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Hi,

The composition transit_H_to_X = transit_Y_to_X * transit_H_to_Y works well for me (Sage 10.2): it yields a result that does not depend on sin(th):

t = t
x = r*cos(ph_/(r*sin(th)))*sin(th_/r)
y = r*sin(th_/r)*sin(ph_/(r*sin(th)))
z = r*cos(th_/r)

I think the error in your initial code comes from the spurious sin(th) introduced when setting the inverse:

transit_H_to_X.set_inverse(
    t,
    sqrt(x^2+y^2+z^2),
    r*arccos(z/sqrt(x^2+y^2+z^2)),
    r*sin(th)*atan2(y,x)        #  <-- there should be no sin(th) here, nor r
)

The correct definition of the inverse is the one used in your answer:

transit_H_to_X.set_inverse(
    t,
    sqrt(x^2+y^2+z^2),
    sqrt(x^2+y^2+z^2)*arccos(z/sqrt(x^2+y^2+z^2)),
    sqrt(x^2+y^2+z^2)*sin(arccos(z/sqrt(x^2+y^2+z^2)))*atan2(y,x)
)

Best wishes,

Eric.