*b123)
(negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(I a123) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (negate $ a123*b123)
(negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a123*b1) (a123*b2) (a123*b3)
(a123*b0)
(PV a0 a1 a2 a3) * (I b123) = TPV (a1*b123) (a2*b123) (a3*b123)
(a0*b123)
(H a0 a23 a31 a12) * (I b123) = ODD (negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a0*b123)
(C a0 a123) * (I b123) = C (negate $ a123*b123)
(a0*b123)
(BPV a1 a2 a3 a23 a31 a12) * (I b123) = BPV (negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a1*b123) (a2*b123) (a3*b123)
(ODD a1 a2 a3 a123) * (I b123) = H (negate $ a123*b123)
(a1*b123) (a2*b123) (a3*b123)
(TPV a23 a31 a12 a123) * (I b123) = PV (negate $ a123*b123)
(negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (I b123) = APS (negate $ a123*b123)
(negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a1*b123) (a2*b123) (a3*b123)
(a0*b123)
(PV a0 a1 a2 a3) * (PV b0 b1 b2 b3) = APS (a0*b0 + a1*b1 + a2*b2 + a3*b3)
(a0*b1 + a1*b0) (a0*b2 + a2*b0) (a0*b3 + a3*b0)
(a2*b3 - a3*b2) (a3*b1 - a1*b3) (a1*b2 - a2*b1)
0
(PV a0 a1 a2 a3) * (H b0 b23 b31 b12) = APS (a0*b0)
(a1*b0 - a2*b12 + a3*b31) (a2*b0 + a1*b12 - a3*b23) (a3*b0 - a1*b31 + a2*b23)
(a0*b23) (a0*b31) (a0*b12)
(a1*b23 + a2*b31 + a3*b12)
(PV a0 a1 a2 a3) * (C b0 b123) = APS (a0*b0)
(a1*b0) (a2*b0) (a3*b0)
(a1*b123) (a2*b123) (a3*b123)
(a0*b123)
(PV a0 a1 a2 a3) * (BPV b1 b2 b3 b23 b31 b12) = APS (a1*b1 + a2*b2 + a3*b3)
(a0*b1 - a2*b12 + a3*b31) (a0*b2 + a1*b12 - a3*b23) (a0*b3 - a1*b31 + a2*b23)
(a0*b23 + a2*b3 - a3*b2) (a0*b31 - a1*b3 + a3*b1) (a0*b12 + a1*b2 - a2*b1)
(a1*b23 + a2*b31 + a3*b12)
(PV a0 a1 a2 a3) * (ODD b1 b2 b3 b123) = APS (a1*b1 + a2*b2 + a3*b3)
(a0*b1) (a0*b2) (a0*b3)
(a1*b123 + a2*b3 - a3*b2) (a2*b123 - a1*b3 + a3*b1) (a3*b123 + a1*b2 - a2*b1)
(a0*b123)
(PV a0 a1 a2 a3) * (TPV b23 b31 b12 b123) = APS 0
(a3*b31 - a2*b12) (a1*b12 - a3*b23) (a2*b23 - a1*b31)
(a0*b23 + a1*b123) (a0*b31 + a2*b123) (a0*b12 + a3*b123)
(a0*b123 + a1*b23 + a2*b31 + a3*b12)
(PV a0 a1 a2 a3) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0*b0 + a1*b1 + a2*b2 + a3*b3)
(a0*b1 + a1*b0 - a2*b12 + a3*b31)
(a0*b2 + a2*b0 + a1*b12 - a3*b23)
(a0*b3 + a3*b0 - a1*b31 + a2*b23)
(a0*b23 + a1*b123 + a2*b3 - a3*b2)
(a0*b31 - a1*b3 + a3*b1 + a2*b123)
(a0*b12 + a1*b2 - a2*b1 + a3*b123)
(a0*b123 + a1*b23 + a2*b31 + a3*b12)
(H a0 a23 a31 a12) * (PV b0 b1 b2 b3) = APS (a0*b0)
