fwiw, I wrote a few Mickey-Mouse verbs yesterday while waiting for my
virus-checker
to go through its paces. Mainly to get my head round the Cayley-Dickson
complex duple
representations. Here's a proto-script below my sign-off; for starters,
I've assumed
2-column arrays, but rank 1 might not be the way to go (?) - I haven't
got into
arrays of quaternions yet!
Mike
Apologies for any line-wrapping - it looks ok this end!
=======================================================================================
NB. Cayley-Dickson construction
NB. based on entries in
NB. https://en.wikipedia.org/wiki/Quaternion
NB. and
NB. https://en.wikipedia.org/wiki/Cayley-Dickson_construction
NB. Quaternion quadruple (a b c d) with a,b,c,d e. ℝ is stored as a
complex duple ajb cjd e. ℂ2
NB. conjugate of complex duple q = (a,b) is q* = (a*, -b) where * is
complex conjugate
NB. Addition (a,b) + (c,d) = (a+b, c+d) ... so dyad + works
without needing modification
NB. Multiplication (a,b) * (c,d) = (ac-b.d*, ad+b.c*), where * is comp conj
NB. = (a,b) +/ . * (c d*)
NB. (d c*) <== 2x2 complex
matrix
NB. Magnitude
NB. magnitude squared of (a,b), ||(a,b)||^2 = |a|^2 + |b|^2 e. ℝ
where |a| is magnitude of complex a
NB. Reciprocal (a,b)^_1 = (a*, -b)% ||(a,b)||^2
NB. Norm
NB. (magnitude, 0) e. ℝ2 e. ℂ2 - not sure why it needs to be
a quaternion!?
NB. complex verbs
cconj =: +
cnormsq =: *+
cnorm =: %: @ cnormsq
NB. Quaternion verbs
qconj =: ((cconj@[,-@])/)"1 NB. conjugate of (a,b)
qmult =: (+/ . * (,:_1 1 * |. @: cconj))"1 NB. multiply (a,b)*(c,d)
qmagsq =: (+/ @: cnormsq)"1 NB. Magnitude squared
qmag =: %: @ qmagsq NB. Magnitude
qnorm =: (0,~ qmag)"1 NB. Norm
qscalar=: (-: @ (+ qconj))"1 NB. "Scalar" part
qvector=: (-: @ (- qconj))"1 NB. "Vector" part
qrecip =: ((cconj@[, -@])/ % qmagsq)"1 NB. Reciprocal
qdivl =: qmult qrecip NB. via left quotient
- cribbed from Zhuravlov
qdivr =: qmult~qrecip NB. via right quotient
- cribbed from Zhuravlov
NB. 1j2 3j4 qmult 5j6 7j8
NB. _60j12 30j24
NB.
NB. _60j12 30j24 qdivl 5j6 7j8
NB. 1j2 3j4
NB.
NB. _60j12 30j24 qdivr 1j2 3j4
NB. 5j6 7j8
On 01/11/2018 19:38, Raul Miller wrote:
Rotations can be dizzying to think about, so this sort of issue is
maybe best visited casually over a period of time rather than jammed
into an intensive study session.
That said, the wikipedia page on Gimbal Lock can help motivate an
understanding of why a person might want to use quaternions to
represent rotation in three dimensions.
Thanks,
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