Skip to main content

plotting - Continuously varying tube radius


I'm currently trying to draw tubular neighborhoods of torus knots, which Mathematica's Tube function allows me to do quite easily. My question regards the appearance of the neighborhood: is there any way to use an explicit function to continuously specify the radius of the tube? I've managed to find a few examples with nonconstant radii, but nothing where it varies continuously.


I did manage to find enough examples to figure out how draw these tubular neighborhoods so that they are colored according to an explicit function. If possible, I would like the radius to correspond to the color at every point on the curve. Here's what I've got so far:


Clear[γ, t, w, wColor, wmin, wmax]

(*Define a torus knot γ and a weight function w*)
γ[t_] = {(2 + Cos[3 t]) Cos[2 t], (2 + Cos[3 t]) Sin[2 t], Sin[3 t]};

w[t_] = 2 + Cos[t];

(*All this nonsense makes red the heaviest and blue the lightest*)
wmin = First[FindMinimum[{w[t], 0 <= t <= 2 π}, {t, .1}]];
wmax = First[FindMaximum[{w[t], 0 <= t <= 2 π}, {t, .1}]];
wColor[t_] = (7/10)*(1 - ((w[t] - wmin)/(wmax - wmin)));

ParametricPlot3D[γ[t], {t, 0, 2 π + .01}, 
   ColorFunction -> Function[{x, y, z, t}, Hue[wColor[t]]], 
   ColorFunctionScaling -> False, 

   PlotStyle -> Directive[Opacity[.7], CapForm[None]], 
   PlotRange -> All, Boxed -> False, 
   MaxRecursion -> 0, 
   PlotPoints -> 100, 
   Axes -> None, 
   Method -> {"TubePoints" -> 30}] /. 
   Line[pts_, rest___] -> Tube[pts, 0.2, rest]

In short, I would like to continuously vary the radius of this tube:


colored tube




Answer



You can take a continuous function and evaluate it at the same points that are also used by ParametricPlot3D to create the curve. Here is a way to do it:


rr = Reap[
   ParametricPlot3D[γ[t], {t, 0, 2 Pi + .01}, 
    ColorFunction -> 
     Function[{x, y, z, t}, Hue[wColor[Sow[t, "tValues"]]]], 
    ColorFunctionScaling -> False, 
    PlotStyle -> Directive[Opacity[.7], CapForm[None]], 
    PlotRange -> All, Boxed -> False, MaxRecursion -> 0, 
    PlotPoints -> 100, Axes -> None, Method -> {"TubePoints" -> 30}], 

   "tValues"];
rr[[1]] /. 
 Line[pts_, rest___] :> Tube[pts, 0.2 + .1 Sin[rr[[2]]], rest]

plot


Here I chose a Sin[t] variation of the thickness. To do it, I collect the evaluation points from inside ParametricPlot3D using Sow and Reap.


This list of points is in rr[[2]], whereas rr[[1]] is the plot itself. Then I modify the replacement rule you already had by making the radii of Tube into a list obtained by applying the desired continuous function to rr[[2]].


Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more