I wish to plot implicit functions in MATLAB. Like x^3 + xy + y^2 = 36 , equations which cannot be made into simple parametric form. Is there any simple method ?
Here are a couple of options...
Using ezplot (or fplot recommended in newer versions):
The easiest solution is to use the function ezplot:
ezplot('x.^3 + x.*y + y.^2 - 36', [-10 10 -10 10]);
Which gives you the following plot:
Using contour:
Another option is to generate a set of points where you will evaluate the function f(x,y) = x^3 + x*y + y^2 and then use the function contour to plot contour lines where f(x,y) is equal to 36:
[x, y] = meshgrid(-10:0.1:10); % Create a mesh of x and y points
f = x.^3+x.*y+y.^2; % Evaluate f at those points
contour(x, y, f, [36 36], 'b'); % Generate the contour plot
xlabel('x'); % Add an x label
ylabel('y'); % Add a y label
title('x^3 + x y + y^2 = 36'); % Add a title
The above will give you a plot nearly identical to the one generated by ezplot:
In case you want to plot an implicit surface, for example a Horned cube, you can do something like the following.
The idea is to calculate all values of the function (even if they don't equal to zero) and then create an isosurface that will define your equality. In this example the implicit function equals to zero.
fun=#(x,y,z)(1-x.^8-3.*y.^8-2.*z.^8+5.*x.^4.*z.^2.*y.^2+3.*y.^4.*x.^2.*z.^2) ;
[X,Y,Z]=meshgrid(-2:0.1:2,-2:0.1:2,-2:0.1:2);
val=fun(X,Y,Z);
fv=isosurface(X,Y,Z,val,0);
p = patch(fv);
isonormals(X,Y,Z,val,p)
set(p,'FaceColor' , 'red');
set(p,'EdgeColor' , 'none');
daspect([1,1,1])
view(3); axis tight
camlight
lighting phong
axis off
Additionally there is a Matlab File Exchange submission called ezimplot3D that seems to do the job also, as #knedlsepp suggests.
Implot2 and implot from Matlab Central appear to do the job.
There are two new functions to plot implicit function in R2016b:
fimplicit for f(x,y) = 0
fimplicit3 for f(x,y,z) = 0
Related
I have three vectors, one of X locations, another of Y locations and the third is a f(x, y). I want to find the algebraic expression interpolation polynomial (using matlab) since I will later on use the result in an optimization problem in AMPL. As far as I know, there are not any functions that return the interpolation polynomial.
I have tried https://la.mathworks.com/help/matlab/ref/griddedinterpolant.html, but this function only gives the interpolated values at certain points.
I have also tried https://la.mathworks.com/help/matlab/ref/triscatteredinterp.html as sugested in Functional form of 2D interpolation in Matlab, but the output isn't the coefficents of the polynomial. I cannot see it, it seems to be locked inside of a weird variable.
This is a small program that I have done to test what I am doing:
close all
clear
clc
[X,Y] = ndgrid(1:10,1:10);
V = X.^2 + 3*(Y).^2;
F = griddedInterpolant(X,Y,V,'cubic');
[Xq,Yq] = ndgrid(1:0.5:10,1:0.5:10);
Vq = F(Xq,Yq);
mesh(Xq,Yq,Vq)
figure
mesh(X, Y, V)
I want an output that instead of returning the value at grid points returns whatever it has used to calculate said values. I am aware that it can be done in mathematica with https://reference.wolfram.com/language/ref/InterpolatingPolynomial.html, so I find weird that matlab can't.
You can use fit if you have the curve fitting toolbox.
If it's not the case you can use a simple regression, if I take your example:
% The example data
[X,Y] = ndgrid(1:10,1:10);
V = X.^2 + 3*(Y).^2;
% The size of X
s = size(X(:),1);
% Let's suppose that you want to fit a polynome of degree 2.
% Create all the possible combination for a polynome of degree 2
% cst x y x^2 y^2 x*y
A = [ones(s,1), X(:), Y(:), X(:).^2, Y(:).^2, X(:).*Y(:)]
% Then using mldivide
p = A\V(:)
% We obtain:
p =
0 % cst
0 % x
0 % y
1 % x^2
3 % y^2
0 % x*y
I'm wondering if anyone has any insight into why these two plot commands produce domains that are orders of magnitude different?
syms t
x1Axis = 0:.01:10;
fun1(t) = sin(t)
plot(sin(x1Axis))
hold on
y = sin(x1Axis)
plot(x1Axis, y)
fun1(t) is plotted "symbolically" while y is evaluated and plotted "discreetly". Should I be using a different plot method in the case of the first function?
