Create combined polar coordinates function

Hello, I have been trying to solve for what angles my function r in polar coordinates cuts the unit circle and use the functions below to plot the solution. Now I am trying to come up with a function that given the input of a angle (x) gives the corresponding y value of the difference of the function and the unit circle so that I can calculate an exact solution with fzero. I am very confused on how to do this and been stuck on this for a while. Thanks in advance for any help.

%My function on a seperate script

function [x,y]=polarcoord(f)

theta=linspace(0,2*pi);

x=f(theta).*cos(theta);

y=f(theta).*sin(theta);

end

%Main script

r=@(x)(2+sin(3*x))./sqrt(1+exp(cos(x)));

[x,y]=polarcoord(r); %Main function

[X,Y]=polarcoord(@(x) 1); %Unit circle

Answer 1

[Mathieu NOE]

hello

it seems to me there is a straightforward solution to your problem by using this Fex submission :

Curve intersections - File Exchange - MATLAB Central (mathworks.com)

%Main script
r=@(x)(2+sin(3*x))./sqrt(1+exp(cos(x)));
[x,y]=polarcoord(r); %Main function
[X,Y]=polarcoord(@(x) 1); %Unit circle

P = InterX([x;y],[X;Y]);
plot(x,y,'-*',X,Y,'-*',P(1,:),P(2,:),'ro')
axis square

%My function on a seperate script
function [x,y]=polarcoord(f) 
    theta=linspace(0,2*pi,100);
    x=f(theta).*cos(theta);
    y=f(theta).*sin(theta);
end

NB the crossing points coordonates are interpolated so the angular resolution of your x,y, X,Y data is not (too) critical

zoomed plot :

[Erik]

Thanks for the quick response, however my question is more of how to create the general function as supposed to find the solutions and if I understand InterX properly all it helps with is find the intersecting points. Although this method is interesting.

Do you have any other ideas on how to proceed?

[Mathieu NOE]

This is another way to do it without the Fex InterX

simply look for the zero crossing values (theta) of your radius function minus the radius 1 circle.

this is ploted in figure 1

the cartesian plot is in fig 2

%Main script

theta=linspace(0,2*pi,100);
r=@(x)(2+sin(3*x))./sqrt(1+exp(cos(x)));

[x,y]=pol2cart(theta,r(theta)); %Main function
[X,Y]=pol2cart(theta,1);  %Unit circle

thetac = zero_crossing(theta,r(theta)-1);
[xc,yc]=pol2cart(thetac,r(thetac)); % zero crossing points

figure(1),plot(theta,r(theta)-1,thetac,zeros(size(thetac)),'ro')
title('r(theta)-1')
xlabel('Theta (rad)');
ylabel('amplitude');

figure(2),plot(x,y,'-*',X,Y,'-*',xc,yc,'ro')
axis square

function xc = zero_crossing(x,y)
% zero crossing with linear interpolation
    i = find(y(1:end-1).*y(2:end)<0); % index of zero crossings
    m = (y(i+1)-y(i))./(x(i+1)-x(i)); % slope
    xc =-y(i)./m+x(i);                % x coordinates of zero crossings linear interpolated

end