How to create the waved style for curves in Graphics?

Question

There are the dashed and dotted styles for lines and circles when we plot them with Graphics. For example, Graphics[{Dashed, Line[{{1, 0}, {2, 1}}]}]. Is it possible to construct the line or circle with wave instead of dash or dot? That is, Graphics[{Waved, Line[{{1, 0}, {2, 1}}]}] will produce

and Graphics[{Waved, Circle[{0, 0}, 1]}] will produce

Answer 1

Not a complete answer.

You can make use of the undocumented function Typeset`MakeBoxes mentioned in this post. Here I'll just code waved line and circle as examples:

(* Stolen from Simon's post, notice the tiny modification. *)
SetAttributes[createPrimitive, HoldAll]
createPrimitive[patt_, expr_] := 
 Typeset`MakeBoxes[p : patt, fmt_, Graphics] := 
  With[{e = Cases[expr, Line[_], Infinity]}, 
   Typeset`MakeBoxes[Interpretation[e, p], fmt, Graphics]]

createPrimitive[Waved[a_, f_, pts_: Automatic]@Circle[p : {x0_, y0_} : {0, 0}, r0_: 1], 
 ParametricPlot[{x0 + Cos[t] (r0 + a Sin[f t]), y0 + Sin[t] (r0 + a Sin[f t])}, {t, 0, 
   2 Pi}, PlotPoints -> pts]]

createPrimitive[Waved[a_, f_, pts_: Automatic]@Line[p : {{_, _?NumericQ} ..}], 
 Module[{fx, fy, 
   distance = Prepend[Accumulate@Sqrt[Total@Transpose@((Rest@# - Most@# &@N@p)^2)], 0.], 
   normal}, {fx, fy} = 
   ListInterpolation[#, distance, InterpolationOrder -> 1] & /@ Transpose@N@p; 
  normal = Sqrt[fx'[t]^2 + fy'[t]^2]; 
  ParametricPlot[{fx@t + a Sin[f t] fy'[t]/normal, fy@t - a Sin[f t] fx'[t]/normal}, {t, 
    0, distance[[-1]]}, PlotPoints -> pts]]]

Usage:

Graphics[{Red, Thick, Waved[1/50, 40]@Line[{{1, 0}, {2, 1}, {3, -1}, {4, 0}}], Orange, 
  Waved[1/10, 50, 51]@Circle[{2.5, 0}, 3/2]}]

Remaining Issues

1. The achieved syntax is slightly different from the expected one, not sure if the expected syntax can be achieved with Typeset`MakeBoxes.

2. The waved style is coded separately for every graphics primitive, so creating a complete waved style still requires huge amount of work.

3. ParametricPlot is relatively slow.

4. The wave doesn't look great at corners:

    Graphics[{Red, Thick, Waved[1/10, 40]@Line[{{1, 0}, {2, 1}, {3, 0}}]}]
   

   

Answer 2

It is possible to make ondulations on a BSpline.

Here is a toy example with a closed BSpline ("closed" in order to see the continuity at the ends)

p={{15.7336, -3.557}, {11.1177, -2.53343}, {15.4259, 19.1467},
   {6.60292, 10.5131},{-28.5053, 10.9099}, {-22.7909, -1.35239},
   {-3.22756, -13.0483},{-17.1309, -32.426}, {6.23965, -7.05847},
   {25.0532, -25.0634}};

f = BSplineFunction[p, SplineClosed -> True];  

Show[ParametricPlot[f[x], {x, 0, 1}]]  

absCurv=NDSolveValue[{abcCurv'[x]==Norm[f'[x]],abcCurv[0]==0},{abcCurv},{x,0,1}][[1]];  

length=absCurv[1];

numberOfTurns = 50;

f1[x_]=f[x] - Sin[2 Pi numberOfTurns absCurv[x]/length]  {{0,1}, {-1,0}}.Normalize[f'[x]];

ParametricPlot[f1[x],{x,0,1},PlotPoints-> 1000]

inspiration source 1 (about {{0,1}, {-1,0}}.Normalize[f'[x]])
inspiration source 2 (about absCurv=NDSolveValue[...)