  We study Lagrange spectra arising from intrinsic Diophantine approximation of
circles and spheres. More precisely, we consider three circles embedded in
$\mathbb{R}^2$ or $\mathbb{R}^3$ and three spheres embedded in $\mathbb{R}^3$
or $\mathbb{R}^4$. We present a unified framework to connect the Lagrange
spectra of these six spaces with the spectra of $\mathbb{R}$ and $\mathbb{C}$.
Thanks to prior work of Asmus L.~Schmidt on the spectra of $\mathbb{R}$ and
$\mathbb{C}$, we obtain as a corollary, for each of the six spectra, the
smallest accumulation point and the initial discrete part leading up to it
completely.
