I was wondering how the visible ellipse perimeter changes while rotating the ellipse in space. I am not mathematician, but I suppose the issue is probably more complex than it seems, so let's make some assumptions as follow:
- the major to minor semiaxes ratio is $1.4$
- we use the approximate ellipse perimeter equation $L \approx 2\pi \sqrt{\dfrac{a^2+b^2}{2}}$
- we rotate the ellipse around the major semi-axis (i.e. '$x$-line' - please see the PICTURE).
I would very gladly compare this function with the change in visible semi-minor axis length to see how the differences of the two parameters interplay during rotation.
I am MD measuring congenital heart defects so I do not need the analysis to be very complex - the briefer the better.
Thank you for your interest.
Let $\alpha$ be a plane containing your ellipse $\Gamma$, let $a$ be the major semi-axis lenght of $\Gamma$, $b$ the minor semi-axis lenght of $\Gamma$ and let $\beta$ be another plane containing the major axis of $\Gamma$. Let $\theta$ denote the angle between $\alpha$ and $\beta$. The orthogonal projection of $\Gamma$ on $\beta$ is an ellipse $\Psi$ with the same major-axis lenght and semi-minor axis lenght given by $b\cos(\theta)$.
The visible perimeter is then the perimeter $L'$ of $\Psi$ which is given by:
$$L'\approx 2\pi \sqrt{\dfrac{a^2+b^2\cos^2(\theta)}{2}}$$