Let consider an hyper-ellipse $\mathcal{Q}$ in $\mathbb{R}^n$ given by $x^\top Q x = K$ and a hyperplane $\mathcal{H}$ in $\mathbb{R}^n$ given by $a^\top x = b$. We assume that $Q \in \mathbb{R}^{n \times n}$ is positive definite, $K >0$ and $a \in \mathbb{R}^n$ has unit norm, i.e. $||a|| = 1$.
I would like to get a parametric expression for the intersection $\mathcal{I} = \mathcal{Q} \cap \mathcal{H}$?
Let $L L^\top = Q$ be a Cholesky factorization of $Q$. We introduce the orthonormal columns $H \in \mathbb{R}^{n \times n-1}$ such that $H^\top a = 0$ and $H^\top H = I_{n-1}$.
We write $x \in \mathcal{H}$ as $x = (a a^\top) x + (I - a a^\top) x = \alpha a + H (H^\top x) = \alpha a + H t$ with $t \in \mathbb{R}^{n-1}$ a free parameter.
Plugging this decomposition into $x^\top Q x = K$ gives:
$(L^\top\alpha a + L^\top H t)^\top (\alpha L^\top H a + L^\top H t) = K$
Is there a way to rewrite this expression to highlight that the variable $t$ is on a ellipsoid embedded in $\mathbb{R}^{n-2}$?
Thank you for your help,
