I want to show intersection of these two spheres. How should I do it?

theta=linspace(0,2*pi,40);

phi=linspace(0,pi,40);

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

r=1;

x=r*sin(phi).*cos(theta);

y=r*sin(phi).*sin(theta);

z=r*cos(phi);

mesh(x,y,z)

hold on

theta=linspace(0,2*pi,40);

phi=linspace(0,pi,40);

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

r=1;

x=r*sin(phi).*cos(theta);

y=r*sin(phi).*sin(theta);

z=r*cos(phi);

x=x+0.25;

y=y-0.2;

z=z+0.1;

surf(x,y,z)

Answer 1

[Kye Taylor]

The spheres you describe have equations

1.) x^2 + y^2 + z^2 = 1

2.) (x-0.25)^2 + (y+1/5)^2 + (z-0.1)^2 = 1

Since equations 1 and 2 have same right-hand-side (equal to one), set the left-hand sides equal and you'll end up getting rid of the squared terms to be left with

3.) 5*x+4*y+2*z = 9/8

Equation 3 is the equation for the plane that contains the intersection of the two spheres. To see it add these lines of code to the end of your code above.

[X,Y] = meshgrid(linspace(-1,1,20));
Z = -5/2*X + 2*Y + 9/16;
surf(X,Y,Z)

That gives you the plane that contains the intersection. Realize that the intersection of the spheres is actually a curve that is a circle in this plane. Parametrizing that circle is more complicated.

[Samira]

thank you, Kye!

and how can I show the common volume of these two spheres?!

[Kye Taylor]

My pleasure!

What do you mean by common value? As you move along the curve where the two spheres meet, the values of (x,y,z) will change.

[Jelle]

In case you guys haven't seen it yet, there is a sign mistake in the code. Equation 3 is correct and hence the code should be:

[X,Y] = meshgrid(linspace(-1,1,20));
Z = -5/2*X + 2*Y - 9/16;
surf(X,Y,Z)

This answer is validated by plotting both unit spheres and the plane that contains the intersection.

[X,Y] = meshgrid(linspace(-1,1,20));
Z = -5/2*X + 2*Y - 9/16;
surf(X,Y,Z)
hold on
[x,y,z] = sphere;
surf(x,y,z)
surf((x-0.25),(y+0.2),(z-0.1))
daspect([1 1 1])