Making a spherical cap using equations of sphere

Hello everyone,

I am trying to make a spherical cap using points or nodes and mesh grid . I dont want to use built in functions. I am having some problems. Here is my code

% plotting script
% select : which plot you want
set(groot,'defaultLegendInterpreter','latex');
set(0, 'DefaultFigureRenderer','painters');
set(0, 'DefaultFigureRenderer', 'OpenGL');
set(groot,'defaultAxesTickLabelInterpreter','latex');
set(groot,'defaulttextInterpreter','latex');
%%
% Plot the sphere or spherical patch


p = linspace(-1/2,1/2,10);

radius_1 = p(size(p,2))/2;
radius_2 = radius_1 /2;

[X,Y,Z] = meshgrid(p,p,p);

theta= atan(Y./X);

active = (X.^2+Y.^2 +Z.^2 <= radius_1); 
active_2 = (X.^2+Y.^2 +Z.^2 <= (radius_1 -radius_2)) ;


figure()
plot3(X(active),Y(active),Z(active),'o','MarkerFaceColor','red');
hold on 
plot3(X(active_2),Y(active_2),Z(active_2),'o','MarkerFaceColor','cyan');

I cannot draw the spherical cap. I have the thetha but how do i use it to cut the sphere to make spherical cap and i want to know which nodes are inside the cap and which are outside the cap using mesh grid.

Does anyone knows how to do it?

[Bruno Luong]

X.^2+Y.^2 +Z.^2 is the square of the distance to origin, then you compare with radius and diffderence of radius, it does noot make any interpretable sense to me.

Answer 1

[Bruno Luong]

phi=linspace(pi/2,pi/6,30); % the cap end is determined by pi/6 change accordinglt
theta=linspace(0,2*pi,120);
r=3;

[PHI,THETA]=meshgrid(phi,theta);
X=r*cos(THETA).*cos(PHI);
Y=r*sin(THETA).*cos(PHI);
Z=r*sin(PHI);

surf(X,Y,Z)
axis equal;

[Chris Dan]

Hi,

thank you for you answer. But i dont want to use surf. I want to first make a box mesh and than make a spherical cap inside it, so I can see which points are inside the cap and which are outside.

[Bruno Luong]

But I have no idea what you mean by "spherical cap" since you code wrong equation at first. I only guess you want a "spherical shell".

[Chris Dan]

yes, it is similar to cap in geometry

[Chris Dan]

do you know how it can be done,

I have made the box and the sphere, now I just want to cut that sphere so only spherical shell/ cap look alike is plotted inside the box

[Bruno Luong]

I guess your cap is the set

{ X^2 + Y^2 + Z^2 <= R^2 and Z >= something }

-R < something < R

[Sam Chak]

@hamzah khan, instead of using the built-in function as shown by @Bruno Luong, I think you borrowed the concept like computing the Area of Segment:

So, I guess you are trying to remove a solid Cube from a solid Sphere, leaving you with SIX (6) Spherical Caps? Judging from geometry, I think you will get the Cap with a square Base? If my interpretation is wrong, please sketch a simple diagram so that we can understand your product requirement.

[Chris Dan]

hii,

thanks . this is my code now

clear all; close all;


p = linspace(-1/2,1/2,50);

[X,Y,Z] = meshgrid(p,p,p); % box mesh 

radius_1 = p(size(p,2))/2; 
radius_2 = radius_1/2;

active = (X.^2+Y.^2 +Z.^2 <= radius_1 & Z >= radius_2); % plot sphere


figure()
plot3(X(active),Y(active),Z(active),'o','MarkerFaceColor','red');
hold on 

and it is making the spherical cap, thanks

one last question, if i want to fit in the angle thetha, how can i use it?

theta is the angle between y and x if we look at the sphere is 2D

[Bruno Luong]

Sorry I don't understand what you ask for.

And in your last code you are still compare distance squared with radius without squared.

Answer 2

[Chris Dan]

alternate answer:

clear all; close all;
p = linspace(-1/2,1/2,100);
[X,Y,Z] = meshgrid(p,p,p); % box mesh
R = p(size(p,2))/2;
r = R/1.5;
alpha = deg2rad(45);
f = figure('visible','on');

theta = atan2(sqrt(X.^2+Y.^2),Z);

active = (X.^2 + Y.^2 + Z.^2  <= R^2) & (X.^2 + Y.^2 +Z.^2 >= r^2);
plot3(X(active),Y(active),Z(active),'o','MarkerFaceColor','blue');
active = (X.^2 + Y.^2 + Z.^2 <= R^2) & (X.^2 + Y.^2 +Z.^2 >= r^2) & (abs(theta) <= alpha); 
hold on
plot3(X(active),Y(active),Z(active),'o','MarkerFaceColor','red');

saveas(f,'3d_patch','fig')