In this paper, on the basis of the differential geometry, the geometrical theory of machining free form surface by cylindrical cutter in 5-axis NC machine tool is investigated. The second order osculating condition between the cutter and the theoretical surface is analyzed. By means of the condition of positive semidefinite of a quadratic form, the location of the tool-axis can be acquired. The osculating condition between the cutter and the theoretical surface is equivalent to the osculating condition between the tool-axis and the offset surface of the theoretical surface and has been demonstrated. By means of the proposed machining method, the cusp height can be limited within the third order infinitesimal. The machining precision and the efficiency can be improved, and the potential ability of the 5-axis NC machine tools could be developed.
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