R help - Nov 2011 - 3-D ellipsoid equations

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Hello, 




The parametric equations of an ellipsoid can be written in terms of spherical
coordinates. The three spherical coordinates are converted to Cartesian
coordinates by






X=a cos (α) sin(θ) 

Y=b sin(α) sin(θ) 

Z=c cos(θ) 



for α and θ 

The parameter α varies from 0 to 2 π and θ varies from 0 to π . Here ( X o , Y o
,Z o ) is the center of the ellipsoid, and θ is the angle of rotation. I need to
come up with an expression for the ellipsoid expressed parametrically as the
path of a point in 3- space. My first try is that it is something like the
following:



X(alpha)=Xo+a cos(α) cos( θ )-b sin(α) cos( θ ) + c cos( θ ) 
Y(alpha)=Yo+ cos (α) sin(θ)+b sin(α) cos (θ) 
Z(alpha)=Zo+a cos (α) sin(θ) +b sin(α) cos( θ ) 
Most of the books I have read use eigenvectors. The eigenvectors of course
consist of the direction cosines. My difficulty is going from that approach to
the approach that Alberto Monteiro took in his message on the 9 October 2006. I
understand the R code and am using it for a two-dimensional ellipse problem.
There does not seem to be allowance for the new coordinates of the center of the
ellipsoid under the transformation when using direction cosines. By that I mean
adding the centroid coordinates would not be necessary as is done in my
"first try".

Can you help me extend this to 3 dimensions? 

Sincerely, 
Mary A. Marion 














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Hello, 




I want to delete prior questions online but am getting an error message? 




Please see R code in enclosed file. I don't understand the error message. 






The parametric equations of an ellipsoid can be written in terms of spherical
coordinates. The three spherical coordinates are converted to Cartesian
coordinates by






X=a cos (?) sin(?) 

Y=b sin(?) sin(?) 

Z=c cos(?) 

The parameter ? varies from 0 to 2 ? and ? varies from 0 to ? . Here ( X o , Y o
,Z o ) is the center of the ellipsoid, and ? is the angle of rotation. I need to
come up with an expression for the ellipsoid expressed parametrically as the
path of a point in 3- space. I think that it is something like the following:



x (alpha)<- x0 + a * cos(theta) * cos(alpha) - b * sin(theta) * sin(alpha) 
y(alpha) <- y0 + a * cos(theta) * sin(alpha) + b * sin(theta) * cos(alpha) 
z (alpha)<- z0 + a * cos(theta) * sin(alpha) + c * sin(theta) * cos(alpha) 




Do I have these equations correct? 


Most of the books I have read use eigenvectors. The eigenvectors of course
consist of the direction cosines. My difficulty is going from that approach to
the approach that Alberto Monteiro took in his message on the 9 October 2006. I
understand the R code and am using it for a two-dimensional ellipse problem.
There does not seem to be allowance for the new coordinates of the center of the
ellipsoid under the transformation when using direction cosines. By that I mean
adding the centroid coordinates would not be necessary.

I need to come up with an example where I do it both ways(as above and using
direction cosines).

My confusion lies in the fact that rather than one rotational angle theta there
are 9 direction cosines. Can you assist with this.



Sincerely, 
Mary A. Marion 




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