How to continuously control line thickness in plotting functions?

Questions
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1

I know that I can control color and opacity of lines in plotting functions – like in the example below. I also would like the line to change its thickness smoothly as we move in the positive x-direction. Is it possible to do via options to Plot? If not, - what could be the easiest approaches?

Plot[Sin[x^2], {x, -3, 3}, PlotStyle -> Thickness[.04], PlotPoints -> 50, 
  ColorFunction -> Function[{x, y}, Directive[ColorData["Rainbow"][x], Opacity[Sin[10 y + 15 x]]]]]

530×342 48.1 KB

2

Not a general solution, but this works some times:

ParametricPlot3D[{{x, Sin[x^2], x}, {x, Cos[x^2], x}}, {x, 0 - 3, 3},
PlotStyle -> {{Opacity[0.5], Tube[x/2000]}, {Opacity[0.3], Tube[x/2000]}},
ViewPoint -> {0, 0, 1000}, PlotRange -> All, ColorFunction -> "Rainbow"]

667×263 49 KB

npts = 12;
ParametricPlot3D[{{Cos[2 t], Sin[2 t], Cos}, {Sin[2 t], Cos[2 t], Sin}}, {t, 0, 2 Pi},
  PlotStyle -> {{Red, Tube[{0.1 Sin[t/(2 Pi)], 0.1}, PlotPoints -> npts, Frame -> "Frenet"]},
                {Yellow, Tube[{0.1 Sin[t/(2 Pi)], 0.1}, Frame -> "Frenet", PlotPoints -> npts]}},
  PlotRange -> All, MaxRecursion -> 0, PlotPoints -> 200]

568×624 75.1 KB

3

It crossed my mind that in some situations we can take advantage of Filling:

Table[Plot[Evaluate[
  Flatten[{#, # + amp (x + 3.5), # - amp (x + 3.5)} & /@ {Cos[x^2], .3 Sin[10 x]}]], {x, -3, 3},
  PlotPoints -> 50, Filling -> {3 -> {2}, 5 -> {6}}, PlotStyle -> {None, Gray, Gray, None, Gray, Gray},
  ColorFunction -> Function[{x, y}, Directive[ColorData["Rainbow"][x], Opacity[x]]],
  AspectRatio -> 1/4, ImageSize -> 500], {amp, .005, .15, .05}]

761×603 95.4 KB

The controlled thickness is only vertical in this case. It will not work in some other cases, for example ParametricPlot does not have Filling option at all. But, in ParametricPlot case, for example, we can use the function itself, because it can show bounded areas:

ParametricPlot[{(v + 1) u Cos[u], (v + 1) u Sin[u]}, {u, 0, 6 Pi}, {v, 0.5, 1}, 
Mesh -> None, MaxRecursion -> 4, Axes -> False, BoundaryStyle -> GrayLevel[.9], 
 ColorFunction -> Function[{x, y, u, v}, Directive[ColorData["Rainbow"][u v], Opacity[v]]]]

557×496 97.8 KB

4

Using MeshShading will also work but not as good as ParametricPlot and using Filling but it doesn’t change the function to use and original input.

Plot[Sin[x^2], {x, -3, 3}, Mesh -> {Range[-3, 3, 6/100]}, MeshStyle -> None,
MeshShading -> (Directive[Hue[Rescale[#, {-3, 3}, {0, 1}]],
AbsoluteThickness[Rescale[#, {-3, 3}, {1, 20}]]] & /@Range[-3, 3, 6/100]), 
PlotRangePadding -> Scaled[0.05]]

5

Min, that is an awful abuse of Mesh. I like it. :smiley:

6

This whole thread demonstrates mostly for better the stupifying generality of Mathematica. Awesome.

7

Very interesting techniques to control the thickness of the curves. It well suited in animating the orbit of celestial objects. Thanks for this sharing.

x[t_] = 50 Cos[t]; y[t_] = 50 Sin[t]; z[t_] = 0 t;
H[t_, a_] = 
  ParametricPlot3D[{x[t], y[t], z[t]}, {t, a, a - 1.8 Pi}, 
   PlotRange -> {{-60, 60}, {-60, 60}, {-5, 5}},
   PlotStyle -> {Red, Opacity[0.6], Tube[t/200]}];
S[t_] = Graphics3D[Sphere[{x[t], y[t], z[t]}, 2]];
Animate[Show [H[t, a], S[a]], {a, 0, 2 Pi}]
8

Hi Mohammed,
Your code didn’t work at first trial, but with little modifications, it worked, hope this is what you intended to show

Open in Wolfram Cloud ↗
9

Hi Ahmed.
Of course there was the objective of the code. Thanks.