# How to fill with color the space between plotted points?

In this program I am rotating a rhombus about the origin, in a normed non-Euclidean flat metric space :

$$||x||_n =(|x_1| {}^n+|x_2| {}^n){}^{1/n}$$ , with $$n=\frac{1}{3}$$

The rotation preserves radial distances from the origin to the points onto the rhombus and of course, forces to change its shape:

  n = 1/3; a = -Sqrt[3]; b = 1.5;
graph1 = Manipulate[ListPlot[
Table[{{(b*(Abs[Cos[θ]^n] + Abs[Sin[θ]^n])^(1/n)*Cos[θ + ϕ])/
(((-a)*Cos[θ] + Sin[θ])*(Abs[Cos[θ + ϕ]^n] + Abs[Sin[θ + ϕ]^n])^(1/n)),
(b*(Abs[Cos[θ]^n] + Abs[Sin[θ]^n])^(1/n)*Sin[θ + ϕ])/
(((-a)*Cos[θ] + Sin[θ])*(Abs[Cos[θ + ϕ]^n] + Abs[Sin[θ + ϕ]^n])^(1/n))},
{(b*(Abs[Cos[θ + Pi/2]^n] + Abs[Sin[θ + Pi/2]^n])^(1/n)*Cos[θ + Pi/2 + ϕ])/
((a*Cos[θ + Pi/2] + Sin[θ + Pi/2])*(Abs[Cos[θ + Pi/2 + ϕ]^n] +
Abs[Sin[θ + Pi/2 + ϕ]^n])^(1/n)),
(b*(Abs[Cos[θ + Pi/2]^n] + Abs[Sin[θ + Pi/2]^n])^(1/n)*Sin[θ + Pi/2 + ϕ])/
((a*Cos[θ + Pi/2] + Sin[θ + Pi/2])*(Abs[Cos[θ + Pi/2 + ϕ]^n] +
Abs[Sin[θ + Pi/2 + ϕ]^n])^(1/n))},
{((-b)*(Abs[Cos[θ + (2*Pi)/2]^n] + Abs[Sin[θ + (2*Pi)/2]^n])^(1/n)*
Cos[θ + (2*Pi)/2 + ϕ])/(((-a)*Cos[θ + (2*Pi)/2] + Sin[θ + (2*Pi)/2])*
(Abs[Cos[θ + (2*Pi)/2 + ϕ]^n] + Abs[Sin[θ + (2*Pi)/2 + ϕ]^n])^(1/n)),
((-b)*(Abs[Cos[θ + (2*Pi)/2]^n] + Abs[Sin[θ + (2*Pi)/2]^n])^(1/n)*
Sin[θ + (2*Pi)/2 + ϕ])/(((-a)*Cos[θ + (2*Pi)/2] + Sin[θ + (2*Pi)/2])*
(Abs[Cos[θ + (2*Pi)/2 + ϕ]^n] + Abs[Sin[θ + (2*Pi)/2 + ϕ]^n])^(1/n))},
{((-b)*(Abs[Cos[θ + (3*Pi)/2]^n] + Abs[Sin[θ + (3*Pi)/2]^n])^(1/n)*
Cos[θ + (3*Pi)/2 + ϕ])/((a*Cos[θ + (3*Pi)/2] + Sin[θ + (3*Pi)/2])*
(Abs[Cos[θ + (3*Pi)/2 + ϕ]^n] + Abs[Sin[θ + (3*Pi)/2 + ϕ]^n])^(1/n)),
((-b)*(Abs[Cos[θ + (3*Pi)/2]^n] + Abs[Sin[θ + (3*Pi)/2]^n])^(1/n)*
Sin[θ + (3*Pi)/2 + ϕ])/((a*Cos[θ + (3*Pi)/2] + Sin[θ + (3*Pi)/2])*
(Abs[Cos[θ + (3*Pi)/2 + ϕ]^n] + Abs[Sin[θ + (3*Pi)/2 + ϕ]^n])^(1/n))}},
{θ, 0, Pi/2, 0.005}], AspectRatio -> None, AxesOrigin -> {0, 0}, Axes -> False,
PlotStyle -> PointSize[0.005], PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}],
{ϕ, 0, 2*Pi, 0.01}]


My question is, how to fill the space inside the ever-changing shape of rhombus as it rotates, to make the graphic image more presentable?

