Let $Q(x),\hat{Q}(x)$ be pair of homogeneous quadratic forms with $x\in \Bbb C^s$.

Let $S_p(x,r)$ be a sphere of radius $r$ with $x\in \Bbb C^s$ centered at $p \in \Bbb C^s$.

Let $Z_Q$, $Z_{\hat{Q}}$ and $Z_S$ be zero sets of $Q(x)$, $\hat{Q}(x)$ and $S_p(x,r)$ respectively.

Let $\hat{v}_s \in Z_S$ be a fixed vector.

We call $[Q(x),\hat{Q}(x)]_{(\hat v_s,p,r)}$ a $(\hat{v}_s,p,r)$-pair if $x \in Z_S\cap Z_Q\iff (\hat{v}_s-x) \in Z_S\cap Z_\hat{Q}$.

Given a homogeneous $Q(x)$, is there always a $\hat{Q}(x)$? If there is a pair, how do one find a homogeneous $\hat{Q}(x)$ that is a pair?