Plotting arrows at the edges of a curve

Question

Inspired by this question at ask.sagemath, what is the best way of adding arrows to the end of curves produced by Plot, ContourPlot

Answer 1

The following seems to work, by sorting the segments first:

f[x_] := {E^-x^2, Sin[10 x], Sign[x], Tan[x], UnitBox[x], 
             IntegerPart[x], Gamma[x],
             Piecewise[{{x^2, x < 0}, {x, x > 0}}], {x, x^2}}; 

arrowPlot[f_] := 
 Plot[{#}, {x, -2, 2}, Axes -> False, Frame -> True, PlotRangePadding -> .2] /.

 {Hue[qq__], a___, x___Line} :> {Hue[qq], a, SortBy[{x}, #[[1, 1, 1]] &]} /. 

 {a___,{Line[x___], d___, Line[z__]}} :> 
                           List[Arrowheads[{-.06, 0}], a, Arrow[x], {d}, 
                                             Arrowheads[{0, .06}], Arrow[z]] /. 

 {a___,{Line[x__]}}:> List[Arrowheads[{-.06, 0.06}], a, Arrow[x]] & /@ f[x];  

arrowPlot[f]

Answer 2

The following construct has the advantage of not messing with the internal structure of the Graphics structure, and is more general than the one suggested in ask.sagemath, as it manage PlotRange and infinities better.

f[x_] = Gamma[x]

{plot, evals} = 
  Reap[Plot[f[x], {x, -2, 2}, Axes -> False, Frame -> True, 
    PlotRangePadding -> .2, EvaluationMonitor :> Sow[{x, f[x]}]]];

{{minX, maxX}, {minY, maxY}} = Options[plot, PlotRange] /. {_ -> y_} -> y; 
ev = Select[evals[[1]], minX <= #[[1]] <= maxX && minY <= #[[2]] <= maxY &];
seq = SortBy[ev, #[[1]] &];
arr = {Arrow[{seq[[2]], seq[[1]]}], Arrow[{seq[[-2]], seq[[-1]]}]};

Show[plot, Graphics[{Red, arr}]]

Edit

As a function:

arrowPlot[f_, interval_] := Module[{plot, evals, within, seq, arr},
   within[p_, r_] :=
    r[[1, 1]] <= p[[1]] <= r[[1, 2]] &&
     r[[2, 1]] <= p[[2]] <= r[[2, 2]];

   {plot, evals} = Reap[
     Plot[f[x], Evaluate@{x, interval /. List -> Sequence},
      Axes -> False,
      Frame -> True,
      PlotRangePadding -> .2,
      EvaluationMonitor :> Sow[{x, f[x]}]]];

   seq = SortBy[Select[evals[[1]],
      within[#, 
        Options[plot, PlotRange] /. {_ -> y_} -> y] &], #[[1]] &];

   arr = {Arrow[{seq[[2]], seq[[1]]}], Arrow[{seq[[-2]], seq[[-1]]}]};
   Show[plot, Graphics[{Red, arr}]]
   ];

arrowPlot[Gamma, {-3, 4}]  

Still thinking what is better for ListPlot & al.

Answer 3

Inspired by both Alexey's comment and belisarius's answers, here's my attempt.

makeArrowPlot[g_Graphics, ah_: 0.06, dx_: 1*^-6, dy_: 1*^-6] := 
 Module[{pr = PlotRange /. Options[g, PlotRange], gg, lhs, rhs},
  gg = g /. GraphicsComplex -> (Normal[GraphicsComplex[##]] &);
  lhs := Or@@Flatten[{Thread[Abs[#[[1, 1, 1]] - pr[[1]]] < dx], 
                      Thread[Abs[#[[1, 1, 2]] - pr[[2]]] < dy]}]&;
  rhs := Or@@Flatten[{Thread[Abs[#[[1, -1, 1]] - pr[[1]]] < dx], 
                      Thread[Abs[#[[1, -1, 2]] - pr[[2]]] < dy]}]&;
  gg = gg /. x_Line?(lhs[#]&&rhs[#]&) :> {Arrowheads[{-ah, ah}], Arrow@@x};
  gg = gg /. x_Line?lhs :> {Arrowheads[{-ah, 0}], Arrow@@x};
  gg = gg /. x_Line?rhs :> {Arrowheads[{0, ah}], Arrow@@x};
  gg
  ]

We can test this on some functions

Plot[{x^2, IntegerPart[x], Tan[x]}, {x, -3, 3}, PlotStyle -> Thick]//makeArrowPlot

And on some contour plots

ContourPlot[{x^2 + y^2 == 1, x^2 + y^2 == 6, x^3 + y^3 == {1, -1}}, 
   {x, -2, 2}, {y, -2, 2}] // makeArrowPlot

One place where this fails is where you have horizontal or vertical lines on the edge of the plot;

Plot[IntegerPart[x],{x,-2.5,2.5}]//makeArrowPlot[#,.03]&

This can be fixed by options such as PlotRange->{-2.1,2.1} or Exclusions->None.

Finally, it would be nice to add an option so that each "curve" can arrow heads only on their boundaries. This would give plots like those in Belisarius's answer (it would also avoid the problem mentioned above). But this is a matter of taste.