# Calculating the sum of the lengths and plotting all curves on the same map

I have a curve like below;

ParametricPlot[
FromPolarCoordinates[{Exp[(t + 0)*0.5], t}] // Evaluate, {t, 0,
Pi - 0}, PlotRange -> All] 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] 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. 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: What I need is something like this: And the sum of those curves in the circle area.

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

• 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]] &? Feb 17, 2019 at 13:47
• @kglr nope. I editted my answer and showed what your solution is like. Feb 17, 2019 at 13:47
• @corey979 Yes! Yes! Exactly like that one and the sum of the curves' length within the circle Feb 17, 2019 at 13:49
• @corey979 However, your solution is not the rotation case. It is the case where radial growth of the spiral increases. Feb 17, 2019 at 13:57
• @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 Feb 17, 2019 at 14:07

ctr = {3, 0};
pp = PolarPlot[Evaluate@Table[Exp[(t + Pi*i/50)*0.5], {i, 1, 100}], {t, -Pi, Pi},
RegionFunction -> (Norm[{#, #2} - ctr] <= radius &)] 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] Total[ArcLength /@ Cases[pp2, _Line, All]]


314.511