I understand that this can be a very hard question to answer..

• We use  Catenate@Transpose@data to construct a cyclic path to construct a polygon to fill the region.
Clear["Global*"];
n = 1/3; a = -Sqrt[3]; b = 1.5;
data[ϕ_] =
Table[{{(b*(Abs[Cos[θ]^n] + Abs[Sin[θ]^n])^(1/n)*
Cos[θ + ϕ])/(((-a)*Cos[θ] +
Sin[θ])*(Abs[Cos[θ + ϕ]^n] +
Abs[Sin[θ + ϕ]^n])^(1/
n)), (b*(Abs[Cos[θ]^n] + Abs[Sin[θ]^n])^(1/n)*
Sin[θ + ϕ])/(((-a)*Cos[θ] +
Sin[θ])*(Abs[Cos[θ + ϕ]^n] +
Abs[Sin[θ + ϕ]^n])^(1/
n))}, {(b*(Abs[Cos[θ + Pi/2]^n] +
Abs[Sin[θ + Pi/2]^n])^(1/n)*
Cos[θ + Pi/2 + ϕ])/((a*Cos[θ + Pi/2] +
Sin[θ + Pi/2])*(Abs[Cos[θ + Pi/2 + ϕ]^n] +
Abs[Sin[θ + Pi/2 + ϕ]^n])^(1/
n)), (b*(Abs[Cos[θ + Pi/2]^n] +
Abs[Sin[θ + Pi/2]^n])^(1/n)*
Sin[θ + Pi/2 + ϕ])/((a*Cos[θ + Pi/2] +
Sin[θ + Pi/2])*(Abs[Cos[θ + Pi/2 + ϕ]^n] +
Abs[Sin[θ + Pi/2 + ϕ]^n])^(1/
n))}, {((-b)*(Abs[Cos[θ + (2*Pi)/2]^n] +
Abs[Sin[θ + (2*Pi)/2]^n])^(1/n)*
Cos[θ + (2*Pi)/2 + ϕ])/(((-a)*
Cos[θ + (2*Pi)/2] +
Sin[θ + (2*Pi)/2])*(Abs[
Cos[θ + (2*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (2*Pi)/2 + ϕ]^n])^(1/
n)), ((-b)*(Abs[Cos[θ + (2*Pi)/2]^n] +
Abs[Sin[θ + (2*Pi)/2]^n])^(1/n)*
Sin[θ + (2*Pi)/2 + ϕ])/(((-a)*
Cos[θ + (2*Pi)/2] +
Sin[θ + (2*Pi)/2])*(Abs[
Cos[θ + (2*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (2*Pi)/2 + ϕ]^n])^(1/
n))}, {((-b)*(Abs[Cos[θ + (3*Pi)/2]^n] +
Abs[Sin[θ + (3*Pi)/2]^n])^(1/n)*
Cos[θ + (3*Pi)/2 + ϕ])/((a*
Cos[θ + (3*Pi)/2] +
Sin[θ + (3*Pi)/2])*(Abs[
Cos[θ + (3*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (3*Pi)/2 + ϕ]^n])^(1/
n)), ((-b)*(Abs[Cos[θ + (3*Pi)/2]^n] +
Abs[Sin[θ + (3*Pi)/2]^n])^(1/n)*
Sin[θ + (3*Pi)/2 + ϕ])/((a*
Cos[θ + (3*Pi)/2] +
Sin[θ + (3*Pi)/2])*(Abs[
Cos[θ + (3*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (3*Pi)/2 + ϕ]^n])^(1/
n))}}, {θ, 0, Pi/2, 0.005}];
Manipulate[
Module[{pts}, pts = data[ϕ];
ListPlot[pts,
Prolog -> {Opacity[.2], Green,
Polygon[Catenate@Transpose@pts]}]], {ϕ, 0, 2*Pi, 0.01}]


