how to generate ellipsoid in n=4

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imola
imola 2014-11-7
编辑： imola 2015-2-17
Dear All,
I want to generate ellipsoid in dimensions n=4,
Regards, Imola

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Matt J
Matt J 2014-11-7
For any NxN positive definite matrix, A
x.'*A*x=1
is an implicit equation for an N-dimensional ellipsoid.
  2 个评论
imola
imola 2014-11-7
编辑：imola 2015-2-17
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
Matt J 2014-11-8
编辑：Matt J 2014-11-8
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.

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