I have found here that an ellipse in the 3D space can be expressed parametrically by
$$\mathbf x (t)=\mathbf c+(\cos t)\mathbf u+(\sin t)\mathbf v$$
with $\mathbf c = (c_1,c_2,c_3)$ being the center of the ellipse and the lenghts of the half-axis being the lengths of the vectors $\mathbf u = (u_1,u_2,u_3)$ and $\mathbf v = (v_1,v_2,v_3)$.
How could these three vectors $\mathbf c$, $\mathbf u$ and $\mathbf v$ be related to the directions of the axis of the ellipse? Is there maybe a parametric equation for the ellipse in an arbitrary plane of the space whose elements have a more intuitive meaning?
Best Answer
In the parametric equation $\mathbf x (t)=\mathbf c+(\cos t)\mathbf u+(\sin t)\mathbf v$, we have: $\mathbf c$ is the center of the ellipse, $\mathbf u$ is the vector from the center of the ellipse to a point on the ellipse with maximum curvature, and $\mathbf v$ is the vector from the center of the ellipse to a point with minimum curvature. I assumed $\|u\| > \|v\|$.