how to pass from x y z plane to toroidal coordinate?

Hey guys,

I am trying to make the transition from x, y, z coordinate to toroidal coordinate.

I tried to write the following code but I didn't achieve to do correctly the transition.

Any idea how to fix fix

Thank you in advance

Code:

clear all

clc

subplot 121

% suppose we have a line in the x,y,z plane

x=linspace(-1,1,10)

y=linspace(-2,2,10)

z=linspace(-3,3,10)

plot3(x,y,z)

grid on

subplot 122

% first creat torous

r_i = 1; % inner radius

r_o = 3; % outer radius

theta = linspace(-pi,pi,20);

phi = linspace(0*pi,2*pi,20);

[thetaa,phii] = meshgrid(phi,theta);

xx = (r_o + r_i.*cos(phii)).*cos(thetaa);

yy = (r_o + r_i.*cos(phii)).*sin(thetaa);

zz = r_i.*sin(phii);

surf(xx,yy,zz,'Facealpha',0,'Edgealpha',0.3);

hold on

% second do the transformation from x,y,z plane to the Toroidal Coordinates

R=(r_i+r_o)/2

f=atan(y./x) %azimuthal angle

r=sqrt( (sqrt(x.^2+y.^2)-R).^2+z.^2) %

t=atan(z./(sqrt(x.^2+y.^2)-R))

plot3(r,t,f,'MarkerSize',50)

[Bjorn Gustavsson]

It is not at all clear how you want to transform that line to your torus-surface. I know how to map the coordinates of the x-y plane (or the complex plane if you want) to a spherical surface:

theta = linspace(0*pi,pi,19);  
phi = linspace(-pi,pi,25);   
[phii,thetaa] = meshgrid(phi,theta);
r_i = 1;
xx = r_i.*cos(phii).*cos(thetaa);
yy = r_i.*cos(phii).*sin(thetaa);
zz = r_i+r_i.*sin(phii);
surf(xx,yy,zz,'Facealpha',0.2,'Edgealpha',0.3);
hold on
x=linspace(-1,1,10);
y=linspace(-1,1,10)+1;
phi_l = atan2(x,y);
theta_l = atan((x.^2+y.^2).^.5/(2*r_i));
plot(x,y,'g')                                                                                                
plot3(r_i*sin(phi_l).*sin(2*theta_l),r_i*cos(phi_l).*sin(2*theta_l),r_i-r_i*cos(2*theta_l),'g','linewidth',2)
azV = -157.04;
elV =  34;
view(azV,elV-5)

You might have to make a more explicit explanation. Perhaps it becomes easier if you plot both line and torus-surface in the same axes?

[Rabih Sokhen]

Hy Bjorn

first thank you for your help.

i would like to know how to change x,y,z to ro, fi,theta in the toroidal coordiante

thanks

Answer 1

[John D'Errico]

You have a line in three dimensionas, and you want to find the equation of that line in a toroidal coordinate system?

First, what is the equation of a line? Most simply, I would do it parametrically. Perhaps like this, using vectors X0 and DX, as

X = X0 + DX*t

In terms of coordinates x,y, and z, you might write:

x = x0 + dx*t

y = z0 + dz*t

z = z0 + dz*t

In this link, we see some explanation of a toroidal coordinate system.

https://en.wikipedia.org/wiki/Toroidal_coordinates

In fact, that link even gives a conversion to the toroidal coordinate system from Cartesian coordiantes, (x,y,z).

[Rabih Sokhen]

Hey John

• You have a line in three dimensions as, and you want to find the equation of that line in a toroidal coordinate system?

--> Yes, exactly

• what is the equation of a line?

--> I don't have a specific equation, we can use any coordinate in the x, y, z plane.

I have seen the following link, but I didn't really know how to make this transition since I don't have d1 and d2 as mentioned in the following link

thank you

[Bjorn Gustavsson]

It should be as simple as stepping through the section Inverse_transformation, there you can take equation by equation to calculate the phi, but you'd use atan2, rho and tau from your x, y, and z points without much problems.

@John D'Errico: It never before occurred to me to move away from the torus-surface - so I learnt something today too. Now I wonder if this principle of moving away extends to other pastry-shaped things?