function Lemniscates
% "Lemniscates" plots one or more examples of each of the following: 
% The Lemniscate of Bernoulli, The Leminiscate of Gerono, The Lemniscate of
% Booth, The Cassinian Oval, and "The Inverse Lemniscate". Just choose an
% example from the drop-down menus and click on it. It's that easy!
% by Andrew Smith, Dec.8 2008 

% close all existing open figures
close all

% define the figure (the GUI itself)
Fig = figure('Units','Normalized',...
    'Position',[.6,.2,.68,.76],...
    'color',[.01 .035 .43],...
    'menubar','none',...
    'name','Lemniscates');

% define the axes
ha = axes('Units','Normalized',...
    'position',...
    [.12 .13 .76 .73],...
    'Box','on',...
    'Fontname','Lucida Sans',...
    'FontSize',12,...
    'FontWeight','d',...
    'Visible','off',...
    'XTick',[],'YTick',[],...
    'GridLineStyle','-.',...
    'Color',[.01 .435 .63]);

% the title text
TitleText = uicontrol('style','text',...
    'Units','Normalized',...
    'pos',[.078,.89,.81,.04],...
    'Fontsize',17,...
    'Fontweight','b',...
    'string','The Lemniscates of Bernoulli, Gerono, Booth, and Cassini',...
    'horiz','right',...
    'FontName','Bookman Old Style',...
    'Background',[.01 .035 .43],...
    'visible','on',...
    'foregroundcolor','y');

% The Menus %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Tools Menu =========================================================================
tmenu = uimenu('Label','Tools','ForegroundColor',[.5 0 .5]);
uimenu(tmenu,'Label','Grid On/Off','Callback',@GridOnOff)
uimenu(tmenu,'Label','Black Axes','Callback',@AxBW);
uimenu(tmenu,'Label','Close','Accelerator','X','Callback','close',...
    'Separator','on');
% Lemniscate of Bernoulli Menu ========================================================
bmenu = uimenu('Label','Lemniscate of Bernoulli','ForegroundColor',[.5 0 .5]);
uimenu(bmenu,'Label','Parametric','Callback',@BernPar,...
    'Separator','on','Accelerator','b');
uimenu(bmenu,'Label','Contour Map 1','Callback',@BernContour1);
uimenu(bmenu,'Label','Gradient Vectors for Contour Map 1','Callback',@BernContourGrad);
uimenu(bmenu,'Label','Contour Map 2','Callback',@BernContour2);
uimenu(bmenu,'Label','Contour Map 3','Callback',@BernContour3);
% Lemniscate of Gerono Menu ===========================================================
gmenu = uimenu('Label','Lemniscate of Gerono','ForegroundColor',[.25 .6 .25]);
uimenu(gmenu,'Label','Parametric','Callback',@GeronoPar,...
    'Separator','on','Accelerator','g');
uimenu(gmenu,'Label','Parametric  2','Callback',@GeronoPar2);
uimenu(gmenu,'Label','Polar  (Cartesian Graph)','Callback',@GeronoPol);
uimenu(gmenu,'Label','Polar  (Polar Graph)','Callback',@GeronoPol2);
uimenu(gmenu,'Label','Contour Map','Callback',@GeronoLevel);
% Lemniscate of Booth Menu =============================================================
bbmenu = uimenu('Label','Lemniscate of Booth','ForegroundColor',[.85 .14 .25]);
uimenu(bbmenu,'Label','Contour Map 1','Callback',@Booth,...
    'Separator','on');
uimenu(bbmenu,'Label','Gradient Vectors for Contour Map 1','Callback',@BoothContourGrad);
% Lemniscate of Cassinni Menu ===========================================================
cmenu = uimenu('Label','Cassinian Oval ','ForegroundColor',[.05 .35 .25]);
uimenu(cmenu,'Label','Cassinian Oval Lemniscate','Callback',@CassOval,...
    'Separator','on'); 
uimenu(cmenu,'Label','Contour Map 1','Callback',@CasContour);
uimenu(cmenu,'Label','Contour Map 2','Callback',@CassOvalContour);
% Inverse Lemniscate of Bernoulli Menu ====================================================
Imenu = uimenu('Label','The Inverse Lemniscate',...
    'ForegroundColor',[.05 .35 .25]);
uimenu(Imenu,'Label','The Inverse Lemniscate','Callback',@InverseLem);

movegui(Fig,'center');

