We propose and demonstrate a linear time-to-space mapping system, which is based on two times electrooptic sinusoidal beam deflection. The direction of each deflection is set to be mutually orthogonal with the relative deflection phase of pi/2 rad so that the circular optical beam trajectory can be achieved. The beam spot at the observation plane moves with an uniform velocity and as a result linear time-to-space mapping (an uniform temporal resolution through the mapping) can be realized. The proof-of-concept experiment are carried out and the temporal resolution of 5 ps has been demonstrated using traveling-wave type quasi-velosity-matched electrooptic beam deflectors. The developed system is expected to be applied to characterization of ultrafast optical signal or optical arbitrary waveform shaping for modulated microwave/millimeter-wave generation.