Find the equation of circle defined on 2-dimensional subspace of $\mathbb{R}^n$

Question

Let $\mathbf{u},\mathbf{v}\in\mathbb{R}^n$ be two $n$-dimensional vectors, which --using the standard basis-- are given as $\mathbf{u}=\left(u_1,\ldots,u_n\right)^\top$ and $\mathbf{v}=\left(v_1,\ldots,v_n\right)^\top$.

Granted that $\mathbf{u}$ and $\mathbf{v}$ are linearly independent, they define a subspace of $\mathbb{R}^n$, the 2-plane $\mathcal{H}$, which we assume that includes the origin, as shown below.

Finally, a circle of radius $r$ is defined on $\mathcal{H}$ centered at the origin, as shown below.

My question is about the expression of the circle equation using the original coordinates (i.e., the standard basis of $\mathbb{R}^n$).

Answer 1

1. In a first step, orthogonalize basis $(u,v) \ \rightarrow \ (u,w)$, by keeping $u$ and replacing $v$ by

$$w=v+au$$

such that $w \perp u$, i.e., such that the dot product $u.w=u.(v+au)$ is $0$, which is possible iff $ u.v+a\|u\|^2=0$, giving

$$a=-\dfrac{u.v}{\|u\|^2}=-\dfrac{\|u\|\|v\|\cos \theta}{\|u\|^2}=-\dfrac{\|v\|\cos \theta}{\|u\|}.$$

2. Then, normalize this basis : $(u,w) \ \rightarrow \ \left(u'=\dfrac{u}{\|u\|},w'=\dfrac{w}{\|w\|}\right)$.

3. Finally, a parametric equation for the circle is:

$$x \ = \ \cos \theta \ u'+ \sin \theta \ w'$$