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Let $G=(V,E)$ be a 3-connected planar graph, with $V=\{1,...,n\}$. Let $M=\left(m_{i,j}\right)$ be a symmetric $n\times n$ matrix with exactly one negative eigenvalue (of multiplicity 1), such that for $i,j$ with $i\ne j$, if $i$ and $j$ are adjacent then $m_{i,j}\<0$ and if $i$ and $j$ are nonadjacent then $m_{i,j}=0$, and such that $M$ has rank $n-3$. Then the null space $kerM$ of $M$ gives an embedding of $G$ in $S^2$ as follows: let $\{a,b,c\}$ be a basis of $kerM$, and for $i\in V$ let $\varphi \left(i\right):=(a_i,b_i,c_i{)}^T$; then $\varphi \left(i\right)\ne 0$, and $\psi \left(i\right):=\varphi \left(i\right)/\parallel \varphi \left(i\right)\parallel$ embeds $V$ in $S^2$ such that connecting, for any two adjacent vertices $i,j$, the points $\psi \left(i\right)$ and $\psi \left(j\right)$ by a shortest geodesic on $S^2$, gives...