plot an ellipsoid with matlab

Hello,

I have an equation of an ellipsoid x^2+y^2+z^2-1=0 and I need to plot without using the ellipsoid function of matlab. thanks

[Roger Stafford]

I'm sorry. I erased my original response and your first comment by mistake. They were:

My original response:

- - - - - -

 [theta,phi] = ndgrid(linspace(0,pi),linspace(0,2*pi));
 x = sin(theta).*cos(phi);
 y = sin(theta).*sin(phi);
 z = cos(theta);
 surf(x,y,z)

Note: Your particular ellipsoid is called a 'sphere'.

- - - - -

Your first comment:

could you, please, explain me again. I did not understand your response.

Suppose that I have another ellipsoid not canonic like x^2/3+Y^2/4+z^2/3-2=0. how I can plot it

Answer 1

[Roger Stafford]

Your second ellipsoid's equation can be rewritten in the form:

 x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

where a, b, and c are defined as below:

 a = sqrt(6);
 b = sqrt(8);
 c = sqrt(6);
 [theta,phi] = ndgrid(linspace(0,pi),linspace(0,2*pi));
 x = a*sin(theta).*cos(phi);
 y = b*sin(theta).*sin(phi);
 z = c*cos(theta);
 surf(x,y,z)

[ProgramMatlab]

Thanks a lot!

If i have a matrix rotation and R and a translation T. How I can applicate these transformation on the ellipsoid equation?

[ProgramMatlab]

Hello, Could you give me the theoretical that let us write

x = a*sin(theta).*cos(phi); y = b*sin(theta).*sin(phi); z = c*cos(theta); thanks!

[Roger Stafford]

To begin with, suppose we have a sphere of radius r centered at the origin

 x^2/r^2 + y^2/r^2 + z^2/r^2 = 1.

The quantities r, theta and phi are spherical coordinates that satisfy the three equations

 x = r*sin(theta)*cos(phi)
 y = r*sin(theta)*sin(phi)
 z = r*cos(theta)

as you can verify by looking up references on spherical coordinates. Now imagine stretching (or contracting) the sphere by a ratio of a/r in the x-direction, by b/r in the y direction, and by c/r in the z direction. Then you would have the three equations

 x = a*sin(theta)*cos(phi)
 y = b*sin(theta)*sin(phi)
 z = c*cos(theta)

which satisfy the equation

 x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

and the sphere has become an ellipsoid. The theta and phi parameters are no longer spherical coordinates but they serve very nicely as the needed two parameters for representing the surface of the ellipsoid for plotting by 'surf'. The two families of curves you see in the surface plot represent contours of constant theta and constant phi, respectively, in accordance with their values in the grid from 'ndgrid'.

Answer 2

[Roger Stafford]

It depends on whether R or T is to be performed first. I will assume you want to first rotate by multiplying by the 3 x 3 matrix R from the left, and then translate the center by adding the 3 x 1 column vector T.

 % Generate x, y, and z as before
 P = bsxfun(@plus,R*[x(:),y(:),z(:)]',T); % Rotate, then translate
 X = reshape(P(1,:),size(x,1),size(x,2));
 Y = reshape(P(2,:),size(y,1),size(y,2));
 Z = reshape(P(3,:),size(z,1),size(z,2));
 surf(X,Y,Z)