Pose　graphs　have　become　an　attractive　representation　for　solving　Simultaneous　Localization　and　Mapping　(SLAM)　problems.　In　this　paper,　we　analyze　the　structure　of　the　nonlinearities　in　the　2D　SLAM　problem　formulated　as　the　optimizing　of　a　pose　graph.　First,　we　prove　that　finding　the　optimal　configuration　of　a　very　basic　pose　graph　with　3　nodes　(poses)　and　3　edges　(relative　pose　constraints)　with　spherical　covariance　matrices,　which　can　be　formulated　as　a　six　dimensional　least　squares　optimization　problem,　is　equivalent　to　solving　a　one　dimensional　optimization　problem.　Then　we　show　that　the　same　result　can　be　extended　to　the　optimizing　of　a　pose　graph　with　＂two　anchor　nodes＂　where　every　edge　is　connecting　to　one　of　the　two　anchor　nodes.　Furthermore,　we　prove　that　the　global　minimum　of　the　resulting　one　dimensional　optimization　problem　must　belong　to　a　certain　interval　and　there　are　at　most　3　minima　in　that　interval.　Thus　the　globally　optimal　pose　configuration　of　the　pose　graph　can　be　obtained　very　easily　through　the　bisection　method　and　closed-form　formulas.　©　2013　Massachusetts　Institute　of　Technology.