This doesn't seem to work. Is it a bug or a limitation of ParametricRegion?
Make a spline function that returns a pair of real values for a parameter 0<s<1:
pts = {{0, 0}, {1, 1}, {2, -1}, {3, 0}};
spl = BSplineFunction[pts]
Valid syntax for ParametricRegion is:
ParametricRegion[{Cos[t], Sin[t]}, {{t, 0, 2 Pi}}]
However, these throw an error:
parRegion = ParametricRegion[spl[t], {{t, 0, 1}}]
parRegion = ParametricRegion[Evaluate@spl[t], {{t, 0, 1}}]
And, more bizarrely, this doesn't throw an error but gives a result that appears to be wrong:
fx[t_] := First[spl[t]]
fy[t_] := Last[spl[t]]
ParametricRegion[{fx[t], fy[t]}, {{t, 0, 1}}]
returns:
ParametricRegion[{{t, t}, 0 <= t <= 1}, {t}]
