Distances　are　functions　of　spatial　positions,　therefore　distance　uncertainties　should　be　resulted　from　transmission　of　spatial　position　uncertainties　via　the　functional　relationships　accordingly.　How　to　model　the　transmission　precisely,　a　challenging　problem　in　GIScience,　has　increasingly　drawn　attentions　during　the　past　several　decades.　Aiming　at　the　limitations　of　presently　available　solutions　to　the　problem,　this　article　derived　probability　dis-tribution　functions　and　corresponding　density　functions　of　distance　uncertainties　in　two-dimensional　space　related　to　one　or　two　uncertain　endpoints　respectively,　under　the　premise　that　real　positions,　corresponding　with　the　observed　position　of　an　uncertain　point,　follow　the　kernel　function　within　its　error　circle.　The　density　functions　were　employed　to　explore　the　diffusing　law　of　uncertainty　information　from　point　positions　to　dis-tances,　which　opened　up　a　new　way　for　thoroughly　solving　problems　of　measuring　distance　uncertainties.　It　turns　out　that　the　proposed　methods　in　this　article　are　more　efficient,　robust　than　the　corresponding　Monte　Carlo　ones,　which　verifies　their　effectiveness　and　advantages.