Rot3

Taylor Expansion in Logmap

Series[t/(2 Sin[t]),{t,0,3}]

Plot[(1/2)+((t^2)/12)-t/(2 Sin[t]),{t,-(10^(-4)),(10^(-4))}]

Plot[ArcCos[(t-1)/2],{t,2.999,3}]

exp=ArcCos[(t-1)/2]/(2 Sin[ArcCos[(t-1)/2]])//Simplify

Series[exp,{t,3,1}]

Plot[{exp-((1/2)-((t-3)/12))},{t,3-(10^(-5)),3}]

FullSimplify[(2*Sin[t]-t*(1+Cos[t]))/(2*Sin[t])]//CForm

Cayley Transform in retract

Skew[xi_,yi_,zi_]:=(({{0, -zi, yi}, {zi, 0, -xi}, {-yi, xi, 0}}))

I3=IdentityMatrix[3];

Cayley[A_]:=(I3-A).Inverse[I3+A]

Cayley[-(({{0, -z, y}, {z, 0, -x}, {-y, x, 0}}))/2]//Simplify

Inverse Cayley Transform in localCoordinates

Cayley[(({{a, b, c}, {d, e, f}, {g, h, i}}))]//Simplify//MatrixForm

FullSimplify[1+e-c g-c e g+c d h-f h+i+e i-b (d-f g+d i)+a (1+e-f h+i+e i)]/.{1+e-f h+i+e i->K}