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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.

Accepted Answer

MathWorks Support Team
MathWorks Support Team on 27 Jun 2009
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
  3 Comments
Stephen23
Stephen23 on 27 Sep 2019
Edited: MathWorks Support Team on 2 Jan 2020
"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:
or from Wolfram Mathematics http://mathworld.wolfram.com/Toroid.html :
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.

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More Answers (2)

Alex Pedcenko
Alex Pedcenko on 5 Nov 2017
Edited: Alex Pedcenko on 27 Sep 2019
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');
torus1.png
  6 Comments
Alex Pedcenko
Alex Pedcenko on 5 Jul 2020
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');

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DGM
DGM on 8 Jan 2022
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.
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
As axis orders are independent and user-defined, the profile and sections do not have to be circular, but can be any superellipse:
radius = [1 1 2 3];
order = [5 3];
radius = [1 1 1 3];
order = [0.8 4];
Also included is a generalized superellipsoid tool.

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