# Parametric equation of an ellipse in the 3D space

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?

• Hint : Points on ellipse can be parameterized as $(a\cos t, b\sin t)$. Jan 21, 2021 at 20:49
• You have two vectors which represent the direction of both major and minor semi-axes and its magnitude. You have a position vector which represents its center and both vectors are orthogonal to the normal vector of the plane that ellipse lies in. This represents all points on the ellipse. Jan 21, 2021 at 20:58

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 $$\|u\| > \|v\|$$.
• If you don't assume that $u \perp v$ you will not have in general maximal/minimal curvature at the extremities of $u$ and $v$. Jan 21, 2021 at 21:53
• You are correct. If they are not perpendicular, they are just some vectors from the center to the curve. But for any given ellipse you can choose $c,u,v$ as I specified to get a parameterization. Jan 21, 2021 at 22:50
• I plotted this parametrization along with the $\mathbf u$ and $\mathbf v$ vectors: it's true that they are not in the direction of the axis of the ellipse unless they are orthogonal (although they are indeed in the plane of the ellipse). math3d.org/FMjy6jzb Jan 22, 2021 at 9:46