The SO(3) group is topologically a 3-dimensional ball of radius [tex]\pi[/tex], if the opposite points on its surface are identified with each other. (the name of it is 3-dimensional projective space). The center of the ball represents the unit element e of the group. An arbitrary point g in the ball represents a rotation with axis g-e and with angle of ||e-g|| (thinking the ball as a part of the 3-dimensional euclidean space). I am curious to know how looks the natural group left action in this ball. This would be completely described if we knew the curve t -> exp(tv)g for one arbitrarily selected v of so(3) and for each g of SO(3). How look these curves in the ball? In the case of g=e (i.e. the center of the ball), this curve is a straight line passing from the center to a point of the surface which is identified with the opposite point and from this opposite point back to the origin. But I can't imagine, what curves we get if we take an arbitrary g point in the ball instead of the center.