We know that the 3-sphere can be expressed with two different sets of parametric equations. There's the standard one:
sin a sin b cos c
sin a sin b sin c
sin a cos b
cos a,
and one based on the Hopf fibration:
cos a cos b
cos a sin b
sin a cos c
sin a sin c
Clearly a similar thing can be done with higher dimensional spheres, as well as rotatopes and toratopes that involve these spheres. The Hopf coordinates can be viewed as a special case of the tiger with both minor radii equal to zero.
Here's a conjecture. There is a unique set of Hopf spherical coordinates for each nD toratope that does not contain a lonely '1'. So for example, in 6D we have Hopf coordinates based on the following toratopes:
(IIIIII)
((IIII)(II))
(((II)(II))(II))
((III)(III))
((II)(II)(II))
The (((II)(II))(II)) Hopf coordinates in 6D are:
cos a cos b cos c
cos a cos b sin c
cos a sin b cos d
cos a sin b sin d
sin a cos e
sin a sin e