% Callbacks % ===============================================================================
%============================================================================================
    function GridOnOff(src, evnt)
        grid
    end
%Close =====================================================================================
    function Close(src, evnt)
        delete(Fig)
    end
% Black/White Axes ============================================================================
    function AxBW(hObject,src, evnt)
        set(gca,'color','k');
    end
% Bernoulli Parametric =======================================================================
    function BernPar(src,evnt)
        hold off
        set(TitleText,'visible','off');
        theta = 0:.003:2*pi ;
        x = cos(theta).*sqrt(cos(2.*theta));
        y = sin(theta).*sqrt(cos(2.*theta));
        h = plot(x,y);grid on
        axis equal
        set(h,'Color',[.6 .11 .45]);
        set(h,'Linewidth',3);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('0 \leq \theta \leq 2\pi','Color','y');
        set(xl,'Fontname','Euclid','Fontsize',18);
        % Title ===================================================================================================================
        TS = ['$$','The~Lemniscate~of~Bernouli:x=cos(\theta){\sqrt{cos(2\theta)}}~~~y=sin(\theta){\sqrt{cos(2\theta)}}','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',19);
        set(L,'color','y');
    end
%  Bernoulli Contours  ================================================================
    function BernContour1(src, evnt)
        hold off
        set(TitleText,'visible','off');
        [x,y] = meshgrid(-2.3403:.01:2.3403);
        z = ((x.^2+y.^2)).^2-4.*(x.^2-y.^2);
        contour(x,y,z,30,'linewidth',2); 
        axis('equal','square');
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        % Title ===========================================================================
        TS = ['$$','The~Lemniscate~of~Bernoulli:~~~(x^2+y^2)^2=a^2(x^2-y^2)','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
% Bernoulli Contour Gradiants =============================================================
    function BernContourGrad(src, evnt)
        hold on
        v = -2.3403:.25:2.3403;
        [x,y] = meshgrid(v);
        z = ((x.^2+y.^2)).^2-4.*(x.^2-y.^2);
        [dx,dy] = gradient(z);
        hold on
        quiver(x,y,dx,dy,.75);
        axis('equal','square');
    end
%  Bernoulli Contours 2 ====================================================================
    function BernContour2(src, evnt)
        hold off
        set(TitleText,'visible','off');
        [x,y] = meshgrid(-pi:.005:pi);
        z = (1./4).*((x.^2+y.^2)+(4.*y.^2./(x.^2+y.^2)));
        set(gca,'Fontsize',9);
        contour(x,y,z,.5,'linewidth',2,'color',[.01 .35 .38]);
        axis('square');
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('-\pi \leq {\it{x,y}} \leq \pi',...
            'Color','y');
        set(xl,'Fontname','Euclid','Fontsize',20);
        %  Title =================================================================================
        TS = ['$$','The~Lemniscate~of~Bernoulli:~~(x^2+y^2)^2+4y^2=~4c~(x^2+y^2),~~c=.5 ','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
%  Bernoulli Contours  ===========================================================================
    function BernContour3(src, evnt)
        hold off
        set(TitleText,'visible','off');
        [x,y] = meshgrid(-2:.01:2);
        z = (x.^2-x.^4-y.^2).^(1/2);grid on
        contour(x,y,z,12);
        colormap('hot');
        axis('equal','square');
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('-1 \leq {\itx} \leq 1     -2 \leq {\ity} \leq 2',...
            'Color','y');
        set(xl,'Fontname','Euclid','Fontsize',18);
        %  Title =================================================================================
        TS = ['$$','f(x,y)=~{\sqrt{x^2-x^4-y^2}}','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',21);
        set(L,'color','y');
    end
% Gerono Parametric ===============================================================================
    function GeronoPar(src,evnt)
        hold off
        set(TitleText,'visible','off');
        t = 0:.003:2*pi ;
        x = t;
        x1 = cos(t);
        y1 = cos(t).*sin(t);
        comet(x1,y1);
