Harmonic　drive　(HD)　is　one　of　the　core　components　of　the　robot　joint.　Studies　show　that　tooth　shape　design　and　meshing　characteristics　of　the　HD　directly　affect　the　motion　control　accuracy　and　vibration　characteristics　of　the　robot.　In　the　present　study,　common　coordinate　systems　are　established　for　three　tooth　profiles　to　analyze　the　differences　between　them.　To　this　end,　expressions　for　the　double　circular　arc　common-tangent　tooth　profile　(DCTP),　cycloid　common　tangent　tooth　profile　(CCTP),　and　the　involute　tooth　profile　(ITP)　are　established　in　the　same　coordinate　system.　By　applying　the　envelope　conjugate　theory,　the　conjugate　existent　domain　(CED)　and　conjugate　tooth　profile　(CTP)　of　the　HD　transmission　are　solved　independently　for　each　profile.　Furthermore,　the　influences　of　tooth　profile　parameters　on　both　the　CTP　and　CED　are　analyzed.　Obtained　results　show　that　both　the　DCTP　and　CCTP　have　more　robust　envelope　processes　when　compared　with　the　ITP.　Moreover,　it　is　found　that　both　profiles　can　achieve　the　second　conjugate　and　two-point　conjugate　engagement　through　applying　variations　in　the　tooth　shape　design　parameters.　It　is　concluded　that　meshing　performances　of　the　DCTP　and　CCTP　are　better　than　that　of　the　ITP,　providing　guidelines　for　the　future　development　of　the　harmonic　reducer　tooth　shape　design.