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If you have three points on a the curve of an ellipse in $\mathbb{R}^3$ and you know the eccentricity are you able to find the center, major, and minor vectors. This information should uniquely determine an ellipse in $\mathbb{R}^3$. The form of the equation of the ellipse I'm looking for is in parametric form: $$ \vec{c} + \vec{M}\cos(\theta) + \vec{m}\sin(\theta) $$ Because we know the eccentricity and the properties of an ellipse we know the following: $$ |\vec{M}|\sqrt{1 - e^2}| = |\vec{m}| \quad , \quad \vec{M}\cdot\vec{m}=0 $$ The problem is a generalization of finding the circumcircle of a triangle in $\mathbb{R}^3$ with a general eccentricity instead of an eccentricity fixed at 1.

Blue
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Aidan R.S.
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    Three points + (non-zero) eccentricity are not enough to determine a (non-circle) conic uniquely. See, for instance, this old answer of mine that shows an infinite family of parabolas (so, all with eccentricity $1$) through three given points. Generally, it takes five pieces of information (eg, five points) to determine a conic. ... BTW: Since your given three points necessarily lie in a plane, you might as well restrict your ambient space to $\mathbb{R}^2$. (Also, the eccentricity of a circle is $0$.) – Blue Jun 14 '25 at 02:40

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