Table of Contents
pow(qq,rp)

— David Wagner 2010/01/13 18:48

Raising a quaternion q to a real power p.

ln(q) p

l = ln(q)p = p (ln vert q vert + {{q_x i + q_y j + q_z k}/{vert q_v vert}} acos {q_w/{vert q vert}} ) = p (ln vert q vert + {{q_x i + q_y j + q_z k}/{vert q_v vert}} atan {{vert q_v vert}/{q_w}} ).

q^p

q^p = e^{p ln q} = e^l = e^{l_w} (cos vert l_v vert + {{l_v}/{vert l_v vert}} sin vert l_v vert)

= e^{p ln vert q vert} (cos vert l_v vert + {{l_v}/{vert l_v vert}} sin vert l_v vert)

= {{vert q vert}^{p}} (cos vert l_v vert + {{l_v}/{vert l_v vert}} sin vert l_v vert)

= {{vert q vert}^{p}} (cos p alpha + {{q_v}/{vert q_v vert}} sin p alpha):

l_v = l_x i + l_y j + l_z k = {p}/{vert q_v vert} arctan{{vert q_v vert}/q_w} lbrace q_x i + q_y j + q_z k rbrace

p alpha = {vert l_v vert} = delim{|}{ p arctan{{vert q_v vert}/q_w} }{|}.

q^½p i (q^½p)*

q^{½ p} i overline{q^{½ p}} = {{vert q vert}^{p}} lbrace

{1/2}( {{l_x}^2-{l_y}^2-{l_z}^2}/{vert l_v vert^2}(1-cos{vert l_v vert} ) + cos{vert l_v vert} +1 )i

+( {l_x l_y}/{vert l_v vert^2} (1-cos{vert l_v vert} ) +{l_z}/{vert l_v vert} sin {vert l_v vert} )j

+({l_x l_z}/{vert l_v vert^2} (1-cos{vert l_v vert} ) -{l_y}/{vert l_v vert} sin {vert l_v vert} )k rbrace

= {{vert q vert}^{p}} lbrace

{1/2}( {{q_x}^2-{q_y}^2-{q_z}^2}/{vert q_v vert^2}(1-cos{p alpha} ) + cos{p alpha} +1 )i

+( {q_x q_y}/{vert q_v vert^2} (1-cos{p alpha} ) +{q_z}/{vert q_v vert} sin {p alpha} )j

+({q_x q_z}/{vert q_v vert^2} (1-cos{p alpha} ) -{q_y}/{vert q_v vert} sin {p alpha} )k rbrace


Personal Tools