Plot elipses with a foci based on equations to plot flowers

Question

Ahmed Mohamed Mansoor 2022-10-6

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1818770-plot-elipses-with-a-foci-based-on-equations-to-plot-flowers

评论： Ahmed Mohamed Mansoor 2022-10-10

I am trying to practice with matlab with various equations out there. Does anyone know how to plot this picture (see image below) based on these equations? I have coded the equations (see below picture), but have no idea how to plot it.

I have coded the equations A(k), B(k), C(k), D(k), E(k) but I do not know where to go from here.

Any help would be appreciated.

close all; clc; clear;
sin48 =@(k) sin((48*pi*k)./1000000);
cos48 =@(k) cos((48*pi*k)./1000000);
sin8  =@(k) sin((8*pi*k)./1000000);
cos8  =@(k) cos((8*pi*k)./1000000);
sin24 =@(k) sin((24*pi*k)./1000000);
cos24 =@(k) cos((24*pi*k)./1000000);
sin72 =@(k) sin((72*pi*k)./1000000);
cos72 =@(k) cos((72*pi*k)./1000000);
cos648=@(k) cos((648*pi*k)./1000000);
cos64 =@(k) cos((64*pi*k)./1000000);
Ek = 0.1 + 0.25*sin48(x).^2 + 0.2*cos18(x).^18 *...
    (1 + (1/3)*cos24(x).^8  + 0.1*cos24(x)^20) + (1/6)*sin8(x).^16 +...
    (8/20)*sin8(x).*(1 - 0.4*cos72(x).^4) + 0.0625*sin(x).^18.*...
    sin24(x).^20.*sin72(x).^30 - (0.1*sin24(x).^6 + (1/6)*sin24(x).^30)...
    (1-sin8(x));
Dk = (pi/2) + (1686*pi*x/1000000);
Ck = 0.003 + 0.097*cos648(x).^40;
Bk = -cos64(x).*Ek - 0.1*cos64(x);
Ak = 1.2*sin64(x)*Ek;

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Sam Chak 2022-10-6

Hmm... Didn't you have few examples posted on the last few days?

Can you put the scatter() and colormap(jet) functions in the code?

Ahmed Mohamed Mansoor 2022-10-6

Yes. However, there are a few issues. These are ellipses with equations involving imaginary numbers.

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Answer 1

Davide Masiello 2022-10-6

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1818770-plot-elipses-with-a-foci-based-on-equations-to-plot-flowers#answer_1068050

在 MATLAB Online 中打开

The code below does the job drawing the figure, but not the coloring.

I can't run it online (takes too long).

clear,clc
figure(1)
hold on
for k = 1:1000000
plotEllipse(k)
end
function plotEllipse(k)
e       = 98/100;
sin48   = sin((48*pi*k)/1000000);
sin8    = sin((8*pi*k)/1000000);
cos8    = cos((8*pi*k)./1000000);
sin24   = sin((24*pi*k)/1000000);
cos24   = cos((24*pi*k)/1000000);
sin72   = sin((72*pi*k)/1000000);
cos72   = cos((72*pi*k)/1000000);
cos648  = cos((648*pi*k)/1000000);
sin64   = sin((64*pi*k)/1000000);
cos64   = cos((64*pi*k)/1000000);
Ek      = 0.1 + 0.25*sin48^2 + 0.2*cos8^18 *...
            (1 + (1/3)*cos24^8  + 0.1*cos24^20) + (1/6)*sin8^16 +...
            (8/20)*sin8*(1 - 0.4*cos72^4) + 0.0625*sin8^18.*...
            sin24^20*sin72^30 - (0.1*sin24^6 + (1/6)*sin24^30)*...
            (1-sin8);
Dk = (pi/2) + (1686*pi*k/1000000);
Ck = 0.003 + 0.097*cos648^40;
Bk = -cos64*Ek - 0.1*cos64;
Ak = 1.2*sin64*Ek;
F1      = Ak + 1i*Bk + Ck*exp(1i*Dk) ;                                        % Focus 1
F2      = Ak + 1i*Bk - Ck*exp(1i*Dk) ;                                        % Focus 2
centre  = mean([F1,F2]);                                                    % ellipse centre
angle   = atan(abs(imag(F1)-imag(F2))/abs(real(F1)-real(F2)));              % angle between ellipse horizontal axis and x-axis
c       = sqrt((real(F1)-real(F2))^2+(imag(F1)-imag(F2))^2)/2;              % distance from centre to foci
a       = c/e;                                                              % ellipse horizontal axis
b       = a^2-c^2;                                                          % ellipse vertical axis
t       = linspace(0,2*pi,100);                                             % parametric coordinate
x0      = a*cos(t);                                                         % x-coordinates of ellipse if centered with reference axis and horizontal
y0      = b*sin(t);                                                         % y-coordinates of ellipse if centered with reference axis and horizontal
x       = x0*cos(angle)-y0*sin(angle)+real(centre);                         % Absolute x-coordinates (rotation+translation)
y       = y0*cos(angle)+x0*sin(angle)+imag(centre);                         % Absolute y-coordinates (rotation+translation)
plot(x,y)
end

