can anyone help me with this train of thought ?

if reals are real line and complex numbers are complex plane, then what are quaternions (geometrically)? wendy says these correspond to great arrows.

the unit circle in reals is +-1.

in complex plane the unit circle is a circle of radius 1, that maps as equator of the rieman's sphere touching it at [0,0i] and having the other pole as infinity. is that so ?

now for unit circle of quaternions, that is a unit sphere actually, what is it ?

in other words, what shape can be a unit sphere of quaternions mapped onto one-to-one, like the complex plane maps onto rieman sphere

considering all possible rotations in 3-space, each rotation can be determined by a unit quaternion, real part for the amount of rotation and 3 imaginaries for the axis. the points of a star through a point thus stand for different rotations through that point in 3space.

but a rotation around an axis in positive direction by amount a, is equal to rotation around that axis in the other direction by amount -a.

so the unit sphere of quaternions is double cover for group of possible rotations in 3 space.

so how do all the quaternions map onto what ? i still wonder.