How to Plot a Bézier Curve for a specified Control Polygon with a restricted interval

Question

I am having some confusion in plotting a Bézier curve for the specified interval $[a, b]$ for a given control point polygon. For example:

pts = {{0, 0, 0}, {1, 0, 0}, {1, 0, 1}, {1, 1, 1}, {1, 1, 0}, {0, 1, 
    0}}; (*my control Polygon*)
f = BezierFuntion[pts]

Show[Graphics3D[{PointSize[Medium], Red, Map[Point, pts]}], 
 Graphics3D[{Gray, Line[pts], Line[Transpose[pts]]}], 
 ParametricPlot3D[
  f[u, v], {u,a , b}, {v, a, b}]]

when I try this, I get various dimension mismatch errors. Can someone help me understand and plot the curve ?

Answer 1

Maybe

pts = {{0, 0, 0}, {1, 0, 0}, {1, 0, 1}, {1, 1, 1}, {1, 1, 0}, {0, 1, 0}};
f = BezierFunction[pts];
Graphics3D[{PointSize[Medium], Red, Point@pts, Thick, Gray, Line@pts, 
           Blue, ParametricPlot3D[f[u], {u, 0, 1}, {v, 0, 1}][[1]]}]