No, you are not plotting the symbolic function correctly.
In your code, the instruction plot(sin(x1Axis)) is not a symbolic plot, but a numeric plot of the sine versus the index of each value.
From the plot documentation page:
plot(Y) creates a 2-D line plot of the data in Y versus the index of
each value.
If Y is a vector, then the x-axis scale ranges from 1 to length(Y).
To plot the symbolic function use fplot.
The following example will allow you to see that both the symbolic and numeric plots are the same:
xmin = 0;
xmax = 10;
% Symbolic Plot.
syms t
fun1(t) = sin(t);
fplot(fun1, [xmin xmax], '-r');
hold on;
% Numeric Plot.
x = xmin:.01:xmax;
y = sin(x);
plot(x, y, '--g');
% Add legend.
legend('Symbolic Plot', 'Numeric Plot', 'Location', 'north');
This is the result:
As we can make contour plot of f=(x.^2) + (y.^2); in 2-D as follows:
[x,y]= meshgrid(-10:10, -10:10);
contour(x,y, (x.^2)+(y.^2));
and we can make contour plot of f=(x.^2) + (y.^2); in 3-D using contour3
Can we make contour plot of f=(x.^2) + (y.^2) + (z.^2); in 3-D
The matlab function isosurface can do what you are asking. However you can also achieve the results you want using other alternatives, like using surf. I will cover both methods.
Method 1: Using isosurface
We need to create the domain for x, y and z and then generate a 3D mesh with those values so that we can evaluate the function f(x,y,z) = x^2 + y^2 + z^2. Then, we shall give a value to the constant k and feed all this information to isosurface, so that we can obtain the family of (x,y,z) values that satisfy the condition: f(x,y,z) = k. Note that this contours are in fact spheres! Finally we can use patch to generate a surface with those values.
It is very interesting to give different values for k iterativelly and see the contours associated with those values.
% Value for x, y and z domain.
a = 10;
% Domain for x ,y and z.
x = linspace(-a,a);
y = linspace(-a,a);
z = linspace(-a,a);
% Generate a 3D mesh with x, y and z.
[x,y,z] = meshgrid(x,y,z);
% Evaluate function (3D volume of data).
f = x.^2 + y.^2 + z.^2;
% Do contours from k = 0 to k = 100 in steps of 1 unit.
for k = 0:100
% Draw the contour that matches k.
p = patch(isosurface(x,y,z,f,k));
isonormals(x,y,z,f,p)
p.FaceColor = 'red';
p.EdgeColor = 'none';
% Adjust figure properties.
title(sprintf('Contours of f(x,y,z) = x^2 + y^2 + z^2\nwith f(x,y,z) = k = %d',k));
xlabel('x-axis');
ylabel('y-axis');
zlabel('z-axis');
axis equal;
grid on;
box on;
axis([-10 10 -10 10 -10 10]);
camlight left;
lighting phong;
% Update figure.
drawnow;
% Clear axes.
cla;
end
This is the output:
Method 2: Using surf
As in the previous method, to contour the function f(x,y,z) = x^2 + y^2 + z^2, we need to match the function to a constant value, this is f(x,y,z) = k, where k is any constant value you choose.
If we isolate z in terms of k, x and y we have: z = ± sqrt(k-x.^2-y.^2), so we have the explicit values for x, y and z. Now, let's give different values for k iterativelly and see the results that we get with the surf function!
% Do contours from k = 0 to k = 100 in steps of 1 unit.
for k = 0:100
% Find the value where: k - x^2 - y^2 = 0
a = sqrt(k);
% Domain for x and y.
x = linspace(-a,a);
y = linspace(-a,a);
[x,y] = meshgrid(x, y);
% Isolate z in terms of k, x and y.
z = sqrt(k-x.^2-y.^2);
% Find complex entries in z.
i = find(real(z)~=z);
% Replace complex entries with NaN.
z(i) = NaN;
% Draw upper hemisphere of surface.
surf(x,y,z,'FaceColor','red','EdgeColor','none');
hold on;
% Draw lower hemisphere of surface.
surf(x,y,-z,'FaceColor','red','EdgeColor','none');
% Adjust figure properties.
title(sprintf('Contours of f(x,y,z) = x^2 + y^2 + z^2\nwith f(x,y,z) = k = %d',k));
xlabel('x-axis');
ylabel('y-axis');
zlabel('z-axis');
axis equal;
grid on;
box on;
axis([-10 10 -10 10 -10 10]);
camlight left;
lighting phong;
% Update figure.
drawnow;
hold off;
end
This is the output:
References
I took some of the ideas from David Arnold's article "Complex Numbers and Plotting in Matlab", which is well worth a read and will help you understand how to plot functions that generate complex numbers.
