Abstract: The construction, manufacture and inspection of involute gears and spline joints can be significantly simplified by substituting the involute profiles with approximate circular arcs. It was noticed that the current least-square method does not always give the best curve fit. On the basis of a thorough investigation of the deviations between the involute and the substitute, an appropriate objective function leading to the best approximate circular arc is proposed. The maximum deviation so obtained is only half of Tsuda, s result, or even less. Three exam ples are given to illustrate its application to form grinding, wire cutting of injection molds and the milling-cutter design.