orthogonal 4d rotational matrices in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg51664] orthogonal 4d rotational matrices in Mathematica
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 27 Oct 2004 23:44:36 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```This was made up as an answer to a sci.math question but they will plug
in to
Mathematica very nicely to do 4d rotations with six angles :

m1 = {{Cos[a], Sin[a], 0, 0}, {-Sin[a], Cos[a], 0, 0}, {0, 0, 1, 0}, {0,
0, 0, 1}}

m2 = {{Cos[b], 0, -Sin[b], 0}, {0, 1, 0, 0}, {Sin[b], 0, Cos[b], 0}, {0,
0, 0, 1}}

m3 = {{1, 0, 0, 1}, {0, Cos[c], Sin[c], 0}, {0, -Sin[c], Cos[c], 0}, {0,
0, 0, 1}}

m4 = {{Cos[d], 0, 0, -Sin[d]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sin[d], 0,
0, Cos[d]}}

m5 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, Cos[e], -Sin[e]}, {0, 0,
Sin[e], Cos[e]}}

m6 = {{1, 0, 0, 0}, {0, Cos[f], 0, Sin[f]}, {0, 0, 1, 0}, {0, -Sin[f],
0, Cos[f]}}

Simplify[Det[m1]]

Simplify[Det[m2]]

Simplify[Det[m3]]

Simplify[Det[m4]]

Simplify[Det[m5]]

Simplify[Det[m6]]

M = m1 . m2 . m3 . m4 . m5 . m6

M={{Cos[a] Cos[b] Cos[d] + Cos[a] Cos[b] Sin[d], Cos[f] (Cos[c] Sin[a] +
Cos[a] Sin[b] Sin[c]) - (Cos[e] (Cos[a] Cos[b] Cos[d] - Cos[a] Cos[b]
Sin[d]) - (-Cos[a] Cos[c] Sin[b] + Sin[a] Sin[c]) Sin[e]) Sin[f],

Cos[e] (-Cos[a] Cos[c] Sin[b] + Sin[a] Sin[c]) + (Cos[a] Cos[b]
Cos[d] - Cos[a] Cos[b] Sin[d]) Sin[e],

Cos[f] (Cos[e] (Cos[a] Cos[b] Cos[d] - Cos[a] Cos[b] Sin[d]) -
(-Cos[a] Cos[c] Sin[b] + Sin[a] Sin[c]) Sin[e]) + (Cos[c] Sin[a] +
Cos[a] Sin[b] Sin[c]) Sin[f]},

{-Cos[b] Cos[d] Sin[a] - Cos[b] Sin[a] Sin[d], Cos[f] (Cos[a] Cos[c] -
Sin[a] Sin[b] Sin[c]) - (Cos[e] (-Cos[b] Cos[d] Sin[a] + Cos[b] Sin[a]
Sin[d]) - (Cos[c] Sin[a] Sin[b] + Cos[a] Sin[c]) Sin[e]) Sin[f],

Cos[e] (Cos[c] Sin[a] Sin[b] + Cos[a] Sin[c]) + (-Cos[b] Cos[d]
Sin[a] + Cos[b] Sin[a] Sin[d]) Sin[e],

Cos[f] (Cos[e] (-Cos[b] Cos[d] Sin[a] + Cos[b] Sin[a] Sin[d]) -
(Cos[c] Sin[a] Sin[b] + Cos[a] Sin[c]) Sin[e]) + (Cos[a] Cos[c] - Sin[a]
Sin[b] Sin[c]) Sin[f]},

{Cos[d] Sin[b] + Sin[b] Sin[d], -Cos[b] Cos[f] Sin[c] - (Cos[e]
(Cos[d] Sin[b] - Sin[b] Sin[d]) - Cos[b] Cos[c] Sin[e]) Sin[f], Cos[b]
Cos[c] Cos[e] + (Cos[d] Sin[b] - Sin[b] Sin[d]) Sin[e],

Cos[f] (Cos[e] (Cos[d] Sin[b] - Sin[b] Sin[d]) - Cos[b] Cos[c]
Sin[e]) - Cos[b] Sin[c] Sin[f]}, {Sin[d], -Cos[d] Cos[e] Sin[f], Cos[d]
Sin[e], Cos[d] Cos[e] Cos[f]}}

Simplify[Det[M]]

Respectfully, Roger L. Bagula