As　an　indispensible　component　in　understanding　spatial　processes,　direction　has　been　concerned　by　researchers　of　related　disciplines　for　a　long　time.　Challenges　still　remain　in　modeling　directional　uncertainties　due　to　randomness　of　positional　information　in　the　input　data.　By　formulating　involving　cumulative　distribution　functions　(CDFs)　and　corresponding　probability　density　functions　(PDFs),　this　paper　modeled　the　directional　uncertainty　between　an　accurate　point　and　an　uncertain　one　as　well　as　that　between　two　uncertain　points　respectively　in　2D　spaces　on　the　condition　of　that　the　real　position　corresponding　to　the　observed　one　of　an　uncertain　point　follows　either　Poisson　distribution　or　the　kernel　function　decreasing　from　the　center　to　periphery　within　its　error　circle.　The　established　models　accurately　quantify　the　correlation　of　the　directional　uncertainty　with　r　(the　radius　of　the　error　circle　of　the　uncertain　point),　L　(the　observed　distance　containing　one　or　two　uncertain　points),　and　their　ratio　(r/L)　for　the　four　different　situations　respectively,　which　opens　up　a　new　way　for　related　studies　including　spatial　query,　analysis　and　reasoning,　etc.　Experimental　results　indicate　that　the　proposed　methods　in　this　article　are　more　efficient,　robust　than　the　corresponding　Monte　Carlo　simulation.　Their　effectiveness　has　also　been　demonstrated　with　practical　cases.　©　The　Author(s)　2025.