This　paper　presents　the　study　of　the　two　methods　for　the　beam-steering　planar　metasurface　antennas　based　on　optical　Risley　Prism　scan　theory:　paraxial　approximation　and　phase　gradient　methods.　A　two-dimensional　(2D)　beam-steering　antenna　is　first　developed,　consisting　of　two　planar　metasurface　transmit　arrays　and　a　linearly　polarized　(LP)　horn　the　two　transmitarrays　have　identical　constant　phase　delay　gradients,　and　the　beam　steering　capability　is　realized　by　mechanically　rotating　the　two　independent　metasurface　transmitarrays　the　antenna　developed　is　then　used　to　compare　and　verify　the　precisions　of　the　paraxial　approximation　method　and　the　phase　method.　Simulation　results　show　that　the　phase　method　is　more　accurate　than　the　paraxial　approximation　method,　especially　with　large　phase　delay　gradients.　©　2021　IEEE.