Objectives:　Distances　are　functions　of　spatial　positions.　Precisely　revealing　the　functional　relationship　which　quantitatively　embodies　the　transmission　of　uncertainty　from　spatial　positions　to　their　distance,　a　key　scientific　problem　in　need　of　being　solved　urgently　in　geomatics,　has　important　theoretical　and　practical　significance.　Methods:　Aiming　at　the　limitation　of　presently　available　solution　of　the　above　mentioned　problem,　under　the　premise　of　that　the　real　position　corresponding　with　the　observed　one　of　an　uncertain　point　follows　the　complete　spatial　random　distribution　within　the　error　circle,　we　have　derived　the　probability　distribution　function　of　the　distance　uncertainty　and　the　corresponding　density　function　containing　an　uncertain　point　and　those　between　two　uncertain　points　respectively　in　two-dimensional　space.　The　latter　has　been　employed　to　explore　the　transmission　law　of　point　uncertainties　to　distance　uncertainties,　opening　up　a　new　way　for　studying　and　solving　the　problem　of　distance　uncertainties.　Results:　The　results　show　that　for　all　cases:　(1)　When　the　radius　of　the　error　circle　(corresponding　to　the　point　position　accuracy)　and　the　observed　distance　between　points　change　simultaneously,　their　ratio　has　a　significant　positive　correlation　with　the　level　of　distance　uncertainties.　(2)　When　the　former　remains　constant,　the　distance　uncertainty　has　a　significant　negative　correlation　with　the　latter.　(3)　When　the　latter　remains　constant,　the　distance　uncertainty　has　a　significant　positive　correlation　with　the　former.　Conclusions:　As　far　as　the　distance　uncertainty　of　cases　containing　an　uncertain　point　and　the　one　of　those　between　two　uncertain　points　are　concerned,　the　latter　is　obviously　greater　than　the　former　when　the　radius　of　the　error　circle　and　the　observed　distance　between　points　are　consistent　for　both　of　them.　Otherwise　they　are　not　comparable.　©　2023　Wuhan　University.　All　rights　reserved.