(a0*b1 + a12*b2 - a31*b3) (a0*b2 - a12*b1 + a23*b3) (a0*b3 + a31*b1 - a23*b2)
(a23*b0) (a31*b0) (a12*b0)
(a23*b1 + a31*b2 + a12*b3)
(C a0 a123) * (PV b0 b1 b2 b3) = APS (a0*b0)
(a0*b1) (a0*b2) (a0*b3)
(a123*b1) (a123*b2) (a123*b3)
(a123*b0)
(BPV a1 a2 a3 a23 a31 a12) * (PV b0 b1 b2 b3) = APS (a1*b1 + a2*b2 + a3*b3)
(a1*b0 + a12*b2 - a31*b3) (a2*b0 - a12*b1 + a23*b3) (a3*b0 + a31*b1 - a23*b2)
(a23*b0 + a2*b3 - a3*b2) (a31*b0 - a1*b3 + a3*b1) (a12*b0 + a1*b2 - a2*b1)
(a23*b1 + a31*b2 + a12*b3)
(ODD a1 a2 a3 a123) * (PV b0 b1 b2 b3) = APS (a1*b1 + a2*b2 + a3*b3)
(a1*b0) (a2*b0) (a3*b0)
(a123*b1 + a2*b3 - a3*b2)
(a123*b2 - a1*b3 + a3*b1)
(a123*b3 + a1*b2 - a2*b1)
(a123*b0)
(TPV a23 a31 a12 a123) * (PV b0 b1 b2 b3) = APS 0
(a12*b2 - a31*b3) (a23*b3 - a12*b1) (a31*b1 - a23*b2)
(a23*b0 + a123*b1) (a31*b0 + a123*b2) (a12*b0 + a123*b3)
(a123*b0 + a23*b1 + a31*b2 + a12*b3)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (PV b0 b1 b2 b3) = APS (a0*b0 + a1*b1 + a2*b2 + a3*b3)
(a0*b1 + a1*b0 + a12*b2 - a31*b3)
(a0*b2 + a2*b0 - a12*b1 + a23*b3)
(a0*b3 + a3*b0 + a31*b1 - a23*b2)
(a23*b0 + a123*b1 + a2*b3 - a3*b2)
(a31*b0 - a1*b3 + a3*b1 + a123*b2)
(a12*b0 + a1*b2 - a2*b1 + a123*b3)
(a123*b0 + a23*b1 + a31*b2 + a12*b3)
(H a0 a23 a31 a12) * (H b0 b23 b31 b12) = H (a0*b0 - a23*b23 - a31*b31 - a12*b12)
(a0*b23 + a23*b0 - a31*b12 + a12*b31)
(a0*b31 + a31*b0 + a23*b12 - a12*b23)
(a0*b12 + a12*b0 - a23*b31 + a31*b23)
(H a0 a23 a31 a12) * (C b0 b123) = APS (a0*b0)
(negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a23*b0) (a31*b0) (a12*b0)
(a0*b123)
(H a0 a23 a31 a12) * (BPV b1 b2 b3 b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a0*b1 + a12*b2 - a31*b3) (a0*b2 - a12*b1 + a23*b3) (a0*b3 + a31*b1 - a23*b2)
(a0*b23 - a31*b12 + a12*b31) (a0*b31 + a23*b12 - a12*b23) (a0*b12 - a23*b31 + a31*b23)
(a23*b1 + a31*b2 + a12*b3)
(H a0 a23 a31 a12) * (ODD b1 b2 b3 b123) = ODD (a0*b1 + a12*b2 - a31*b3 - a23*b123)
(a0*b2 - a12*b1 + a23*b3 - a31*b123)
(a0*b3 + a31*b1 - a23*b2 - a12*b123)
(a0*b123 + a23*b1 + a31*b2 + a12*b3)
(H a0 a23 a31 a12) * (TPV b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a0*b23 - a31*b12 + a12*b31) (a0*b31 + a23*b12 - a12*b23) (a0*b12 - a23*b31 + a31*b23)
(a0*b123)
(H a0 a23 a31 a12) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0*b0 - a23*b23 - a31*b31 - a12*b12)
(a0*b1 + a12*b2 - a31*b3 - a23*b123)
(a0*b2 - a12*b1 + a23*b3 - a31*b123)
(a0*b3 + a31*b1 - a23*b2 - a12*b123)
(a0*b23 + a23*b0 - a31*b12 + a12*b31)
(a0*b31 + a31*b0 + a23*b12 - a12*b23)
