Abstract In this study, dual unitary matrices SUD(2) were obtained. We correspond to one to one elements of the unit dual sphere S D 3 with the dual unitary matrices SUD(2). Thus, we express spherical concepts such as meridians of longitude and parallels of latitude on SUD(2). The equality SO(R3) ≅ S3/{±1} = RP3 known as the real projective spaces was generalized to the dual projective space and then, the equality SO ( D 3 ) ≅ S D 3 / { ± 1 } = DP 3 was acquired. In particular, 2-sphere S2 was obtained by considering dual parts as zero of S D 3 . Hence, it was found that Hop fibriation map of S2 can be used for Twistors in quantum mechanics applications.