How do I plot a toroid in MATLAB?

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I would like to plot a toroid in MATLAB but MATLAB does not have a built in function to do this.

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Answer 1

MathWorks Support Team 2009-6-27

在 MATLAB Online 中打开

1 个投票

You will need to formulate the x, y, and z-coordinate matrices manually and then plot them using the SURF function.

The SURF and MESH functions accept only one set of x, y, and z-coordinates, but in a toroid, (x,y) ordered pairs can have two corresponding z-coordinates. Therefore, to plot a toroid in MATLAB, you will need to plot the top and bottom halves as two separate surfaces on the same plot. For example:

%%Create R and THETA data
theta = 0:pi/10:2*pi;
r = 2*pi:pi/20:3*pi;
[R,T] = meshgrid(r,theta);
%%Create top and bottom halves
Z_top = 2*sin(R);
Z_bottom = -2*sin(R);
%%Convert to Cartesian coordinates and plot
[X,Y,Z] = pol2cart(T,R,Z_top);
surf(X,Y,Z);
hold on;
[X,Y,Z] = pol2cart(T,R,Z_bottom);
surf(X,Y,Z);
axis equal
shading interp

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Stephen23 2019-9-27

编辑：MathWorks Support Team 2026-2-5，8:09

"This is not a torus..."

That is correct: it is not a torus.

However it is a toroid, which is what the title and the answer state it is.

It might help to revise the difference between a toroid (what this question and answer are about) and a torus (which is what your comment is about), e.g. from Wikipedia:

"a toroid is a surface of revolution with a hole in the middle, like a doughnut, forming a solid body... If the revolved figure is a circle, then the object is called a torus."
"a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle."

or from Wolfram Mathematics https://mathworld.wolfram.com/Toroid.html :

"A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus."

The answer creates a toroid from sine curves, just as it states. It does not create a torus, nor does the answer state that it creates a torus.

Alex Pedcenko 2019-9-27

you're right

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Answer 2

Alex Pedcenko 2017-11-5

编辑：Alex Pedcenko 2019-9-27

在 MATLAB Online 中打开

9 个投票

R=5; % outer radius of torus
r=2; % inner tube radius
th=linspace(0,2*pi,36); % e.g. 36 partitions along perimeter of the tube 
phi=linspace(0,2*pi,18); % e.g. 18 partitions along azimuth of torus
% we convert our vectors phi and th to [n x n] matrices with meshgrid command:
[Phi,Th]=meshgrid(phi,th); 
% now we generate n x n matrices for x,y,z according to eqn of torus
x=(R+r.*cos(Th)).*cos(Phi);
y=(R+r.*cos(Th)).*sin(Phi);
z=r.*sin(Th);
surf(x,y,z); % plot surface
daspect([1 1 1]) % preserves the shape of torus
colormap('jet') % change color appearance 
title('Torus')
xlabel('X');ylabel('Y');zlabel('Z');

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Stephen23 2019-9-27

在 MATLAB Online 中打开

Alex Pedcenko's "Answer" moved here:

Well... you can make them 3D if you generate a set of spheres (see: help sphere) with a small radius and position them at these coordinates, but it will be much slower (your PC will need to render huge number of 3D objects) than this method above.

e.g

R=5; % outer radius of torus
r=2; % inner tube radius
th=linspace(0,2*pi,18); % e.g. 18 partitions along perimeter of the tube 
phi=linspace(0,2*pi,36); % e.g. 36 partitions along azimuth of torus
% we convert our vectors phi and th to [n x n] matrices with meshgrid command:
[Phi,Th]=meshgrid(phi,th); 
% now we generate n x n matrices for x,y,z according to eqn of torus
x=(R+r.*cos(Th)).*cos(Phi);
y=(R+r.*cos(Th)).*sin(Phi);
z=r.*sin(Th);
mesh(x,y,z,'EdgeColor','k'); % plot surface
daspect([1 1 1]) % preserves the shape of torus
%colormap('jet') % change color appearance 
title('Torus')
xlabel('X');ylabel('Y');zlabel('Z');
%% spheres:
hold on
[X Y Z]=sphere(10);
R=0.15; % size of the spheres
lighting gouraud
for i = 1:numel(x)
    surf(R*X+x(i),R*Y+y(i),R*Z+z(i),'FaceColor',[0.2 0.2 0.2],'FaceAlpha',0.8,'FaceLighting','gouraud','EdgeColor','none');
end
camlight

Alex Pedcenko 2020-7-5

在 MATLAB Online 中打开

How about 3D spiral?

R=5; % outer radius of torus
a=1; % inner tube smaller radius
b=1; % inner tube larger radius
p=0.5; % pitch in z-direction
N=10; %turns along z
th=linspace(0,2*pi,36); % e.g. 36 partitions along perimeter of the tube 
phi=linspace(0,N*2*pi,36*N); % e.g. 18 partitions along azimuth of torus
% we convert our vectors phi and th to [n x n] matrices with meshgrid command:
[Phi,Th]=meshgrid(phi,th); 
% now we generate n x n matrices for x,y,z according to eqn of torus
x=(R+a.*cos(Th)).*cos(Phi);
y=(R+b.*cos(Th)).*sin(Phi);
z=a.*sin(Th)+p*Phi;
surf(x,y,z); % plot surface
daspect([1 1 1]) % preserves the shape of torus
colormap('jet') % change color appearance 
%shading interp
title('Not a Torus')
xlabel('X');ylabel('Y');zlabel('Z');

Stephen23 2020-7-5

Looks good... please animate and upload a GIF!

For inspiration: https://www.mathworks.com/matlabcentral/fileexchange/25086-extrude-a-ribbon-tube-and-fly-through-it

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Answer 3

DGM 2022-1-8

在 MATLAB Online 中打开

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MATLAB may not have a built-in function, but that doesn't mean there aren't any functions out there that can conveniently do the work.

https://www.mathworks.com/matlabcentral/fileexchange/58413-plot-superquadratic-surfaces

I'm sure this isn't the only thing on the File Exchange, but it's the one I use. Syntax is similar to sphere() or ellipsoid(), returning three matrices which can be fed to surf() or mesh(). The input arguments are the center location, radii, order, and number of points.

center = [0 0 0];
radius = [1 1 1 3];
order = 2;
npoints = 100;
[x y z] = supertoroid(center,radius,order,npoints);
surf(x,y,z)
shading flat
axis equal
colormap(parula)
view(-16,27)
camlight