I would like to plot this function of Two Variables you can find it here
$$z^2=t(t-i) \Longleftrightarrow x^2+y^2=4x^2y^2 \Longleftrightarrow y=\dfrac{\pm x}{\sqrt{4x^2-1}} \mbox{ with } |x|>\frac{1}{2}$$
would someone show me step by step how to plot this in matlab
is there any script or toolbox in http://www.mathworks.com/matlabcentral/fileexchange
which make plot of that kind of curves quickly
this is by geogebra
This is by wolframe
You can use symbolic variables with ezplot.
syms x y % makes symbolic variables
h1 = ezplot('-4*x^2*y^2+x^2+y^2'); % plots the equation
axis equal
set(h1, 'Color', 'k');
Or you can define a function,
f = #(x,y) -4.*x.^2.*y.^2+x.^2+y.^2;
h1 = ezplot(f);
set(h1, 'Color', 'k');
It won't be easy to have the axis in the middle, I hope it's not necessary to have that.
Edit
You can download oaxes here
syms x y
h1 = ezplot('-4*x^2*y^2+x^2+y^2');
axis equal
set(h1, 'Color', 'm');
oaxes('TickLength',[3 3],'Arrow','off','AxisLabelLocation','side',...
'LineWidth',1)
Edit
For 3D plot try this,
% First line provides a grid of X and Y varying over -5 to 5 with .5 as step-size
[X,Y] = meshgrid(-5:.5:5);
% instead of "=0", Z takes the values of the equation
Z = -4 .* X.^2 .* Y.^2 + X.^2 + Y.^2;
surf(X,Y,Z) % makes a 3D plot of X,Y,Z
You can also try contourf(X,Y,Z) for 2D plot.
I wish to plot implicit functions in MATLAB. Like x^3 + xy + y^2 = 36 , equations which cannot be made into simple parametric form. Is there any simple method ?
Here are a couple of options...
Using ezplot (or fplot recommended in newer versions):
The easiest solution is to use the function ezplot:
ezplot('x.^3 + x.*y + y.^2 - 36', [-10 10 -10 10]);
Which gives you the following plot:
Using contour:
Another option is to generate a set of points where you will evaluate the function f(x,y) = x^3 + x*y + y^2 and then use the function contour to plot contour lines where f(x,y) is equal to 36:
[x, y] = meshgrid(-10:0.1:10); % Create a mesh of x and y points
f = x.^3+x.*y+y.^2; % Evaluate f at those points
contour(x, y, f, [36 36], 'b'); % Generate the contour plot
xlabel('x'); % Add an x label
ylabel('y'); % Add a y label
title('x^3 + x y + y^2 = 36'); % Add a title
The above will give you a plot nearly identical to the one generated by ezplot:
In case you want to plot an implicit surface, for example a Horned cube, you can do something like the following.
The idea is to calculate all values of the function (even if they don't equal to zero) and then create an isosurface that will define your equality. In this example the implicit function equals to zero.
fun=#(x,y,z)(1-x.^8-3.*y.^8-2.*z.^8+5.*x.^4.*z.^2.*y.^2+3.*y.^4.*x.^2.*z.^2) ;
[X,Y,Z]=meshgrid(-2:0.1:2,-2:0.1:2,-2:0.1:2);
val=fun(X,Y,Z);
fv=isosurface(X,Y,Z,val,0);
p = patch(fv);
isonormals(X,Y,Z,val,p)
set(p,'FaceColor' , 'red');
set(p,'EdgeColor' , 'none');
daspect([1,1,1])
view(3); axis tight
camlight
lighting phong
axis off
Additionally there is a Matlab File Exchange submission called ezimplot3D that seems to do the job also, as #knedlsepp suggests.
Implot2 and implot from Matlab Central appear to do the job.
There are two new functions to plot implicit function in R2016b:
fimplicit for f(x,y) = 0
fimplicit3 for f(x,y,z) = 0