\$Version

(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)

Clear["Global*"]

n = 1/3; a = -Sqrt[3]; b = 3/2;

graph1 = Manipulate[
ListLinePlot[
Evaluate[
Table[#, {θ, 0, Pi/2,
0.005}] & /@ {{(b*(Abs[Cos[θ]^n] +
Abs[Sin[θ]^n])^(1/n)*
Cos[θ + ϕ])/(((-a)*Cos[θ] +
Sin[θ])*(Abs[Cos[θ + ϕ]^n] +
Abs[Sin[θ + ϕ]^n])^(1/
n)), (b*(Abs[Cos[θ]^n] + Abs[Sin[θ]^n])^(1/
n)*Sin[θ + ϕ])/(((-a)*Cos[θ] +
Sin[θ])*(Abs[Cos[θ + ϕ]^n] +
Abs[Sin[θ + ϕ]^n])^(1/
n))}, {(b*(Abs[Cos[θ + Pi/2]^n] +
Abs[Sin[θ + Pi/2]^n])^(1/n)*
Cos[θ + Pi/2 + ϕ])/((a*Cos[θ + Pi/2] +
Sin[θ + Pi/2])*(Abs[
Cos[θ + Pi/2 + ϕ]^n] +
Abs[Sin[θ + Pi/2 + ϕ]^n])^(1/
n)), (b*(Abs[Cos[θ + Pi/2]^n] +
Abs[Sin[θ + Pi/2]^n])^(1/n)*
Sin[θ + Pi/2 + ϕ])/((a*Cos[θ + Pi/2] +
Sin[θ + Pi/2])*(Abs[
Cos[θ + Pi/2 + ϕ]^n] +
Abs[Sin[θ + Pi/2 + ϕ]^n])^(1/
n))}, {((-b)*(Abs[Cos[θ + (2*Pi)/2]^n] +
Abs[Sin[θ + (2*Pi)/2]^n])^(1/n)*
Cos[θ + (2*Pi)/2 + ϕ])/(((-a)*
Cos[θ + (2*Pi)/2] +
Sin[θ + (2*Pi)/2])*(Abs[
Cos[θ + (2*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (2*Pi)/2 + ϕ]^n])^(1/
n)), ((-b)*(Abs[Cos[θ + (2*Pi)/2]^n] +
Abs[Sin[θ + (2*Pi)/2]^n])^(1/n)*
Sin[θ + (2*Pi)/2 + ϕ])/(((-a)*
Cos[θ + (2*Pi)/2] +
Sin[θ + (2*Pi)/2])*(Abs[
Cos[θ + (2*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (2*Pi)/2 + ϕ]^n])^(1/
n))}, {((-b)*(Abs[Cos[θ + (3*Pi)/2]^n] +
Abs[Sin[θ + (3*Pi)/2]^n])^(1/n)*
Cos[θ + (3*Pi)/2 + ϕ])/((a*
Cos[θ + (3*Pi)/2] +
Sin[θ + (3*Pi)/2])*(Abs[
Cos[θ + (3*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (3*Pi)/2 + ϕ]^n])^(1/
n)), ((-b)*(Abs[Cos[θ + (3*Pi)/2]^n] +
Abs[Sin[θ + (3*Pi)/2]^n])^(1/n)*
Sin[θ + (3*Pi)/2 + ϕ])/((a*
Cos[θ + (3*Pi)/2] +
Sin[θ + (3*Pi)/2])*(Abs[
Cos[θ + (3*Pi)/2 + ϕ]^n] +
Abs[Sin[θ + (3*Pi)/2 + ϕ]^n])^(1/n))}}],
Axes -> False,
PlotStyle -> style,
AspectRatio -> 1,
Filling -> Axis,
PlotRange -> {{-4, 4}, {-4, 4}}],
{{ϕ, 1}, 0, 2*Pi, 0.01, Appearance -> "Labeled"},
{{style, ColorData[97][1]}, {ColorData[97][1],
Automatic -> "Multi"}},
TrackedSymbols :> {ϕ, style}]