Re: [Algorithms] Representing Animation Key Frame withQuaternion+Translation Vector

[Marc B. R. <mar...@or...>]

I should have also mentioned that (unit) quaternions are sometimes called
euler parameters and that some robotics text toss all kinds of
representations into dual-quaternions:

(WARNING: I grabbed the stuff below from some really old notes I made to
myself...might be loaded with errors)

Using set notation, a quaternion Q={V,t}, where V is the bivector and t is
the scalar, and 'e' is the
Dual number basis (ee = 0):

  Q = q       = {R,r}+e{0,0}    quaternion q
  T = 1 + e t = {0,1}+e{t,0}    translation by 2t
  P = 1 + e p = {0,1}+e{p,0}    point p
  E = n + e d = {n,0}+e{d,0}    plane n+d==0
  L = t + e m = {t,0}+e{m,0}    plucker line: t is direction vector, m is
moment about origin

Screw motion:
  S = Cos[a]+Sin[a]t + e (-d Sin[a] + Sin[a]m + d Cos[a]t)
    = {Sin[a]t, Cos[a]} + e{Sin[a] m + d Cos[a]t , -d Sin[a]}

  screw axis        = (t,m)     {plucker}
  angle of rotation = 2a
  displacment       = 2d

General motion:
  G = q + e u, (q.q)=1 (q.u)=0               (mech engineer term 'G' unit or
normalized)

Involutions (conjugates):                    (where a* is quaternion
conjugate)
  X* = (a* + e b*) = {-R,r}+e{-S,s}
  X~ = (a  - e b ) = { R,r}-e{ S,s} 
  X' = (a* - e b*) = {-R,r}-e{-S,s} 

Incidence
  EP + PE  = 0          point on plane
  LP - PL' = 0          point on line
  LP + PL' = 0          line on plane