# Shading an area enclosed by a loop

I am plotting the folium of Descartes as a parametric plot, and trying to find the arclength and the area enclosed by the loop. I got both of them covered result-wise, but my professor wants us to shade the area enclosed by the loop, and bold the arclength(the whole loop). I have been trying to figure that out for quite some time now but just can't seem to get it to work.

1. One way I found out you can do that is by using RegionPlot[] and fill the parameters. In my case nothing is shaded just my plot is being printed. Link

2. I found this question on here Link but I didn't find it helpful, because some of the solutions include Filling which is not available for ParametricPlot[]

My code:

x[t_] := (12 t)/(1 + t^3)
y[t_] := (12 t^2)/(1 + t^3)


Vertical tangent lines

dir = {0, 1};
sol = Solve[Det[{{x'[t], y'[t]}, dir}] == 0, t, Reals];


Plot with the "shaded area" that doesn't work

Show[ParametricPlot[{x[t], y[t]}, {t, -100, 100},
PlotRange -> {{-10, 10}, {-10, 10}}, AspectRatio -> Automatic,
PlotStyle -> Red, AxesLabel -> {"x", "y"},
PlotLabel -> "Folium of Descartes",
BoundaryStyle -> Directive[Red, Thick]],
RegionPlot[
ParametricRegion[{x[t],
y[t]}, {{t, 0, Infinity(*sol = 0.793701*)}}],
PlotStyle -> LightBlue, BoundaryStyle -> None]]


It also wouldn't accept sol as an argument. I could be wrong but I think I am supposed to integrate from t = 0 to t = Infinity and not t = sol (0.739...) right?

As for the bolding part of the graph I was thinking about just using Style and Bold for an interval of values for t where they make a loop. Would that work?

Any kind of help is appreciated Thank you!

• It is impossible to find such loop since only when t->Infinity, the parametric curve go back to {0,0}. Limit[{x[t], y[t]}, t -> Infinity] Apr 9, 2023 at 2:29
• Yeah after playing around with this for some time I noticed that. Manipulate[ ParametricPlot[{x[t], y[t]}, {t, -100, 100}, PlotRange -> {{-10, 10}, {-10, 10}}, PerformanceGoal -> "Quality", AxesLabel -> {x, y}, Epilog -> {Red, PointSize -> .05, Point[{x[t1], y[t1]}]}], {t1, 0, 0.739}] Thank you Apr 9, 2023 at 2:31

maybe something like:

pp = ParametricPlot[{ x[t], y[t]}, {t, -100, 100},
PlotRange -> {{-10, 10}, {-10, 10}},
PlotPoints -> 400,
AspectRatio -> Automatic,
MeshFunctions -> {# #2 &},
Mesh -> {{0}},
MeshShading -> { Automatic, Green},
PlotStyle -> Red]


Normal @ pp /. {Green, l_Line} :>
{l, Polygon @@ l, Black,
Text[Style[Row[{"arc length: ", ArcLength@l}], 14], {1, 9}, Left],
Text[Style[Row[{"area: ", Area[Polygon @@ l]}], 14], {1, 7.5}, Left]}
`