Abstract: The concept of quartic Hermite curve is presented for an application to meet some special engineering requisitions, based on the theoretic methods of Hermite interpolation curve. The quartic Hermite curve can fit to positional vector and first derivative vector of two given points, and make sure that the curve could be constructed to be on the given conical surface. The interpolation curve on conical surface can be used to describe the boundary lines of tooth surface, such as that of hypoid bevel gear.