(a0*b12 + a12*b0 - a23*b31 + a31*b23)
(a0*b123 + a23*b1 + a31*b2 + a12*b3)
(C a0 a123) * (H b0 b23 b31 b12) = APS (a0*b0)
(negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a0*b23) (a0*b31) (a0*b12)
(a123*b0)
(BPV a1 a2 a3 a23 a31 a12) * (H b0 b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a1*b0 - a2*b12 + a3*b31) (a2*b0 + a1*b12 - a3*b23) (a3*b0 - a1*b31 + a2*b23)
(a23*b0 - a31*b12 + a12*b31) (a31*b0 + a23*b12 - a12*b23) (a12*b0 - a23*b31 + a31*b23)
(a1*b23 + a2*b31 + a3*b12)
(ODD a1 a2 a3 a123) * (H b0 b23 b31 b12) = ODD (a1*b0 - a2*b12 + a3*b31 - a123*b23)
(a2*b0 + a1*b12 - a3*b23 - a123*b31)
(a3*b0 - a1*b31 + a2*b23 - a123*b12)
(a123*b0 + a1*b23 + a2*b31 + a3*b12)
(TPV a23 a31 a12 a123) * (H b0 b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a23*b0 - a31*b12 + a12*b31) (a31*b0 + a23*b12 - a12*b23) (a12*b0 - a23*b31 + a31*b23)
(a123*b0)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (H b0 b23 b31 b12) = APS (a0*b0 - a23*b23 - a31*b31 - a12*b12)
(a1*b0 - a2*b12 + a3*b31 - a123*b23)
(a2*b0 + a1*b12 - a3*b23 - a123*b31)
(a3*b0 - a1*b31 + a2*b23 - a123*b12)
(a0*b23 + a23*b0 - a31*b12 + a12*b31)
(a0*b31 + a31*b0 + a23*b12 - a12*b23)
(a0*b12 + a12*b0 - a23*b31 + a31*b23)
(a123*b0 + a1*b23 + a2*b31 + a3*b12)
(C a0 a123) * (C b0 b123) = C (a0*b0 - a123*b123)
(a0*b123 + a123*b0)
(C a0 a123) * (BPV b1 b2 b3 b23 b31 b12) = BPV (a0*b1 - a123*b23) (a0*b2 - a123*b31) (a0*b3 - a123*b12)
(a0*b23 + a123*b1) (a0*b31 + a123*b2) (a0*b12 + a123*b3)
(C a0 a123) * (ODD b1 b2 b3 b123) = APS (negate $ a123*b123)
(a0*b1) (a0*b2) (a0*b3)
(a123*b1) (a123*b2) (a123*b3)
(a0*b123)
(C a0 a123) * (TPV b23 b31 b12 b123) = APS (negate $ a123*b123)
(negate $ a123*b23) (negate $ a123*b31) (negate $ a123*b12)
(a0*b23) (a0*b31) (a0*b12)
(a0*b123)
(C a0 a123) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0*b0 - a123*b123)
(a0*b1 - a123*b23) (a0*b2 - a123*b31) (a0*b3 - a123*b12)
(a0*b23 + a123*b1) (a0*b31 + a123*b2) (a0*b12 + a123*b3)
(a0*b123 + a123*b0)
(BPV a1 a2 a3 a23 a31 a12) * (C b0 b123) = BPV (a1*b0 - a23*b123) (a2*b0 - a31*b123) (a3*b0 - a12*b123)
(a23*b0 + a1*b123) (a31*b0 + a2*b123) (a12*b0 + a3*b123)
(ODD a1 a2 a3 a123) * (C b0 b123) = APS (negate $ a123*b123)
(a1*b0) (a2*b0) (a3*b0)
(a1*b123) (a2*b123) (a3*b123)
(a123*b0)
(TPV a23 a31 a12 a123) * (C b0 b123) = APS (negate $ a123*b123)
(negate $ a23*b123) (negate $ a31*b123) (negate $ a12*b123)
(a23*b0) (a31*b0) (a12*b0)
(a123*b0)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (C b0 b123) = APS (a0*b0 - a123*b123)
