Equation of the ellipse for musical notation (the quarter note/crotchet and shorter)

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Note heads are often represented with a slightly rotated ellipse, as shown here for instance (first image). Does anyone happen to know the equation of the ellipsis, and the rotation that's applied to it ? The only thing I thought about was trying to bruteforce it until I get something that looks alike.

PS : I hesitated between this website and SO. Yet, even if it's for programming purposes, it would have been off-topic on SO because the problem is not the development part, but rather the mathematical one.