Controlling Hopf Fiber
- Author:
- Ingo Dahn
The sliders define a point P on the unit sphere by its latitude and longitude. The calculated quaternion r describes a rotation of around the axis bisecting the angle SOP for the point S with coordinates (1,0,0) by an angle of 180°, taking S to P.
The Hopf fiber of P, defined by all quaternions of the form , describes a circle on the unit sphere in a 4-dimensional space. This fiber is mapped by stereographic projection from (1,0,0,0) to the green circle in the 3-dimensional space.
Observe the change of the circle with the change of the position of P! Animate by right clicking on the first two items on the input bar.
Switch the trace of the green circle on or off.
With the red dot you may reveal the unit circle in the -plane as a further hint
Can you change the latitude and longitude of P such that the Hopf fiber contains the root of the projection, i.e. the quaternion 1? What does this mean for the green projection of the fiber of P?
What do you observe w.r.t the red circle?