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Davide Masiello 2022-10-6

在 MATLAB Online 中打开

little example up to k = 10000

clear,clc

figure(1)

hold on

for k = 1:10000

plotEllipse(k)

end

function plotEllipse(k)

e = 98/100;

sin48 = sin((48*pi*k)/1000000);

sin8 = sin((8*pi*k)/1000000);

cos8 = cos((8*pi*k)./1000000);

sin24 = sin((24*pi*k)/1000000);

cos24 = cos((24*pi*k)/1000000);

sin72 = sin((72*pi*k)/1000000);

cos72 = cos((72*pi*k)/1000000);

cos648 = cos((648*pi*k)/1000000);

sin64 = sin((64*pi*k)/1000000);

cos64 = cos((64*pi*k)/1000000);

Ek = 0.1 + 0.25*sin48^2 + 0.2*cos8^18 *...

(1 + (1/3)*cos24^8 + 0.1*cos24^20) + (1/6)*sin8^16 +...

(8/20)*sin8*(1 - 0.4*cos72^4) + 0.0625*sin8^18.*...

sin24^20*sin72^30 - (0.1*sin24^6 + (1/6)*sin24^30)*...

(1-sin8);

Dk = (pi/2) + (1686*pi*k/1000000);

Ck = 0.003 + 0.097*cos648^40;

Bk = -cos64*Ek - 0.1*cos64;

Ak = 1.2*sin64*Ek;

F1 = Ak + 1i*Bk + Ck*exp(1i*Dk) ; % Focus 1

F2 = Ak + 1i*Bk - Ck*exp(1i*Dk) ; % Focus 2

centre = mean([F1,F2]); % ellipse centre

angle = atan(abs(imag(F1)-imag(F2))/abs(real(F1)-real(F2))); % angle between ellipse horizontal axis and x-axis

c = sqrt((real(F1)-real(F2))^2+(imag(F1)-imag(F2))^2)/2; % distance from centre to foci

a = c/e; % ellipse horizontal axis

b = a^2-c^2; % ellipse vertical axis

t = linspace(0,2*pi,100); % parametric coordinate

x0 = a*cos(t); % x-coordinates of ellipse if centered with reference axis and horizontal

y0 = b*sin(t); % y-coordinates of ellipse if centered with reference axis and horizontal

x = x0*cos(angle)-y0*sin(angle)+real(centre); % Absolute x-coordinates (rotation+translation)

y = y0*cos(angle)+x0*sin(angle)+imag(centre); % Absolute y-coordinates (rotation+translation)

plot(x,y)

end

Davide Masiello 2022-10-6

在 MATLAB Online 中打开

This is faster

clear,clc
tic
n       = 50;                                   % number of points for each ellipse
kmax    = 10000;
x       = zeros(n,kmax);
y       = zeros(n,kmax);
for k = 1:kmax
    [x(:,k),y(:,k)] = ellipseCoordinates(k,n);
end
plot(x,y)
toc
function [x,y] = ellipseCoordinates(k,n)
e       = 98/100;
sin48   = sin((48*pi*k)/1000000);
sin8    = sin((8*pi*k)/1000000);
cos8    = cos((8*pi*k)/1000000);
sin24   = sin((24*pi*k)/1000000);
cos24   = cos((24*pi*k)/1000000);
sin72   = sin((72*pi*k)/1000000);
cos72   = cos((72*pi*k)/1000000);
cos648  = cos((648*pi*k)/1000000);
sin64   = sin((64*pi*k)/1000000);
cos64   = cos((64*pi*k)/1000000);
Ek      = 0.1 + 0.25*sin48^2 + 0.2*cos8^18 *...
            (1 + (1/3)*cos24^8  + 0.1*cos24^20) + (1/6)*sin8^16 +...
            (8/20)*sin8*(1 - 0.4*cos72^4) + 0.0625*sin8^18.*...
            sin24^20*sin72^30 - (0.1*sin24^6 + (1/6)*sin24^30)*...
            (1-sin8);
Dk = (pi/2) + (1686*pi*k/1000000);
Ck = 0.003 + 0.097*cos648^40;
Bk = -cos64*Ek - 0.1*cos64;
Ak = 1.2*sin64*Ek;
F1      = Ak + 1i*Bk + Ck*exp(1i*Dk) ;                                        % Focus 1
F2      = Ak + 1i*Bk - Ck*exp(1i*Dk) ;                                        % Focus 2
centre  = mean([F1,F2]);                                                    % ellipse centre
angle   = atan(abs(imag(F1)-imag(F2))/abs(real(F1)-real(F2)));              % angle between ellipse horizontal axis and x-axis
c       = sqrt((real(F1)-real(F2))^2+(imag(F1)-imag(F2))^2)/2;              % distance from centre to foci
a       = c/e;                                                              % ellipse horizontal axis
b       = a^2-c^2;                                                          % ellipse vertical axis
    
