Rules for multiplying quaternion exponentials

doug  <sweetser@world.com>

Start with two arbitary quatenrions.

\(A\  = \ q[2, 1, 3, 4]; 
B\  = \ q[1, 2, 3, 5]; \)

The product of two quaternions AB is composed of two parts, and even and an odd part, the latter of which changes its sign for BA.

(A.B).{1, 0, 0, 0}(even  =  (A.B  +  B.A)/2).{1, 0, 0, 0}(odd  =  (A.B  -  B.A)/2).{1, 0, 0, 0}

(B.A).{1, 0, 0, 0}(even  -  odd).{1, 0, 0, 0}

Any quaternion can be expressed in polar form.  This involves three numbers:  The amplitude, which is the absolute value of a quaternion (square root of the norm), an angle theta, which is the ArcCos of the scalar divided by the absolute value, and a pure quaternion vector with a norm of 1.

\(R\_A\  = \ abs[A]; 

    θ\_A\  = \ 
      ArcCos[\((\(A\  + \ qconj[A]\)\/\(2\ abs[A]\))\)[\([1, 1]\)]]; 

    V\_A\  = \ \(A\  - qconj[A]\)\/\(\ abs[A\  - \ qconj[A]]\); 

    R\_A\ qexp[θ\_A\ V\_A].{1, 0, 0, 0}\)

\(R\_B\  = \ abs[B]; 

    θ\_B\  = \ 
      ArcCos[\((\(B\  + \ qconj[B]\)\/\(2\ abs[B]\))\)[\([1, 1]\)]]; 

    V\_B\  = \ \(B\  - qconj[B]\)\/\(\ abs[B\  - \ qconj[B]]\); 

    R\_B\ qexp[θ\_B\ V\_B].{1, 0, 0, 0}\)

The even part of the quaternion product is formed by using the even part of AB in the polar form.

\(R\_Even\  = \ abs[\(A.B\  + \ B.A\)\/2]; 

    θ\_Even\  = 
	\ 
      ArcCos[\((
            \(\((A.B\  + \ B.A)\)\  + \ qconj[A.B\  + \ B.A]\)\/\(2\ 
                abs[A.B\  + \ B.A]\))\)[\([1, 1]\)]]; 

    V\_Even\  = \ 
      \(\((A.B\  + \ B.A)\)\  - qconj[A.B\  + \ B.A]\)\/\(\ 
        abs[\((A.B\  + \ B.A)\)\  - \ qconj[A.B\  + \ B.A]]\); 

    R\_Even\ qexp[θ\_Even\ V\_Even].{1, 0, 0, 0}\)

The odd part of the quaternion product is formed by using the odd part of AB--which is a pure quaternion--in the polar form, with an angle theta = Pi/2.

\(R\_Odd\  = \ abs[\(A.B\  - \ B.A\)\/2]; 
θ\_Odd\  = \ π\/2; 
    
V\_Odd\  = \ \(A.B\  - \ B.A\)\/\(\ abs[A.B\  - \ B.A]\); 

    R\_Odd\ qexp[θ\_Odd\ V\_Odd].{1, 0, 0, 0}\)

See if the scalar parts add if the vector is equal to i.

\(a = q[2, 4, 0, 0]; 
b = q[3, 5, 0, 0]; \)

(Subscript[θ, complex])  =  
      ArcCos[(
            ((a.b  +  a.b  +  qconj[a.b  +  a.b])/(2 
                abs[a.b  +  a.b])))[[1, 1]]]((Subscript[V, complex])  =  
          ((a.b  +  a.b  - qconj[a.b  +  a.b])/( 
            abs[a.b  +  a.b  -  qconj[a.b  +  a.b]]))).{1, 0, 0, 0}

(Subscript[θ, a])  =  
      ArcCos[(((a  +  qconj[a])/(2 abs[a])))[[1, 1]]](Subscript[θ, b])  =  
      ArcCos[(((b  +  qconj[b])/(2 abs[b])))[[1, 1]]]N[(Subscript[θ, a])  +  (Subscript[θ, b])  -  ArcCos[-(7/(Sqrt[170]))]]

Look for a simpler rule for the amplitude.  The product Ra Rb is diminished by the exponential part.

(Subscript[R, Even])  - 
      N[(Subscript[R, A]) (Subscript[R, B]) 
          abs[((qexp[(Subscript[θ, A]) (Subscript[V, A])].qexp[(Subscript[θ, B]) (Subscript[V, B])] +  
                qexp[(Subscript[θ, B]) (Subscript[V, B])].qexp[(Subscript[θ, A]) (Subscript[V, A])])/2)]]

(Subscript[R, Odd])  - 
      N[(Subscript[R, A]) (Subscript[R, B]) 
          abs[((qexp[(Subscript[θ, A]) (Subscript[V, A])].qexp[(Subscript[θ, B]) (Subscript[V, B])] -  
                qexp[(Subscript[θ, B]) (Subscript[V, B])].qexp[(Subscript[θ, A]) (Subscript[V, A])])/2)]]

((((Subscript[R, Even])^2)   +  ((Subscript[R, Odd])^2))/((((Subscript[R, A]) (Subscript[R, B])))^2))

Rules for the Eulcidean product

The Euclidean product is formed by taking the conjugate of the first quaternion and muliplying it by the second.

(qconj[A].B).{1, 0, 0, 0}(even  =  (qconj[A].B  +  qconj[B].A)/2).{1, 0, 0, 0}(odd  =  (qconj[A].B  -  qconj[B].A)/2).{1, 0, 0, 0}

The even part is the norm with a zero vector, and the odd part has a zero scalar.  This property clearly separates the even and odd.  The even part has V = 0, so e^0 = 1, and the only contribution comes from the norm of the even part.

(Subscript[R, Even])  =  abs[((qconj[A].B  +  qconj[B].A)/2)]

The odd part has no scalar, so theta = pi/2.

\(R\_Odd\  = \ abs[\(qconj[A].B\  - \ qconj[B].A\)\/2]; 

    θ\_Odd\  = \ π\/2; 

    V\_Odd\  = \ 
      \(qconj[A].B\  - \ qconj[B].A\)\/\(\ 
        abs[qconj[A].B\  - \ qconj[B].A]\); 

    R\_Odd\ qexp[θ\_Odd\ V\_Odd].{1, 0, 0, 0}\)

(qconj[A].A).{1, 0, 0, 0}