Finding the major and minor axes lengths of an ellipse given parametric equations

Question

The traditional parametric equations of an ellipse centered at the origin are given by: $$ \{x(t)= a\cos(t)\cos(B) - b\sin(t)\sin(B); y(t)= a\cos(t)\sin(B) + b\sin(t)\cos(B)\} $$ where B is the angle of rotation of the major axis from the x-axis. Here, the major and minor axes lengths are given as a and b. However I have a different parametric form of an ellipse given by: $$ \{x(t)=A\cos(t) + B\sin(t); y(t)=C\cos(t) + D\sin(t)\}. $$ Each equation represents sinusoids along the the x and y axes with independent magnitudes and phases, thus forming an ellipse as the locus $(x(t),y(t))$.

The major and minor axes lengths are not apparent in this form. My approach has been to set the derivative $dx/dy = -(x/y)$ since the slope of the tangent is perpendicular to the radius only at the intersections with the major and minor axes. My result is $t=.5(\tan^{-1}((AB+CD)/2(A^2+C^2-B^2-D^2))$. This value of t does not quite match my numerical simulation (although it is close) and I am perplexed. Any help is greatly appreciated.

Thanks, Bob T.

Answer 1

Hint:

From your equations, if they represents the same ellipse, we have: $$ A=a\cos \beta \qquad B=b\sin \beta $$ $$ C=a\sin \beta \qquad D=b\cos \beta $$ so: $$ \frac{C}{A}=\frac{B}{D}=\tan \beta $$

where $\beta$ is your angle $B$.

Answer 2

With Mathematica’s help, $t$ can be eliminated to get the implicit equation $$\left(C^2+D^2\right)x^2-2(AC+BD)xy=\left(AD-BC\right)^2-\left(A^2+B^2\right) y^2.$$ From here, you can use standard methods to find the principal axes and semi-axis lengths. The latter are eigenvectors of $${1\over(AD-BC)^2}\begin{bmatrix}C^2+D^2 & -(AC+BD) \\ -(AC+BD) & A^2+B^2\end{bmatrix},$$ i.e., $$\left[\left(A^2+B^2\right)-\left(C^2+D^2\right) \pm \sqrt{\left(A^2+B^2+C^2+D^2\right)^2-4\left(BC-AD\right)^2}, 2(AC+BD)\right]^T.$$ The semi-axis lengths are the reciprocal square roots of the corresponding eigenvalues.

Answer 3

I found my mistake. I carelessly placed the 2 in the denominator of my equation:

t_axes=.5(arctan((AB+CD)/2(A^2+C^2^−D^2)) should be t_axes=.5(arctan(2(AB+CD)/(A^2+C^2−B^2−D^2))

This correctly gives the value of t that can be used to find the points of intersection of the principal axes with the ellipse. Thanks for the comments, which helped me get a new perspective on the problem.