  Let $N$ be an $n$-dimensional $C^\infty$ manifold and let
$\widetilde{\varphi}: N\to \mathbb{R}^{n+1}$, $\widetilde{\nu}: N\to S^n$ be
$C^\infty$ mappings. We first give a necessary and sufficient condition for the
hyperplane family $\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})}$
defined by $\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})} =\cup_{x\in
N}\left\{X\in \mathbb{R}^{n+1}\; |\; \left(X-\widetilde{\varphi}(x)\right)\cdot
\widetilde{\nu}(x)=0\right\}$ to create an envelope (Theorem \ref{theorem1}).
As a by-product of the proof of Theorem 1, when the given hyperplane family
$\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})}$ creates an envelope
$\widetilde{f}: N\to \mathbb{R}^{n+1}$, an explicit expression of the envelope
$\widetilde{f}$ is obtained in terms of $\widetilde{\varphi}$ and
$\widetilde{\nu}$ (Corollary \ref{corollary2}). %at the end of \S
\ref{section3}. The vector formula given in Corollary 2 holds even at a
singular point of $\widetilde{\nu}$ so long as the hyperplane family
$\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})}$ creates an envelope. In
this sense, Corollary \ref{corollary2} may be regarded as a complete
generalization of the celebrated Cahn-Hoffman vector formula. Moreover, we give
a criterion when and only when %the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi}, \widetilde{\nu}\right)}$ creates a
unique envelope (Theorem 2).
