f=BezierFunction[{{0,0,0,0}, {1,1,1,1}, {2,2,2,2}}]

f[0.3]

γ=Interpolation[{
{0,{0,0,0,0.1}, {{1,0,0,0}}},
{0.8,{0.8,0,0,0.1}},
{1,{1,0,0,0.2}},
{2,{2,0.5,0,0.1}},
{3,{3,0,0,0.1}},
{4,{1,0,0,0.1}},
{5,{2,0.5,0,0.1}},
{6,{3,0,0,0.1}},
{6.2,{3.2,0,0,0.1}},
{7,{4,0,0,0.1}}
}]

ParametricPlot[Take[γ[t], 2], {t,0,3.5}, PlotPoints ->20, Axes -> False]

γc={
0->{0,0,0,0.1`},
0.8`->{0.8`,0,0,0.1`},
1->{1,0,0,0.2`},
2->{2,0.5`,0,0.1`},
3->{3,0,0,0.1`},
4->{1,0,0,0.1`},
5->{2,0.5`,0,0.1`},
6->{3,0,0,0.1`},
6.2`->{3.2`,0,0,0.1`},
7->{4,0,0,0.1`}
};
(Subscript[γ, i_]) := Interpolation[{#[[1]], #[[2,i]]}& /@ γc];

(Subscript[γ, 1])

ParametricPlot[{(Subscript[γ, 1])[t], (Subscript[γ, 2])[t]}, {t,0,7}]