A kinematic scheme for thread milling is proposed, allowing the tool to orient relative to the thread being formed by rotating around two axes. A generalized mathematical model of thread forming by a tool in a general position has been developed, enabling the calculation of geometric errors for threads of arbitrary profiles. Numerical values of the geometric error of the profile have been established based on the developed model and demonstrated using the example of a metric thread, corresponding to the adopted degree of accuracy. The proposed model facilitates the effective forming of threads with a small angle on one of the thread’s sides, significantly expanding the application scope of thread milling.