  We investigate how to find generic and globally rigid realizations of graphs
in $\mathbb{R}^d$ based on elementary geometric observations. Our arguments
lead to new proofs of a combinatorial characterization of the global rigidity
of graphs in $\mathbb{R}^2$ by Jackson and Jord\'an and that of body-bar graphs
in $\mathbb{R}^d$ recently shown by Connelly, Jord\'an, and Whiteley. We also
extend the 1-extension theorem and Connelly's composition theorem, which are
main tools for generating globally rigid graphs in $\mathbb{R}^d$. In
particular we show that any vertex-redundantly rigid graph in $\mathbb{R}^d$ is
globally rigid in $\mathbb{R}^d$, where a graph $G=(V,E)$ is called
vertex-redundantly rigid if $G-v$ is rigid for any $v\in V$.
