Visualizing a complicated Region

Question

I was given the problem to find the area bounded by the polar curve

$$r=\frac{1}{2}+\frac{3}{2}\cos(\theta)$$

which looks like

                                      

To be clear, the region meant is the lighter of the two here

                                      

I know the integrals needed to actually find the area, and that really isn't the question anymore. Instead, visualization is. I believe that can define the region that I am interested in with the code block (a and b were defined for generality)

a=1/2;
b=3/2;
v={(a c+b c Cos[t])Cos[t],(a c+b c Cos[t])Sin[t]};
R1=ParametricRegion[v,{{c,0,1},{t,0,(2\[Pi])/3}}];
R2=ParametricRegion[v,{{c,0,1},{t,2\[Pi])/3,(4\[Pi])/3}}];
R3=ParametricRegion[v,{{c,0,1},{t,(4\[Pi])/3,2\[Pi]}}];

R=RegionDifference[RegionUnion[R1,R3],R3];

I can look at these individually, but when I use Region to visualize the whole thing, it doesn't return even a picture, just some internal expression. How can I visualize just the region of interest?

Answer 1

We can construct such a region with CrossingPolygon and in this instance Polygon will work too.

c = 1;
pts = Table[v, {t, Subdivide[0., 2π, 200]}];
pts[[-1]] = pts[[1]];

Graphics[{EdgeForm[ColorData[97][1]], Opacity[.3], ColorData[97][2], Polygon[pts]}]

CrossingCount[Polygon[pts], {0.5, 0}]

2

Answer 2

I think your plot is not quite correct, as the region should only extend to $x=2$ on the horizontal axis.

Here's how to plot it using RegionPlot:

RegionPlot[-1/2 + 3/2 x/Sqrt[x^2 + y^2] < Sqrt[x^2 + y^2] < 1/2 + 3/2 x/Sqrt[x^2 + y^2],
  {x, -1/2, 2}, {y, -5/4, 5/4}, PlotPoints -> 100]

The trick is to use $r=\sqrt{x^2+y^2}$ and $\cos(\theta)=x/r$.