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given an ellipsoid in parametric form int, I was trying to get to the classical equation inx,y. Things are very straightforward, as long as the ellipse radii are aligned with the principal axes. Instead, I am trying to find the equation of a "rotated" ellipse, given a parametrization int.

I tried the following... Let's define the position vector:

[tex]\mathbf{r}(t) = \mathbf{a}cos(t) + \mathbf{b}sin(t)[/tex]

where:

[tex]\mathbf{a}=a_1\mathbf{e_1} + a_2\mathbf{e_2}[/tex]

[tex]\mathbf{b}=b_1\mathbf{e_1} + b_2\mathbf{e_2}[/tex]

and we have that [tex]<\mathbf{a},\mathbf{b}>=0[/tex], that is, the directional radii are perpendicular but not aligned to the main axes.

Since [tex]x = <\mathbf{r},\mathbf{e_1}>[/tex], and [tex]y = <\mathbf{r},\mathbf{e_2}>[/tex], we have:

[tex]x = a_1cos(t) + b_1sin(t)[/tex]

[tex]y = a_2cos(t) + b_2sin(t)[/tex]

At this point I got stuck, because I can't manage to get rid oft. When the ellipse is aligned to the main axes we have [tex]b_1=0[/tex], and [tex]a_2=0[/tex], and everything becomes easy by squaring the terms.

I know that the final result should be of the form: [tex]\mathbf{x^T}A\mathbf{x}[/tex] where A is symmetric positive definite, but I can't really get there.

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# Equation of ellipsoid

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