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For the basic definitions on rigidity of frameworks and graphs, see the page [[Rigid graph]], or see the survey of Whiteley <ref name="Wi2"/> for more informationa detailed overview.

For Given a ''d''-dimensional framework ''(G,p)'', we introduce the ''d''-dimensional associated [[rigidity matrix]] ''R(G,p)'' to describe defines infinitesimal rigidity. An '''infinitesimal motion''' of a framework is an assignment <math>s:V \rightarrow \mathbb{R}^d</math> of vectors to the nodes such that the corresponding motion preserves edge lengths. More precisely, <math>( p(u)-p(v)) ( s(u)-s(v)) =0</math> for every edge ''uv'', that is, each edge is orthogonal to the difference of the infinitesimal motions of its endpoints. This can be written as <math>R(G,p)s=0</math>. Some trivial infinitesimal motions can be derived from isometries of <math>\mathbb{R}^d</math> that do not change orientation. These form a vector space of dimension <math>d+1 \choose 2</math> with <math>d</math> independent shifts and <math>d-1 \choose 2</math> independent rotations.

==Rigidity in 2 dimensions-dimensional rigidity==