Let the n-sphere of radius $r$ centered at $(0,0,...,0,y)\in\mathbb{R}^{n+1}$ be defined by $$ \mathcal{S} \iff {x_1}^2 + {x_2}^2 + ... + {x_n}^2 + (x_{n+1}-y)^2 = r^2 $$ and consider the function $d$ which to any point in the unit-ball $B(0,r)\subset \mathbb{R}^n$ associates the dependent coordinate $x_{n+1}\leq y$ in the lower hemisphere of $\mathcal{S}$: $$ d\ :\ v\in B(0,r)\ \mapsto\ y -\sqrt{r^2 - \|v\|^2} $$
For a given $v = (v_1,...,v_n)\in B(0,r)$, consider now the function $$ \forall t\in I_v\subset\mathbb{R},\quad \psi_v(t) = \big( tv_1, tv_2, ..., tv_n, d(tv) \big) $$
Is the image of $\psi_v$ a circle?
By definition of $d$ and for a given $v$, the image of $\psi_v$ will always be a half-circle of radius $r$ centered at $y$ in the plane $(x-y)\cdot(\psi_v(0)\wedge\psi_v(1)) = 0$.