1. Matrix Help please!!!

Hi everyone,

I have a 3x3 orthogonal matrix. Its first column represemts a point in 3d Cartesian coordinates. Its second column represents a major axis and the third a minor axis.

Let' say I have the following 3x3 matrix:
:[-0.51746 0.024186 0.85537
-0.84703 0.1275 -0.51602
-0.12154 -0.99154 -0.045487]

As I understand the first column represents $\displaystyle \theta=92.607 , \phi=328.9$ in degrees.

Am I right to say that the two other columns (major, minor) can be represented by $\displaystyle (\theta major, \phi major) and (\theta minor, \phi minor)$ respectively in degrees?

I have read that is necessary to specify 3 angles to determine this 3x3 matrix. Two for the first column (namely theta,phi) and a further one to specify the major and minor axes. However, I cannot understand this point at all.

I know it is correct but I cannot understand why. I would very much appreciate anyones help.

Many Thanks

Best Regards

Alex

2. Originally Posted by tecne
Hi everyone,

I have a 3x3 orthogonal matrix. Its first column represemts a point in 3d Cartesian coordinates. Its second column represents a major axis and the third a minor axis.

Let' say I have the following 3x3 matrix:
Code:
:[-0.51746      0.024186        0.85537
-0.84703      0.1275           -0.51602
-0.12154     -0.99154        -0.045487]
As I understand the first column represents $\displaystyle \theta=92.607 , \phi=328.9$ in degrees.

Am I right to say that the two other columns (major, minor) can be represented by $\displaystyle (\theta major, \phi major) and (\theta minor, \phi minor)$ respectively in degrees?

I have read that is necessary to specify 3 angles to determine this 3x3 matrix. Two for the first column (namely theta,phi) and a further one to specify the major and minor axes. However, I cannot understand this point at all.

I know it is correct but I cannot understand why. I would very much appreciate anyones help.

Many Thanks

Best Regards

Alex
The columns of an orthogonal matrix are unit vectors, so expressing them in spherical polars requires just $\displaystyle \theta$, $\displaystyle \phi$ as $\displaystyle r=1$. Similarly expressing them in direction cosines (essentialy the same thing) requires only two parametres.

RonL

3. Hi,

I think I am still confused. How is then possible to express the major and minor axis in terms of these two angles?

I thought that the centre of the ellipse correpsonds to the 3d point, found in the first column of the 3x3 orthogonal matrix.

I know that the 2nd column represents the major axis. If we have an arbitrary orientation of this ellipse, then the second column should be translated as a separate point in 3d space.

By using the cartesian to spherical coordinate transformation:

$\displaystyle sin\theta$ $\displaystyle cos\phi$
$\displaystyle sin\theta$ $\displaystyle sin\phi$
$\displaystyle cos\theta$

For example the second column given in the matrix below

:[-0.51746 0.024186 0.85537
-0.84703 0.1275 -0.51602
-0.12154 -0.99154 -0.045487]

is expressed in Cartesian coordinates. Can we not express this in spherical coordinates? Am I wrong?

Thanks

Regards

Alex

4. Originally Posted by tecne
Hi,

I think I am still confused. How is then possible to express the major and minor axis in terms of these two angles?

I thought that the centre of the ellipse correpsonds to the 3d point, found in the first column of the 3x3 orthogonal matrix.

I know that the 2nd column represents the major axis. If we have an arbitrary orientation of this ellipse, then the second column should be translated as a separate point in 3d space.

By using the cartesian to spherical coordinate transformation:

$\displaystyle sin\theta$ $\displaystyle cos\phi$
$\displaystyle sin\theta$ $\displaystyle sin\phi$
$\displaystyle cos\theta$

For example the second column given in the matrix below

:[-0.51746 0.024186 0.85537
-0.84703 0.1275 -0.51602
-0.12154 -0.99154 -0.045487]

is expressed in Cartesian coordinates. Can we not express this in spherical coordinates? Am I wrong?

Thanks

Regards

Alex
First as these are all unit vectors all they are capable of doing is telling you the orientation of the ellipse not the length of the major or minor axes.

The first column can only represent the position of the centre if it is on the unit circle.

I would sugest you repost your question in the form you recieved it, it seems very likely that some confusion has crept in

RonL

R=$\displaystyle \begin{array}{ccc}-0.51746&0.024186&0.85537\\-0.84703&0.1275&-0.51602\\-0.12154&-0.99154&-0.045487 \end{array}$
We know that the second and third columns represent the orientation of the major and minor axes respectively. Since the major axis is expressed in Cartesian coordinates $\displaystyle \begin{array}{ccc}[0.024186&0.1275&-0.99154]^T\end{array}$ can we not express it in terms of two angles $\displaystyle (\theta major = 172.5435, \phi major =79.2589)$? Am I wrong to say that?
For example we know that the first column can be expressed as $\displaystyle \theta center=92.607 , \phi center=328.9$ after the transformation to spherical coordinates.