# Tracing the existing curve with star symbol

b1={1.8743,1.8784,1.88248,1.89049,1.89828,1.90587,1.91327,1.96335,2.03035,2.12536,2.23701,2.30098,2.34255,2.37175,2.3934,2.42334,2.44307,2.48725,2.51208,2.5173,2.52799,2.53164,2.53348,2.53533,2.53625,2.53745,2.53793,2.53894,2.53909,2.53543};
ks1={0.01,1.,2.,4.,6.,8.,10.,25.,50.,100.,200.,300.,400.,500.,600.,800.,1000.,2000.,4000.,5000.,10000.,15000.,20000.,30000.,40000.,70000.,100000.,1.*10^6,1.*10^8,1.*10^12};
data1 = Transpose[{ks1, b1}];
s1 = ListLogLinearPlot[data1, Joined -> True,
PlotStyle -> {Black, Thickness[0.01]}, AxesStyle -> Black,
PlotRange -> All]


How to put star symbol on same plot which traces the curve. I know with the Plot marker we can achive this, but the thing is I want the tracing at equidistance from start to end. With my data points, I am not getting the tracing properly. How to achive this.

• "I want the tracing at equidistance from start to end." - you did quite a bit of a jump here. You have data points, but placing a star at each of these data points is not what you want? What exactly do you mean by "equidistance", distance along the curve that interpolates your data points? – J. M. is away Jan 5 at 14:01
• I mean I want to trace the curve which is formed by a data points, between the data the line is traced. now all I have is a line. I want to place star from start to end with placement being at an equal distance – acoustics Jan 5 at 14:06
• I meant to say just treat the curve as a contious curve and I wanted to mark the star from start to end – acoustics Jan 5 at 14:15
• A related question. – J. M. is away Jan 5 at 14:31

One can use the undocumented MeshFunctions -> {"ArcLength"} option setting to place equispaced (by the arclength of the interpolating curve) markers:

Normal[ListLogLinearPlot[Transpose[{ks1, b1}], Joined -> True,
Mesh -> 6, MeshFunctions -> {"ArcLength"}, MeshStyle -> Red,
PlotStyle -> {Black, Thickness[0.01]}]] /.
Point[pt_] :> Inset[Style["\[FivePointedStar]", Large], pt]


• Can we make the star unfilled with colour – acoustics Jan 5 at 14:33
• Yes, just replace "\[FivePointedStar]" with "☆". Or, take your pick among the symbols listed here. – J. M. is away Jan 5 at 14:37
• Hm. I cannot help but to me, this does not look equidistantly spaced with respect to arclength... – Henrik Schumacher Jan 5 at 16:32
• @Henrik, I will try to investigate tomorrow, as I have just borrowed a friend's computer to write some answers (and check on you guys ;)), and I have to go back home. The problem is that it's not immediately clear how ListLogLinearPlot[] is interpolating the points (and it wasn't the cubic interpolant when I did a check before posting). – J. M. is away Jan 5 at 16:46

As Henrik Schumacher correctly points out in the comments, the stars on the plot produced by the naive application of the undocumented MeshFunctions -> {"ArcLength"} option

do not look equidistantly spaced with respect to arclength

It took me a while to figure out what happens. The reason is that scales in the vertical and horizontal directions of the plot differ by an order of magnitude. I tried other plotting functions like ListPlot, Plot, ParametricPlot and found that all of them also suffer from this issue.

The following approach solves the problem almost exactly. Small inaccuracy comes from the fact that I use AspectRatio of the whole graphics instead of the actual aspect ratio of the plotting range with PlotRangePadding included. But visually it is hardly noticeable. From the other hand, it isn't too difficult to improve this approach for obtaining exact result.

We start by defining our data1:

b1={1.8743,1.8784,1.88248,1.89049,1.89828,1.90587,1.91327,1.96335,2.03035,2.12536,2.23701,2.30098,2.34255,2.37175,2.3934,2.42334,2.44307,2.48725,2.51208,2.5173,2.52799,2.53164,2.53348,2.53533,2.53625,2.53745,2.53793,2.53894,2.53909,2.53543};
ks1={0.01,1.,2.,4.,6.,8.,10.,25.,50.,100.,200.,300.,400.,500.,600.,800.,1000.,2000.,4000.,5000.,10000.,15000.,20000.,30000.,40000.,70000.,100000.,1.*10^6,1.*10^8,1.*10^12};
data1 = Transpose[{ks1, b1}];


Now define future AspectRatio of the plot and log-transformed data1 (the Graphics produced by ListLogLinearPlot contains log-transformed data: it is just Ticks what creates the illusion of logarithmic plot):

aspectRatio = 1./GoldenRatio;
data1Log = {Log@#1, #2} & @@@ data1;


Now create rescaling function and its inverse:

t = RescalingTransform[
MinMax /@ Transpose[data1Log], {{0, 1}, {0, aspectRatio}}];
tInv = InverseFunction[t];


This function makes vertical and horizontal scales almost equal (if necessary one can improve it for obtaining exact result).

Then we use MeshFunctions -> {"ArcLength"} for generating the equidistantly spaced points with respect to arclength:

s2 = ListLinePlot[t[data1Log], PlotRange -> All,
MeshFunctions -> {"ArcLength"}, Mesh -> True];


Extracting the equidistant points and performing the inverse coordinate transform:

pts = tInv @@@ Cases[Normal[s2], _Point, -1];


Now we can use these coordinates for placing the stars in Epilog:

ListLogLinearPlot[data1, Joined -> True,
PlotStyle -> {Black, Thickness[0.01]}, AxesStyle -> Black, PlotRange -> All,
Epilog -> {Red, Translate[Inset[Style["☆", 15, Bold], {0, 0}], pts]},
AspectRatio -> aspectRatio]


Voila! :)

UPDATE: I just accidentally found that I knew about this feature of MeshFunctions -> {"ArcLength"} already several years ago but completely forgot due to non-use of this functionality. I can add to that old post that now we have the wonderful GraphicsInformation function by Carl Woll which simplifies things considerably.