t       = linspace(0,2*pi,n);                                               % parametric coordinate
x0      = a*cos(t);                                                         % x-coordinates of ellipse if centered with reference axis and horizontal
y0      = b*sin(t);                                                         % y-coordinates of ellipse if centered with reference axis and horizontal
x       = x0*cos(angle)-y0*sin(angle)+real(centre);                         % Absolute x-coordinates
y       = y0*cos(angle)+x0*sin(angle)+imag(centre);                         % Absolute y-coordinates
x       = x';
y       = y';
end

Davide Masiello 2022-10-6

在 MATLAB Online 中打开

Actually, the whole thing can be done without loop

clear,clc
tic
n       = 50;       
k       = 1:1000000;
e       = 98/100; 
Ek      = 0.1 + 0.25*sin((48*pi*k)/1000000).^2 + 0.2*cos((8*pi*k)/1000000).^18.*...
            (1 + (1/3)*cos((24*pi*k)/1000000).^8  + 0.1*cos((24*pi*k)/1000000).^20) + (1/6)*sin((8*pi*k)/1000000).^16 +...
            (8/20)*sin((8*pi*k)/1000000).*(1 - 0.4*cos((72*pi*k)/1000000).^4) + 0.0625*sin((8*pi*k)/1000000).^18.*...
            sin((24*pi*k)/1000000).^20.*sin((72*pi*k)/1000000).^30 - (0.1*sin((24*pi*k)/1000000).^6 + (1/6)*sin((24*pi*k)/1000000).^30).*...                                                                                                
            (1-sin((8*pi*k)/1000000));
Dk = (pi/2) + (1686*pi*k/1000000);
Ck = 0.003 + 0.097*cos((648*pi*k)/1000000).^40;
Bk = -cos((64*pi*k)/1000000).*Ek - 0.1*cos((64*pi*k)/1000000);
Ak = 1.2*sin((64*pi*k)/1000000).*Ek;
F1      = Ak + 1i*Bk + Ck.*exp(1i*Dk) ;                                        % Focus 1
F2      = Ak + 1i*Bk - Ck.*exp(1i*Dk) ;                                        % Focus 2
centre  = mean([F1;F2],1);                                                    % ellipse centre
angle   = atan(abs(imag(F1)-imag(F2))./abs(real(F1)-real(F2)));              % angle between ellipse horizontal axis and x-axis
c       = sqrt((real(F1)-real(F2)).^2+(imag(F1)-imag(F2)).^2)/2;              % distance from centre to foci
a       = c/e;                                                              % ellipse horizontal axis
b       = a.^2-c.^2;                                                          % ellipse vertical axis
    
t       = linspace(0,2*pi,n)';                                               % parametric coordinate
x0      = a.*cos(t);                                                         % x-coordinates of ellipse if centered with reference axis and horizontal
y0      = b.*sin(t);                                                         % y-coordinates of ellipse if centered with reference axis and horizontal
x       = x0.*cos(angle)-y0.*sin(angle)+real(centre);                         % Absolute x-coordinates
y       = y0.*cos(angle)+x0.*sin(angle)+imag(centre);                         % Absolute y-coordinates
plot(x,y)
toc

Davide Masiello 2022-10-10

编辑：Davide Masiello 2022-10-10

@Ahmed Mohamed Mansoor Happy to help, please post the plot if you ever manage to produce it. Also, since you are interested in this cool plots, I suggest you check this out.

https://ch.mathworks.com/matlabcentral/communitycontests/contests/5/entries

Ahmed Mohamed Mansoor 2022-10-10

@Davide Masiello The coincidence!!!! I was literally looking at them right now and was just mesmerised at the different plots. There's just so much to learn!!

and yes i'll try to get the plot when possible.

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Plot elipses with a foci based on equations to plot flowers

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