        h = plot(x1,y1);grid on
        xl = xlabel('0 \leq {\itt} \leq 2\pi','Color','y');
        set(xl,'Fontname','Euclid','Fontsize',18);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        axis equal
        set(h,'Color',[.23 .64 .68]);
        grid on
        set(h,'Linewidth',3);
        set(gca,'Fontsize',11);
        % Title ===================================================================================
        TS1 = sym(sin(t)+cos(t));
        TS2 = sym(sin(t).^2);
        TS = ['$$','The~Lemniscate~of~Gerono:~~x=cos(t)~~~~y=cos(t)sin(t)','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
%  Gerono Parametric 2  ==========================================================================
    function GeronoPar2(src, evnt)
        set(TitleText,'visible','off');
        n = 4;
        t = linspace(0,2*pi,2000);
        a = 1;
        x = a.*cos(t);
        y = a.*cos(t).*sin(t);
        p = plot(x,y,'linewidth',2); grid off
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        axis equal
        hold on
        for a = 1:n
            x = a.*cos(t);
            y = a.*cos(t).*sin(t);
            f1 = plot(x,y);
            set(f1,'Color',[.13 .55 .46],'Linewidth',2)
        end
        c = xlabel('0 \leq \theta \leq 2\pi');
        set(c,'FontName','Euclid');
        set(c,'FontWeight','b');
        set(c,'Fontsize',18);
        set(c,'color','y');
        % Title   % ===========================================================================================
        TS = ['$$','The~Lemniscate~of~Gerono:x=a~cos(\theta)~~y=a~cos(\theta)sin(\theta)~~~a = 1,2,3,4','$$'];
        L = title(TS);
        set(L,'interpreter','latex');
        set(L,'FontName','Euclid');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
% Gerono Polar ==================================================================================================
    function GeronoPol(src,evnt)
        hold off
        set(TitleText,'visible','off');
        t = 0:.001:2*pi ;
        r = (sec(t).^4.*cos(2.*t)).^(1/2);
        x = r.*cos(t);
        y = r.*sin(t);
        plot(x,y,'color',[.782 .12 .22],'Linewidth',3);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('0 \leq \theta \leq 2\pi,   a = 1');
        set(xl,'Fontname','Euclid','Fontsize',18,'Color','y');
        axis equal;
        TS = ['$$','The~Lemniscate~of~Gerono:~~r(\theta)=\pm \sqrt{a^2\frac{cos(2\theta)}{cos(\theta)^4}}','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',19);
        set(L,'color','y');
    end
% Gerono Polar 2 ===================================================================================================
    function GeronoPol2(src,evnt)
        hold off
        set(TitleText,'visible','off');
        t = 0:.01:2*pi ;
        x = t;
        a = 2;
        r = sqrt(cos(a.*t));
        p = polar(t,r);
        set(p,'color',[.782 .12 .22],'Linewidth',2);
        set(gca,'Fontsize',9);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('0 \leq \theta \leq 2\pi','Color','y');
        set(xl,'Fontname','Euclid','Fontsize',18);
        TS = ['$$','r(\theta)=\sqrt{cos(2\theta)}','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',19);
        set(L,'color','y');
    end
%  Gerono Contours  ==================================================================
    function GeronoLevel(src, evnt)
        hold off
        set(TitleText,'visible','off');
        [x,y] = meshgrid(-3.31:.01:3.31);
        z = x.^4-(4.*(x.^2-y.^2));
        contour(x,y,z,30,'linewidth',2); axis('equal');
        colormap('colorcube')
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('-\pi \leq {\it{x,y}} \leq \pi',...
            'Color','y');
        set(xl,'Fontname','Bookman Old Style','Fontsize',18);
        % Title ====================================================================
        TS = ['$$','The~Lemniscate~of~Gerono:~~x^4=a^2(x^2-y^2)','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
%  Booth Contours  =======================================================================
    function Booth(src, evnt)
        hold off
        set(TitleText,'visible','off');
        [x,y] = meshgrid(-pi:.01:pi);
        z = (1./4).*((x.^2+y.^2)+(4.*y.^2./(x.^2+y.^2)));
        set(gca,'Fontsize',9);
        contour(x,y,z,0:.5:10,'linewidth',2);
        axis('equal','square');
        colormap('colorcube')