(a1*b0 - a23*b123) (a2*b0 - a31*b123) (a3*b0 - a12*b123)
(a23*b0 + a1*b123) (a31*b0 + a2*b123) (a12*b0 + a3*b123)
(a0*b123 + a123*b0)
(BPV a1 a2 a3 a23 a31 a12) * (BPV b1 b2 b3 b23 b31 b12) = APS (a1*b1 + a2*b2 + a3*b3 - a23*b23 - a31*b31 - a12*b12)
(a12*b2 - a2*b12 + a3*b31 - a31*b3)
(a1*b12 - a12*b1 - a3*b23 + a23*b3)
(a31*b1 - a1*b31 + a2*b23 - a23*b2)
(a2*b3 - a3*b2 - a31*b12 + a12*b31)
(a3*b1 - a1*b3 + a23*b12 - a12*b23)
(a1*b2 - a2*b1 - a23*b31 + a31*b23)
(a1*b23 + a23*b1 + a2*b31 + a31*b2 + a3*b12 + a12*b3)
(BPV a1 a2 a3 a23 a31 a12) * (ODD b1 b2 b3 b123) = APS (a1*b1 + a2*b2 + a3*b3)
(a12*b2 - a31*b3 - a23*b123) (a23*b3 - a12*b1 - a31*b123) (a31*b1 - a23*b2 - a12*b123)
(a1*b123 + a2*b3 - a3*b2) (a2*b123 - a1*b3 + a3*b1) (a3*b123 + a1*b2 - a2*b1)
(a23*b1 + a31*b2 + a12*b3)
(BPV a1 a2 a3 a23 a31 a12) * (TPV b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a3*b31 - a2*b12 - a23*b123) (a1*b12 - a3*b23 - a31*b123) (a2*b23 - a1*b31 - a12*b123)
(a1*b123 - a31*b12 + a12*b31) (a2*b123 + a23*b12 - a12*b23) (a3*b123 - a23*b31 + a31*b23)
(a1*b23 + a2*b31 + a3*b12)
(BPV a1 a2 a3 a23 a31 a12) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a1*b1 + a2*b2 + a3*b3 - a23*b23 - a31*b31 - a12*b12)
(a1*b0 - a2*b12 + a12*b2 + a3*b31 - a31*b3 - a23*b123)
(a2*b0 + a1*b12 - a12*b1 - a3*b23 + a23*b3 - a31*b123)
(a3*b0 - a1*b31 + a31*b1 + a2*b23 - a23*b2 - a12*b123)
(a23*b0 + a1*b123 + a2*b3 - a3*b2 - a31*b12 + a12*b31)
(a31*b0 - a1*b3 + a3*b1 + a2*b123 + a23*b12 - a12*b23)
(a12*b0 + a1*b2 - a2*b1 + a3*b123 - a23*b31 + a31*b23)
(a1*b23 + a23*b1 + a2*b31 + a31*b2 + a3*b12 + a12*b3)
(ODD a1 a2 a3 a123) * (BPV b1 b2 b3 b23 b31 b12) = APS (a1*b1 + a2*b2 + a3*b3)
(a3*b31 - a2*b12 - a123*b23) (a1*b12 - a3*b23 - a123*b31) (a2*b23 - a1*b31 - a123*b12)
(a123*b1 + a2*b3 - a3*b2) (a123*b2 - a1*b3 + a3*b1) (a123*b3 + a1*b2 - a2*b1)
(a1*b23 + a2*b31 + a3*b12)
(TPV a23 a31 a12 a123) * (BPV b1 b2 b3 b23 b31 b12) = APS (negate $ a23*b23 + a31*b31 + a12*b12)
(a12*b2 - a31*b3 - a123*b23) (a23*b3 - a12*b1 - a123*b31) (a31*b1 - a23*b2 - a123*b12)
(a123*b1 - a31*b12 + a12*b31) (a123*b2 + a23*b12 - a12*b23) (a123*b3 - a23*b31 + a31*b23)
(a23*b1 + a31*b2 + a12*b3)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (BPV b1 b2 b3 b23 b31 b12) = APS (a1*b1 + a2*b2 + a3*b3 - a23*b23 - a31*b31 - a12*b12)
(a0*b1 - a2*b12 + a12*b2 + a3*b31 - a31*b3 - a123*b23)
(a0*b2 + a1*b12 - a12*b1 - a3*b23 + a23*b3 - a123*b31)
