How to generate 3D helix in MATLAB or Python?

I have written a code to generate x, y, z points of a helix and got this result:

Code:

clear all; delete all, clc;
% Spiral constants
THETA_0 = 5; % constant
THETA_1 = 10.3; % starting angle
A = 3.762;
B = 0.001317;
C = 7.967;
D = 0.1287;
E = 0.003056;

s=2;
% Calculate (x,y,z) coordinates of points defining the spiral path
theta = THETA_1:.1:910.3;  % range, starting degree angle:final degree angle
for i = 1:length(theta)
    if (theta(i)<=99.9)
        R(i) = C*(1-D*log(theta(i)-THETA_0));
    else
%       theta_mod = 0.0002*theta(i)^2+.98*theta(i);
        R(i) = A*exp(-B*theta(i));
    end
    % scaling 
    x(i) = s*R(i)*cosd(theta(i));
    y(i) = s*R(i)*sind(theta(i));
    z(i) = s*E*(theta(i)-THETA_1);
end

helix=animatedline('LineWidth',2);
axis equal;
axis vis3d;
% set (gca,'XLim', [-5 5],'YLim', [-10 10], 'ZLim',[0 6])
view(43,24);
hold on;
for i=1:length(z)
    addpoints(helix, x(i),y(i),z(i));
    head=scatter3 (x(i),y(i),z(i));
    drawnow
%   pause(0.01);
    delete(head);
end

and I want a helical structure around it similar to this

Your second picture give you:

• The parametric equations for x,y and z.
• The restriction on u and v

You have all the necessary informations to create your geometry:

%plot the mesh
u=linspace(0,4*pi,50);
v=linspace(0,2*pi,50);
[u,v]=meshgrid(u,v);
x=(1.2+0.5*cos(v)).*cos(u);
y=(1.2+0.5*cos(v)).*sin(u);
z=0.5*sin(v)+u/pi;
surf(x,y,z)
hold on

%plot the 3d line
u = linspace(0,4*pi,40)
x=1.2.*cos(u);
y=1.2.*sin(u);
z=u/pi;
plot3(x,y,z,'r');

axis equal

Now you just have to adjust the parametric equations to fit your line.

EDIT: To apply the same solution to your specific case you simply have to replace u and v with your theta and R variable in the meshgrid function:

THETA_0 = 5; % constant
THETA_1 = 10.3; % starting angle
A = 3.762;
B = 0.001317;
C = 7.967;
D = 0.1287;
E = 0.003056;

s=2;
% Calculate (x,y,z) coordinates of points defining the spiral path
theta = THETA_1:5:910.3;
for i = 1:length(theta)
    if (theta(i)<=99.9)
        R(i) = s*C*(1-D*log(theta(i)-THETA_0));
    else
        R(i) = s*A*exp(-B*theta(i));
    end

end

x = R.*cosd(theta);
y = R.*sind(theta);
z = E.*(theta-THETA_1);
plot3(x,y,z,'r','linewidth',2)

hold on
[u,v]=meshgrid(theta,R);
x=(R+0.5*cos(v)).*cosd(u);
y=(R+0.5*cos(v)).*sind(u);
z=0.5*sin(v)+E.*(u-THETA_1);
mesh(x,y,z,'facecolor','none')

axis equal

Result:

by the way I'm not a big fan of mixing cosd and cos in the same script, but do what you want.