Curve in Spherical Coordinates 

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X cSpherical[t_] := {t, Pi ((1 + Sin[16 t + 4 t^2])/2), 1 + 2 t + 3 t^2}

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X cCartesian[t_] = CoordinateTransform["Spherical" -> "Cartesian", cSpherical[t]]

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A plot of the curve inside the unit sphere. Notice the coloring, which indicates it starts quite slowly and then speeds up as it gets closer to the surface:

In[3]:=

X Show[Graphics3D[{Opacity[.25], Sphere[]}], ParametricPlot3D[cCartesian[t], {t, 0, 1}, PlotStyle -> Thick, ColorFunction -> (Hue[.8 #4] &)], ImageSize -> Medium]

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