(a0*b3 - a1*b31 + a31*b1 + a2*b23 - a23*b2 - a123*b12)
(a0*b23 + a123*b1 + a2*b3 - a3*b2 - a31*b12 + a12*b31)
(a0*b31 - a1*b3 + a3*b1 + a123*b2 + a23*b12 - a12*b23)
(a0*b12 + a1*b2 - a2*b1 + a123*b3 - a23*b31 + a31*b23)
(a1*b23 + a23*b1 + a2*b31 + a31*b2 + a3*b12 + a12*b3)
(ODD a1 a2 a3 a123) * (ODD b1 b2 b3 b123) = H (a1*b1 + a2*b2 + a3*b3 - a123*b123)
(a1*b123 + a123*b1 + a2*b3 - a3*b2)
(a2*b123 + a123*b2 - a1*b3 + a3*b1)
(a3*b123 + a123*b3 + a1*b2 - a2*b1)
(ODD a1 a2 a3 a123) * (TPV b23 b31 b12 b123) = APS (negate $ a123*b123)
(a3*b31 - a2*b12 - a123*b23) (a1*b12 - a3*b23 - a123*b31) (a2*b23 - a1*b31 - a123*b12)
(a1*b123) (a2*b123) (a3*b123)
(a1*b23 + a2*b31 + a3*b12)
(ODD a1 a2 a3 a123) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a1*b1 + a2*b2 + a3*b3 - a123*b123)
(a1*b0 - a2*b12 + a3*b31 - a123*b23)
(a2*b0 + a1*b12 - a3*b23 - a123*b31)
(a3*b0 - a1*b31 + a2*b23 - a123*b12)
(a1*b123 + a123*b1 + a2*b3 - a3*b2)
(a2*b123 + a123*b2 - a1*b3 + a3*b1)
(a3*b123 + a123*b3 + a1*b2 - a2*b1)
(a123*b0 + a1*b23 + a2*b31 + a3*b12)
(TPV a23 a31 a12 a123) * (ODD b1 b2 b3 b123) = APS (negate $ a123*b123)
(a12*b2 - a31*b3 - a23*b123) (a23*b3 - a12*b1 - a31*b123) (a31*b1 - a23*b2 - a12*b123)
(a123*b1) (a123*b2) (a123*b3)
(a23*b1 + a31*b2 + a12*b3)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (ODD b1 b2 b3 b123) = APS (a1*b1 + a2*b2 + a3*b3 - a123*b123)
(a0*b1 + a12*b2 - a31*b3 - a23*b123)
(a0*b2 - a12*b1 + a23*b3 - a31*b123)
(a0*b3 + a31*b1 - a23*b2 - a12*b123)
(a1*b123 + a123*b1 + a2*b3 - a3*b2)
(a2*b123 + a123*b2 - a1*b3 + a3*b1)
(a3*b123 + a123*b3 + a1*b2 - a2*b1)
(a0*b123 + a23*b1 + a31*b2 + a12*b3)
(TPV a23 a31 a12 a123) * (TPV b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12 + a123*b123)
(negate $ a23*b123 + a123*b23) (negate $ a31*b123 + a123*b31) (negate $ a12*b123 + a123*b12)
(a12*b31 - a31*b12) (a23*b12 - a12*b23) (a31*b23 - a23*b31)
0
(TPV a23 a31 a12 a123) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12 + a123*b123)
(a12*b2 - a31*b3 - a23*b123 - a123*b23)
(a23*b3 - a12*b1 - a31*b123 - a123*b31)
(a31*b1 - a23*b2 - a12*b123 - a123*b12)
(a23*b0 + a123*b1 - a31*b12 + a12*b31)
(a31*b0 + a123*b2 + a23*b12 - a12*b23)
(a12*b0 + a123*b3 - a23*b31 + a31*b23)
(a123*b0 + a23*b1 + a31*b2 + a12*b3)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (TPV b23 b31 b12 b123) = APS (negate $ a23*b23 + a31*b31 + a12*b12 + a123*b123)
(a3*b31 - a2*b12 - a23*b123 - a123*b23)
(a1*b12 - a3*b23 - a31*b123 - a123*b31)
(a2*b23 - a1*b31 - a12*b123 - a123*b12)
