Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$-grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$, let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$. We study the $G_0$-equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$. It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$, then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$-orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$, then both are $G_0$-prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$. Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$, then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$. As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G-orbits in place of ${\mathcal O}_{\textsf{min}}$.