Orbits: SO(3)
A basic example of a 3-dimensional orbit is a sphere. A sphere is the orbit of any point on that sphere. Said differently, a sphere is its own orbit, under the action of the SO(3) group. Take, for example, a sphere of radius 1, centered at the origin. Applying every member of SO(3) to a point on the sphere should yield another point on the sphere. The O(3) group is the group of 3x3 matrices with determinant 1 or -1. The SO(3), or "3-dimensional special orthogonal" group is the group of matrices from the O(3) group whose determinant is 1. These matrices are the 3-d rotation matrices.
Use the sliders to manipulate the position of the green vector. The red vector represents the initial position. In addition, the following controls are available to manipulate the image:
- Mouse: rotate camera
- Shift + mouse: zoom in/out (try zooming into the sphere for a good view!)
- Ctrl + mouse: pan x/y
- Shift + ctrl + mouse: adjust camera's focal length