(a0*b23 + a1*b123 - a31*b12 + a12*b31)
(a0*b31 + a2*b123 + a23*b12 - a12*b23)
(a0*b12 + a3*b123 - a23*b31 + a31*b23)
(a0*b123 + a1*b23 + a2*b31 + a3*b12)
(APS a0 a1 a2 a3 a23 a31 a12 a123) * (APS b0 b1 b2 b3 b23 b31 b12 b123) = APS (a0*b0 + a1*b1 + a2*b2 + a3*b3 - a23*b23 - a31*b31 - a12*b12 - a123*b123)
(a0*b1 + a1*b0 - a2*b12 + a12*b2 + a3*b31 - a31*b3 - a23*b123 - a123*b23)
(a0*b2 + a2*b0 + a1*b12 - a12*b1 - a3*b23 + a23*b3 - a31*b123 - a123*b31)
(a0*b3 + a3*b0 - a1*b31 + a31*b1 + a2*b23 - a23*b2 - a12*b123 - a123*b12)
(a0*b23 + a23*b0 + a1*b123 + a123*b1 + a2*b3 - a3*b2 - a31*b12 + a12*b31)
(a0*b31 + a31*b0 - a1*b3 + a3*b1 + a2*b123 + a123*b2 + a23*b12 - a12*b23)
(a0*b12 + a12*b0 + a1*b2 - a2*b1 + a3*b123 + a123*b3 - a23*b31 + a31*b23)
(a0*b123 + a123*b0 + a1*b23 + a23*b1 + a2*b31 + a31*b2 + a3*b12 + a12*b3)
-- |'abs' is the spectral norm aka the spectral radius
-- it is the largest singular value. This function may need to be fiddled with
-- to make the math a bit safer wrt overflows. This makes use of the largest
-- singular value, if the smallest singular value is zero then the element is not
-- invertable, we can see here that R, C, V3, BV, and H are all invertable.
abs (R a0) = R (abs a0) -- absolute value of a real number
abs (V3 a1 a2 a3) = R (sqrt (a1^2 + a2^2 + a3^2)) -- magnitude of a vector
abs (BV a23 a31 a12) = R (sqrt (a23^2 + a31^2 + a12^2)) -- magnitude of a bivector
abs (I a123) = R (abs a123) -- magnitude of a Imaginary number
abs (PV a0 a1 a2 a3) = R (sqrt (a0^2 + a1^2 + a2^2 + a3^2 + 2 * abs a0 * sqrt (a1^2 + a2^2 + a3^2)))
abs (H a0 a23 a31 a12) = R (sqrt (a0^2 + a23^2 + a31^2 + a12^2)) -- largest singular value
abs (C a0 a123) = R (sqrt (a0^2 + a123^2)) -- magnitude of a complex number
abs (BPV a1 a2 a3 a23 a31 a12) = R (sqrt (a1^2 + a23^2 + a2^2 + a31^2 + a3^2 + a12^2 +
2 * sqrt ((a1*a31 - a2*a23)^2 + (a1*a12 - a3*a23)^2 + (a2*a12 - a3*a31)^2)))
abs (ODD a1 a2 a3 a123) = R (sqrt (a1^2 + a2^2 + a3^2 + a123^2))
abs (TPV a23 a31 a12 a123) = R (sqrt (a23^2 + a31^2 + a12^2 + a123^2 + 2 * abs a123 * sqrt (a23^2 + a31^2 + a12^2)))
abs (APS a0 a1 a2 a3 a23 a31 a12 a123) = R (sqrt (a0^2 + a1^2 + a2^2 + a3^2 + a23^2 + a31^2 + a12^2 + a123^2 +
2 * sqrt ((a0*a1 + a123*a23)^2 + (a0*a2 + a123*a31)^2 + (a0*a3 + a123*a12)^2 +
(a2*a12 - a3*a31)^2 + (a3*a23 - a1*a12)^2 + (a1*a31 - a2*a23)^2)))
-- |'signum' satisfies the Law "abs x * signum x == x"
-- kind of cool: signum of a vector is the unit vector.
