Placing a smooth curve inside the tube

Question

How can I draw a smooth curve with a given start and end point and limit it to being inside a piecewise-linear pipe, as shown below?

_{(source: yaroslavvb.com )}

coords = {1 -> {0, 2}, 2 -> {1/3, 1}, 3 -> {0, 0}, 
   4 -> {(1/3 + 2)/2, 1}, 5 -> {2, 1}, 6 -> {2 + 1/3, 0}, 
   7 -> {2 + 1/3, 2}};
gp = GraphPlot[graph, VertexCoordinateRules -> coords];
pr = {{-1, 3 + 1/3}, {-1 - 1/6, 3 + 1/6}};
scale = 50;
is = -scale*(Subtract @@@ pr);
lineThickness = 2/3;
graph = {1 -> 2, 3 -> 2, 2 -> 4, 4 -> 5, 5 -> 6, 5 -> 7};
path = {3, 2, 4, 5, 7};
lp = Graphics[{Blue, Opacity[.5], 
    AbsoluteThickness[lineThickness*scale], Line[path /. coords]}];
Show[lp, gp, PlotRange -> pr, ImageSize -> is]

Answer 1

Maybe something like this:

coords = {2 -> {1/3, 1}, 1 -> {0, 0}, 3 -> {(1/3 + 2)/2, 1}, 
   4 -> {2, 1}, 5 -> {2 + 1/3, 2}};
pr = {{-1, 3 + 1/3}, {-1 - 1/6, 3 + 1/6}};
scale = 50;
is = -scale*(Subtract @@@ pr);
lineThickness = 2/3;
graph = {1 -> 2, 2 -> 3, 3 -> 4, 4 -> 5};
gp = GraphPlot[graph, VertexCoordinateRules -> coords];
path = {1, 2, 3, 4, 5};

f = BezierFunction[
   SortBy[coords /. Rule[x_, List[a_, b_]] -> List[a, b], First]];
pp = ParametricPlot[f[t], {t, 0, 1}];

lp = Graphics[{Blue, Opacity[.5], 
    AbsoluteThickness[lineThickness*scale], Line[path /. coords]}];
Show[pp, lp, gp, PlotRange -> pr, ImageSize -> is]  

You can get better control over the path by adding / removing breakpoints for Bezier. As I recall, “Bspline is contained in the convex hull of its control points,” so you can add control points inside your thick lines (for example, up and down midpoints in the actual set of points) to link Bezier more and more.

Edit

- . , , :

coords = {2 -> {1/3, 1}, 1 -> {0, 0}, 3 -> {(1/3 + 2)/2, 1}, 
   4 -> {2, 1}, 5 -> {2 + 1/3, 2}};
pr = {{-1, 3 + 1/3}, {-1 - 1/6, 3 + 1/6}};
scale = 50;
is = -scale*(Subtract @@@ pr);
lineThickness = 2/3;
graph = {1 -> 2, 2 -> 3, 3 -> 4, 4 -> 5};
gp = GraphPlot[graph, VertexCoordinateRules -> coords];
path = {1, 2, 3, 4, 5};

kk = SortBy[coords /. Rule[x_, List[y_, z_]] -> List[y, z], 
  First]; f = BezierFunction[kk];
pp = ParametricPlot[f[t], {t, 0, 1}, Axes -> False];

mp = Table[{a = (kk[[i + 1, 1]] - kk[[i, 1]])/2 + kk[[i, 1]],
    Interpolation[{kk[[i]], kk[[i + 1]]}, InterpolationOrder -> 1][
      a] + lineThickness/2}, {i, 1, Length[kk] - 1}];
mp2 = mp /. {x_, y_} -> {x, y - lineThickness};
kk1 = SortBy[Union[kk, mp, mp2], First]
g = BezierFunction[kk1];
pp2 = ParametricPlot[g[t], {t, 0, 1}, Axes -> False];

lp = Graphics[{Blue, Opacity[.5], 
    AbsoluteThickness[lineThickness*scale], Line[path /. coords]}];
Show[pp, pp2, lp, gp, PlotRange -> pr, ImageSize -> is]

2

, , :

g1 = Graphics[BSplineCurve[kk1]]; 
Show[lp, g1, PlotRange -> pr, ImageSize -> is]    

, ( )