In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting (by example) at identity (center of ball), through south pole, jump to north pole and ending again at the identity rotation (i.e., a series of rotation through an angle φ where φ runs from 0 to 2 π). This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". This is a closed loop, since the north pole and the south pole are identified. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. These identifications illustrate that SO(3) is connected but not simply connected. In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R 3 so the latter can also serve as a topological model for the rotation group.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |