This　paper　proves　that　the　optimization　problem　of　one-step　point　feature　Simultaneous　Localization　and　Mapping　(SLAM)　is　equivalent　to　a　nonlinear　optimization　problem　of　a　single　variable　when　the　associated　uncertainties　can　be　described　using　spherical　covariance　matrices.　Furthermore,　it　is　proven　that　this　optimization　problem　has　at　most　two　minima.　The　necessary　and　sufficient　conditions　for　the　existence　of　one　or　two　minima　are　derived　in　a　form　that　can　be　easily　evaluated　using　observation　and　odometry　data.　It　is　demonstrated　that　more　than　one　minimum　exists　only　when　the　observation　and　odometry　data　are　extremely　inconsistent　with　each　other.　A　numerical　algorithm　based　on　bisection　is　proposed　for　solving　the　one-dimensional　nonlinear　optimization　problem.　It　is　shown　that　the　approach　extends　to　joining　of　two　maps,　thus　can　be　used　to　obtain　an　approximate　solution　to　the　complete　SLAM　problem　through　map　joining.　(C)　2013　Elsevier　Ltd.　All　rights　reserved.