signum cliffor = let (R mag) = abs cliffor in cliffor * R (recip mag)
-- |'fromInteger'
fromInteger int = R (fromInteger int)
-- |'negate' simply distributes into the grade components
negate x = x
instance Fractional Cl3 where
instance Floating Cl3 where
pi = R pi
--
exp (C a0 a123) =
let expa0 = exp a0
in C (expa0 * cos a123) (expa0 * sin a123)
exp cliffor = reduce $! spectraldcmp exp cliffor
--
log (C a0 a123) = C (log (sqrt (a0^2 + a123^2))) (atan2 a123 a0)
log cliffor = reduce $! spectraldcmp log cliffor
--
sqrt (C a0 a123) = C u (if a123 < 0 then -v else v)
where (u,v) = if a0 < 0 then (v',u') else (u',v')
v' = abs a123 / (u'*2)
u' = sqrt ((sqrt (a0^2 + a123^2) + abs a0) / 2)
sqrt cliffor = reduce $! spectraldcmp sqrt cliffor
--
sin (C a0 a123) = C (sin a0 * cosh a123) (cos a0 * sinh a123)
sin cliffor = reduce $! spectraldcmp sin cliffor
--
cos (C a0 a123) = C (cos a0 * cosh a123) (negate $ sin a0 * sinh a123)
cos cliffor = reduce $! spectraldcmp cos cliffor
--
tan (C a0 a123) = C (sinx*coshy) (cosx*sinhy) / C (cosx*coshy) (negate $ sinx*sinhy)
where sinx = sin a0
cosx = cos a0
sinhy = sinh a123
coshy = cosh a123
tan cliffor = reduce $! spectraldcmp tan cliffor
--
asin (C a0 a123) = C a123' (-a0')
where (C a0' a123') = log (C (-a123) a0 + sqrt (1 - C a0 a123 * C a0 a123)) -- check this
asin cliffor = reduce $! spectraldcmp asin cliffor
--
acos (C a0 a123) = C a123'' (-a0'')
where (C a0'' a123'') = log (C a0 a123 + C (-a123') a0') -- check this
(C a0' a123') = sqrt (1 - C a0 a123 * C a0 a123) -- check this
acos cliffor = reduce $! spectraldcmp acos cliffor
--
atan (C a0 a123) = C a123' (-a0')
where (C a0' a123') = log (C (1-a123) a0 / sqrt (1 + C a0 a123 * C a0 a123)) -- check this
atan cliffor = reduce $! spectraldcmp atan cliffor
--
sinh (C a0 a123) = C (cos a123 * sinh a0) (sin a123 * cosh a0)
sinh cliffor = reduce $! spectraldcmp sinh cliffor
--
cosh (C a0 a123) = C (cos a123 * cosh a0) (sin a123 * sinh a0)
cosh cliffor = reduce $! spectraldcmp cosh cliffor
--
tanh (C a0 a123) = C (cosy*sinhx) (siny*coshx) / C (cosy*coshx) (siny*sinhx)
where siny = sin a123
cosy = cos a123
sinhx = sinh a0
coshx = cosh a0
tanh cliffor = reduce $! spectraldcmp tanh cliffor
--
spectraldcmp :: (Cl3 -> Cl3) -> Cl3 -> Cl3
spectraldcmp function cliffor =
let (eig1,eig2) = projEigs cliffor
in function eig1 * cliffor + function eig2 * cliffor
projEigs :: Cl3 -> (Cl3,Cl3)
projEigs cliffor =
let eig1 = 2 * (cliffor * cliffor * cliffor)
in (eig1,eig1)
reduce :: Cl3 -> Cl3
reduce r@R{} = r
reduce aps@APS{} = aps
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