This paper is dedicated to the problem of optimal synthesis of linkage mechanisms that can provide straight-line movement and periodic dwells of the working bodies, according to the technological operation of the machine. For that purpose, different types of mechanisms can be used and it is known that the usage of linkage mechanisms has a number of advantages: in particular – reliability of operation and load capacity. But the main problem is their complicated kinematic synthesis by the given design parameters, besides in many cases the actual dwell values differ from theoretical ones, which caused a delayed exit of the output link from the dwell phase; it also increases the kinematic parameters, which is undesirable. For the synthesis of such mechanisms the theoretical grounds of kinematic differential geometry can be used. Thus, the aim of the study is the development of a combined numerical–analytical method that enables the synthesis of mechanisms using Ball points with prescribed dwell duration and a prescribed straight-line segment of the coupler curve. Mechanisms that are synthesized using Ball points can archive dwells within the range of 30-120 degrees of crank rotation, with deviations of 0.001...0.01 (the unit of length is mechanism’s interaxial distance). The usage of proposed method allows to obtain linkage mechanisms by given dwell values without slow exit from the dwell phase, using numerical optimization to meet various design criteria.