The paper has proposed to determine the equation of flanks of generation hook with oblong cycloidal arc taking into account two types of cutting tool: a cutting tool that generates the hole of the tooth, respectively a cutting tool that generates the tooth. The curved cylindrical teeth with cycloidal flanks is generated by rolling with straight mobile line and continuous division by using a milling cutter with 1, 2, 3, 4 or 5 groups of knifes equidistantly placed, fixed on an adaptable device on a Phauter toothing machine. The two curves which defines the flank are simultaneously generated by correlated motions. In order to make the analytical study of motions and geometry of the flanks, several 3D coordinating systems were first defined, each of them being bounded by an element that participates in the generating process. There were established the equations of cutting edged of the tool in S_s system, the equations of the flanks of the generating hook are obtained by changing the reference system of edges of the tool from S_s to S_(CG). The line of the flanks is studied in the reference plane and in parallel plans with this, and. the profile of the flanks is determined in perpendicular plans on the workpiece axis.. It was made a calculation program in MATLAB, that allowed to obtain data, which were used to plot the line and the profile of the flanks of the generating hook By analysing the obtained shape they result conclusions used for the manufacturing of the involute-tooth gear with curved oblong cycloidal arc.
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