how to generate ellipsoid in n=4

Dear All,

I want to generate ellipsoid in dimensions n=4,

Regards, Imola

Answer 1

[Matt J]

For any NxN positive definite matrix, A

    x.'*A*x=1

is an implicit equation for an N-dimensional ellipsoid.

[imola]

Dear Matt,

Thanks for replying, the equation

(x-v).'*A*(x-v)=1

is general formula., it is written in Wikipedia that The parameters may be interpreted as spherical coordinates. and when I search for the last I found that we can find the spherical coordinates as follow

X1=r*cos(t1)
X2=r*sin(t1)cos(t2)
X3=r*sin(t1)sin(t2)cos(t3)
.
.
.
Xn-1=r*sin(t1)...sin(tn-2)cos(tn-1)

Xn=r*sin(t1)...sin(tn-2)sin(tn-1)

I really hope you agree with me it true, that will save me.

regards,

Imola

[Matt J]

so can I use them as parameters for the ellipsoid in higher dimensions but I just change the radius

You can if the ellipsoid is unrotated/translated. The equation for an unrotated ellipsoid centered at the origin is

      sum (X(i)/e(i)).^2=1

You can see by direct substitution that the equation will be satisfied by an X of the form

    X(1)=e(1)*cos(t1)
    X(2)=e(2)*sin(t1)cos(t2)
    X(3)=e(3)*sin(t1)sin(t2)cos(t3)
    .
    .
    .
    X(n-1)=e(n-1)*sin(t1)...sin(tn-2)cos(tn-1)
    X(n)=e(n)*sin(t1)...sin(tn-2)sin(tn-1)

If the ellipsoid is rototranslated, you must apply a further transformation to X,

     X'=R*X+t.

where R is an NxN orthogonal matrix and t a translation vector.