2
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I have a curve like below;

ParametricPlot[
 FromPolarCoordinates[{Exp[(t + 0)*0.5], t}] // Evaluate, {t, 0, 
  Pi - 0}, PlotRange -> All]

not rotated

This curve rotates around z axis. Therefore for a rotation of pi/2 gives something like this:

ParametricPlot[
 FromPolarCoordinates[{Exp[(t + Pi/2)*0.5], t}] // 
  Evaluate, {t, -Pi/2, Pi - Pi/2}, PlotRange -> All]

rotated spiral

What I want is to plot all the curves in a specified circle when it rotates with 3.6 degrees. And I want the sum of curves in the circle.

enter image description here

I tried this code but it didn't work:

ParametricPlot[Sum[HeavisideTheta[1 -
      ((Exp[(t + Pi*i/50)*0.5]*Cos[t] - 
           1)^2 + (Exp[(t + Pi*i/50)*0.5]*Sin[t])^2)]*
    FromPolarCoordinates[{Exp[(t + Pi*i/50)*0.5], t}], {i, 1, 100}] //
   Evaluate, {t, -Pi*i/50, Pi - Pi*i/50}, PlotRange -> All]

I also tried this code:

ParametricPlot[ Evaluate@Sum[ HeavisideTheta[ 1 - ((Exp[(t + Pi*i/50)*0.5]Cos[t] - 1)^2 + (Exp[(t + Pii/50)*0.5]Sin[t])^2)] FromPolarCoordinates[{Exp[(t + Pi*i/50)*0.5], t}], {i, 1, 100}], {t, -Pi, Pi}, PlotRange -> All]

But it gives something like below:

not OK

What I need is something like this:

OK

And the sum of those curves in the circle area.

How should I write the code to achieve what I want?

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  • 1
    $\begingroup$ Something like this: ParametricPlot[ Table[FromPolarCoordinates[{Exp[(t + 2 k Pi/100)*0.5], t}], {k, 1, 100, 5}], {t, -Pi, Pi}, PlotRange -> All, PlotStyle -> Red, RegionFunction -> Function[{x, y}, (x - 3)^2 + y^2 < 9]] // Show[#, ContourPlot[(x - 3)^2 + y^2 == 9, {x, 0, 6}, {y, -3, 3}, ContourStyle -> Blue]] &? $\endgroup$ – corey979 Feb 17 at 13:47
  • $\begingroup$ @kglr nope. I editted my answer and showed what your solution is like. $\endgroup$ – Alper91 Feb 17 at 13:47
  • $\begingroup$ @corey979 Yes! Yes! Exactly like that one and the sum of the curves' length within the circle $\endgroup$ – Alper91 Feb 17 at 13:49
  • $\begingroup$ @corey979 However, your solution is not the rotation case. It is the case where radial growth of the spiral increases. $\endgroup$ – Alper91 Feb 17 at 13:57
  • $\begingroup$ @corey979 I am sorry corey what you provide was right, if I know where my limits are. And this is the code I wanted like you suggested: ParametricPlot[ Table[FromPolarCoordinates[{Exp[(t + 2 k Pi/100)*0.5], t}], {k, 1, 100, 5}], {t, -Pi, Pi}, PlotRange -> All, PlotStyle -> Red, RegionFunction -> Function[{x, y}, (x - (Exp[Pi*0.5] + 1)/2)^2 + y^2 < ((Exp[Pi*0.5] - 1)/2)^2]] // Show[#, ContourPlot[(x - (Exp[Pi*0.5] + 1)/2)^2 + y^2 == ((Exp[Pi*0.5] - 1)/2)^2, {x, 0, 6}, {y, -3, 3}, ContourStyle -> Blue]] & However, I need also sum of the lengths $\endgroup$ – Alper91 Feb 17 at 14:07
8
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ctr = {3, 0};
radius = 3;
pp = PolarPlot[Evaluate@Table[Exp[(t + Pi*i/50)*0.5], {i, 1, 100}], {t, -Pi, Pi}, 
 RegionFunction -> (Norm[{#, #2} - ctr] <= radius &)]

enter image description here

Total[ArcLength /@ Cases[pp, _Line, All]]
(* or Total[RegionMeasure /@ Cases[pp, _Line, All]] *)

314.511

If ypu have to use ParametricPlot, you can do

pp2  = ParametricPlot[Evaluate[Table[E^(((j*Pi)/50 + t)/2) { Cos[t], Sin[t]}, {j, 1, 100}]], 
 {t, -Pi, Pi}, 
 RegionFunction -> (Norm[{#, #2} - ctr] <= radius &), PlotRange -> All]

enter image description here

 Total[ArcLength /@ Cases[pp2, _Line, All]]

314.511

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