▁mot ions . ▁In ▁this ▁case ▁the ▁framework ▁is ▁said ▁to ▁be ▁first - order ▁( or ▁inf init es im ally ) ▁rig id . ▁More ▁generally , ▁an ▁edge ▁ ▁belongs ▁to ▁the ▁mat roid ▁closure ▁operation ▁of ▁a ▁set ▁ ▁if ▁and ▁only ▁if ▁there ▁does ▁not ▁exist ▁a ▁continuous ▁motion ▁of ▁the ▁framework ▁that ▁changes ▁the ▁length ▁of ▁ ▁but ▁leaves ▁the ▁lengths ▁of ▁the ▁edges ▁in ▁ ▁un changed . ▁ ▁Although ▁defined ▁in ▁different ▁terms ▁( column ▁vectors ▁versus ▁row ▁vectors , ▁or ▁forces ▁versus ▁mot ions ) ▁static ▁rig id ity ▁and ▁first - order ▁rig id ity ▁reduce ▁to ▁the ▁same ▁properties ▁of ▁the ▁underlying ▁matrix ▁and ▁therefore ▁co inci de ▁with ▁each ▁other . ▁In ▁two ▁dimensions , ▁the ▁generic ▁rig id ity ▁mat roid ▁also ▁describes ▁the ▁number ▁of ▁degrees ▁of ▁freedom ▁of ▁a ▁different ▁kind ▁of ▁motion , ▁in ▁which ▁each ▁edge ▁is ▁const rained ▁to ▁stay ▁parallel ▁to ▁its ▁original ▁position ▁rather ▁than ▁being ▁const rained ▁to ▁maintain ▁the ▁same ▁length ; ▁however , ▁the ▁equivalence ▁between ▁rig id ity ▁and ▁parallel ▁motion ▁breaks ▁down ▁in ▁higher ▁dimensions . ▁ ▁Un ique ▁real ization ▁ ▁A ▁framework ▁has ▁a ▁unique ▁real ization ▁in ▁d - dimensional ▁space ▁if ▁every ▁pla cement ▁of ▁the ▁same ▁graph ▁with ▁the ▁same ▁edge ▁lengths ▁is ▁con gru ent ▁to ▁it . ▁Such ▁a ▁framework ▁must ▁necessarily ▁be ▁rig id , ▁because ▁otherwise ▁there ▁exists ▁a ▁continuous ▁motion ▁bringing ▁it ▁to ▁a ▁non - con gru