Big circles on S^3

[Petter Johansson]

*Let S^k be the unit sphere (of dimension k) in \R^{k+1}.

*We say that a sphere is big it its radius is 1.

Each 4-subset of the points {x_1,x_2,x_3,x_4,x_5} on S^3 determines a sphere of dimension 2 given by the intersection of S^3 with the hyperplane spanned by the four points. There are five such spheres,

S_1(x_2,x_3,x_4,x_5)
S_2(x_1,x_3,x_4,x_5)
..
S_5(x_1,x_2,x_3,x_4)

Assume that there are 4 big circles on S^3, one through each point x_j, j<5. My question is: is it possible for any k to move x_1,...,x_5 along these big circles in such a way that the first sphere S_j to be big is S_k?

The interesting case is when the big circles are in generic position.

More generally, do this work on S^2n for 2n+1 points of which 2n are moveable along big circles?

Regards,

Petter