The the interpolate on the spherical coordinate system

I want to make a structure(eg: r=cos^10(theta)) in the spherical coordinate system and plot the combination of (orginal pattern+rotota 90 pattern)

So I planed to transfer it to cartesian coordinate system firstly. (In the cartesian coordinate system, the pattern rotation is much easier).

close all;
clear all;
THETA_1 = (1e-9:0.01:pi);
PHI_1 = 1e-9:0.01:2*pi;
[theta_1,phi_1] =meshgrid(THETA_1,PHI_1);


%%
% Pattern and rotate
Pattern=cos(theta_1).^10;
patch_dB = 10*log10(abs(Pattern).^2);
Rlim = -30;
patch_dB_temp = patch_dB - Rlim;
patch_dB_temp(patch_dB_temp < 0) = 0;
X_patch_1 = Pattern.*sin(theta_1).*cos(phi_1);
Y_patch_1 = Pattern.*sin(theta_1).*sin(phi_1);
Z_patch_1 = Pattern.*cos(theta_1);
figure(1)
surf(X_patch_1,Y_patch_1,Z_patch_1,'EdgeColor','interp');
figure(3)
surf(X_patch_1,-Z_patch_1,Y_patch_1,'EdgeColor','interp');

Now, we can obtain the rotate 90 degree plotting(figure 3) and original plotting(figure 1).

However, the data point location is also changed and I can not combine them together.

Acutally, we need to get the same data point location with rotate 90 degree pattern(as followed), so i tried to get the corresponding value in the previous point location by interpolating.

But it shows the error:

Error using interp2>makegriddedinterp (line 237)

Input grid is not a valid MESHGRID.

%% Transfer to spherical coordinate system and check the plotting
[azimuth,elevation,r]= cart2sph(X_patch_1,-Z_patch_1,Y_patch_1);
figure(2)
theta_2=pi/2-elevation;
phi_2=azimuth+pi;
X_patch_2 = r.*sin(theta_2).*cos(phi_2);
Y_patch_2 = r.*sin(theta_2).*sin(phi_2);
Z_patch_2 = r.*cos(theta_2);
% plot3(X_patch_2,Y_patch_2,Z_patch_2,'o')

surf(X_patch_2,Y_patch_2,Z_patch_2,'EdgeColor','interp');
%%
%%Sample the value
THETA = (1e-9:0.01:pi);
PHI = (1e-9:0.01:2*pi);
[theta,phi] =meshgrid(THETA,PHI);
interp2(theta_2,phi_2,r,theta,phi)

[Walter Roberson]

the part that has to be a mesh is theta_2 and phi_2

Answer 1

[Matt J]

I want to make a structure(eg: r=cos^10(theta)) in the spherical coordinate system and plot the combination of (orginal pattern+rotota 90 pattern)

You could just use hgtransform()

h1=surf(X_patch_1,Y_patch_1,Z_patch_1,'EdgeColor','interp');

t2 = hgtransform('Parent',gca, 'Matrix',makehgtform('xrotate',pi/2));
h2 = copyobj(h1,t2);