I am trying to plot a density plot on a surface of a sphere, where the density function `Sqrt[1-x^2]` is symmetric around `0`. (There, `x=Sin[θ] Cos[ϕ]`).

So far I have:

    ParametricPlot3D[{Sin[θ] Cos[ϕ], 
    Sin[θ] Sin[ϕ], Cos[θ]}, {θ, 0, π}, {ϕ, 0, 2 π}, ColorFunction -> Function[{x, y, z, θ, ϕ}, ColorData["Rainbow"][Sqrt[1 - x^2]]]]

But somehow when the plot is displayed, the colors at `x=1` and `x=-1` do not match. Does it have something to do with the `ColorFunction`?