#### lavinia

Science Advisor

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To continue this example, the loop that connects ##0## to ##2π## is a circle and cannot be shrunk to a point without breaking the circle,Consider the simpler situation of rotations in the plane where the rotation group is (part of) ##SO(2)## which has ##2(2-1)/2 = 1## degrees of freedom, i.e. one parameter (while ##SO(3)## has ##3(3-1)/2 = 3## parameters). The parameter space is the line ##[0,2 \pi]##. A continuous path in parameter space is just a parameter ##\theta## moving along the interval ##[0,2\pi]##, e.g. if you rotate around a circle, the path in parameter space is just the angle ##\theta## moving from ##0## to a bunch of other angles...

Regarding "rotation space" and "orientation space", I am not sure these things make sense and again it seems like you are thinking of rigid bodies where one fixes the position of the center of mass then the orientation of the rigid body about that center of mass.

Unlike in the case of ##SO(3)## no multiple of this loop is contractible. In ##SO(3)## the double of any loop is contractible ("null homotopic").