Filling a 2D area between 3 curves (Hypocycloid)

Hello!

My task is to plot an usual hypocycloide that is given in parametric form and to fill a specific area (have a look at the pictures). The plotting of the hypocycloide (including the big fix circle and the small rolling circle) is no problem, but the filling of the area is a problem for me. I tried a lot myself using the area, fill and patch function, but I could not receive a result.

The code is the following:

syms R r phi 


%radius big (fix) circle
R=4;
%radius small (rolling) circle
r=1;
%parametric form hypocyloide (point at the rolling circle)
x(phi)=(R-r)*cos(phi)+r*cos(phi*((R-r)/r))
y(phi)=(R-r)*sin(phi)-r*sin(phi*((R-r)/r))

%parametric form big (fix) circle
x1(phi)=R*cos(phi);
y1(phi)=R*sin(phi);

%these points should give the position of the rolling circle at a specfic
%phi value (here: phi = 0.25*pi)
mx=(R-r)*cos(0.25*pi);
my=(R-r)*sin(0.25*pi);
%parametric form small (rolling) circle
x2=mx+x1*r/R;
y2=my+y1*r/R;

%plotting of hypocycloide, bigger circle, smaller circle 
fplot(x,y,[0 2*pi])
hold on
fplot(x1,y1,[0 2*pi])
fplot(x2,y2,[0 2*pi])
hold off

This is how the plot looks like:

And this is the area that I want to fill (in grey). How can I fill it?

I hope that someone can help me.

Looking forward to some hints,

Mick

Answer 1

[Matt J]

%radius big (fix) circle
R=4;
%radius small (rolling) circle
r=1;

phi=linspace(0,360,1000)*pi/180; phi(end)=[];

%parametric form hypocyloide (point at the rolling circle)
x=(R-r)*cos(phi)+r*cos(phi*((R-r)/r));
y=(R-r)*sin(phi)-r*sin(phi*((R-r)/r));

%parametric form big (fix) circle
x1=R*cos(phi);
y1=R*sin(phi);

%these points should give the position of the rolling circle at a specfic
%phi value (here: phi = 0.25*pi)
mx=(R-r)*cos(0.25*pi);
my=(R-r)*sin(0.25*pi);
%parametric form small (rolling) circle
x2=mx+x1*r/R;
y2=my+y1*r/R;

%plotting of hypocycloide, bigger circle, smaller circle 
plot(x,y)
hold on
plot(x1,y1)
plot(x2,y2)
hold off

hcyc=polyshape(x,y);
circ1=polyshape(x1,y1);
circ2=polyshape(x2,y2);

reg=regions(  polybuffer( subtract(circ1,union(hcyc,circ2)) ,-0.001 ));
hold on
plot(reg(3),'FaceColor',[1,1,1]/2)
hold off
axis equal,axis([-4 +4 -4 +4]),

[Mick Raer]

Thank you!

[Mick Raer]

Do you know how to calcualte the area units of the grey area?

[Matt J]

Easy:

area(reg(3))

[Mick Raer]

Perfect! Thank u so much!

[Matt J]

For a more accurate area calculation, you should probably undo the polybuffer operation,

reg3=polybuffer(reg(3),0.001);

Area=area(reg3),