Sorry in advance for the terminology, I'm more programmer than mathematician.
I have an ellipsoid, quadratic surface, from a 3 axis magnetometer.
I know the semi principle axes for the ellipsoid, vectors r1, r2 and r3.
The orthogonal vectors r1, r2 and r3 do not align with the x,y,z world axes.
The ellipsoid is centred at 0,0,0.
Assuming vector lengths r1 > r2 > r3, I wish to "inflate" the ellipsoid into a sphere with radius equal to the length of r1, by scaling the ellipsoid along the r2 and r3 vectors.
The final sphere remains centred at 0,0,0.
There is no rotation of the final sphere with respect to the original ellipsoid, just an odd inflation.
Q: Is it possible to construct a single 3x3 matrix to do this scaling, without using 3(?) rotations to align the ellipsoid axes with the world xyz axes pre scaling and then undoing those rotations post scaling?
It is OK if this is not possible. It might save a bit of coding space.
Examples, explanations, etc. much appreciated.
This looks like the ellipsoid is already aligned with the world xyz axes.
This seems to have a rotation in the creation of the ellipsoid from sphere.