The reciprocating motion type ball reducer has been developed to realize the non-backlash element for the positioning mechanism. This type ball reducer consists of an input rotor with half speed reciprocating cycle wave groove, a fixed internal rotor, many reciprocating motion sliders with freely rotating balls, and an output rotor with wave groove. Both of the input rotor and the output rotor have the wave groove just like a "trochoidal-wave" on these cylindrical surfaces. This reducer generates the reduced rotation using the differential motion. Especially, the input rotor generates the half speed cycle reciprocating motion; it is different from the conventional reducers. It is similar to the grooved cam mechanism which has a translating follower. These sliders are allocated in a uniform angular pitch. And these are preloaded structurally to eliminate backlash. Also these grooves conform to the loci of each ball attached on the linear motion slider. In this report, this reducer is introduced, and the motion principle is analyzed using polar complex vectors. Also, the design parameters are investigated using the pressure angle evaluation method.