In 3 dimensions, assume an axis that is transformed through rotations into all points on a spherical cap (with angle $\theta$ between the axis and a vector starting from the origin and pointing towards a point on the cap boundary). During these transformations an area is covered by the other 2 axis. What is the form of this area? My guess is that this area is a strip of length $2\theta$ over the great circle defined by the 2 axes initial configuration.
Edit: My exact question can be stated as follows: I want to find the possible locations of axis $x,y$ occurred by tilting the $z$-axis along the spherical cap shown in the figure. These transformations include all rotations along the $z$-axis (covering respectively great circles in the $xy$-plane). This amounts to finding all vectors which are normal to vectors starting from the origin and pointing towards a point on the spherical cap surface.