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('-\pi \leq {\it{x,y}} \leq \pi',...
            'Color','y');
        set(xl,'Fontname','Euclid','Fontsize',22);
        % Title ==========================================================================
        TS = ['$$','The~Lemniscate~of~Booth:~~(x^2+y^2)^2+4y^2=~4c~(x^2+y^2)','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',19);
        set(L,'color','y');
    end
% Booth Contour Gradiants ===================================================================
    function BoothContourGrad(src, evnt)
        hold on
        v = -pi:.25:pi;
        [x,y] = meshgrid(v);
        z = (1./4).*((x.^2+y.^2)+(4.*y.^2./(x.^2+y.^2)));
        [dx,dy] = gradient(z);
        hold on
        quiver(x,y,dx,dy,.75);
        axis('equal','square');
    end
% Cassinian Oval Parametric ====================================================================
    function CassOval(src, evnt)
        hold off
        set(TitleText,'visible','off');
        clc
        a = 1;
        b = 1;
        t = 0:.001:2*pi ;
        M = 2*a.^2*cos(2*t)+2*sqrt(-a.^4+b.^4+a.^4*cos(2*t).^2);
        x = sqrt((M./2).^2).*cos(t);
        y = sqrt((M./2).^2).*sin(t);
        comet(x,y);
        p = plot(x,y);grid off
        set(p,'color',[.22 .552 .23],'Linewidth',3);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        c = xlabel('The "Cassinian Oval" Lemniscate,    a = b = 1');
        set(c,'Fontname','Bookman Old Style','Fontsize',16,'color','y');
        % Title ====================================================================================================================================================================
        TS = ['$$','x = {\sqrt\frac{[2a^2cos(2t)+2{\sqrt{-a^4+b^4+a^4cos(2t)^2}~]^2}}4}~cos(t),~y = {\sqrt\frac{[2a^2cos(2t)+2{\sqrt{-a^4+b^4+a^4cos(2t)^2}~]^2}}4}~sin(t)','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',12);
        set(L,'color','y');
    end
% Cassinian Oval Contours ===================================================================
    function CasContour(src, evnt)
        hold off
        set(TitleText,'visible','off');
        a = 5;
        [x,y] = meshgrid( -7.2599:.01:7.2599);
        z = sqrt((x-a).^2+y.^2).*sqrt((x+a).^2+y.^2);
        [c,h] = contour(x,y,z,25,'linewidth',2);
        axis('equal','square');
        colormap('colorcube')
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('a = 5',...
            'Color','y');
        set(xl,'Fontname','Euclid','Fontsize',18);
        %      Title ===========================================================================
        TS = ['$$','The~Cassinian~Oval:~~f(x,y)=\sqrt{(x-a)^2+y^2}~\sqrt{(x+a)^2+y^2}','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
% Cassinian Oval Contours 2 ===================================================================
    function CassOvalContour(src, evnt)
        set(TitleText,'visible','off');
        a = 2;
        b = 2;
        [x,y] = meshgrid(linspace(-5,5,800));
        colormap('colorcube');axis equal
        z = ((x-a).^2+y.^2).*((x+a).^2+y.^2)-b^4;
        contour(x,y,z,0:5:50,'linewidth',2);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        xl = xlabel('a = b = 2',...
            'Color','y');
        set(xl,'Fontname','Euclid','Fontsize',18);
        %      Title ==================================================================
        TS = ['$$','The~Cassinian~Oval:~~((x-a)^2+y^2)~((x+a)^2+y^2)=b^4','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',18);
        set(L,'color','y');
    end
% Inverse Lemniscate ==================================================================
    function InverseLem(src,evnt)
        hold off
        set(TitleText,'visible','off');
        t = 0:.001:6*pi ;
        x = t;
        r = cos(2.*t).^5+sin(6*pi.*t).^12;
        x = r.*cos(t);
        y = r.*sin(t);
        comet(x,y)
        plot(x,y,'color',[.782 .12 .22],'Linewidth',1);
        set(gca,'xcolor',[.5 .5 .5],'ycolor',[.5 .5 .5],'Fontsize',10);
        axis('equal','square');
        xl = xlabel('0 \leq \theta \leq 6\pi');
        set(xl,'Fontname','Euclid','Fontsize',18,'Color','y');
        TS = ['$$','The~"Inverse~Lemniscate"~~~r(\theta) = cos^{5}(2\theta)+sin^{12}(6\theta)','$$'];
        L = title(TS);
        set(L,'Interpreter','latex');
        set(L,'FontWeight','b');
        set(L,'Fontsize',19);
        set(L,'color','y');
    end
end