The tooth surface equations of spiral bevel gears are expressed in vector form, and a lot of coordinate transformation work needs to be done in the machining process. As shown in the figure, in the coordinate system Oxyz and the coordinate system Ox’y’z’, the coordinate system Ox’ y’z’is rotated around the X axis by the coordinate system Oxyz θ After corner.

The coordinate of point P in the coordinate system Oxyz is [x rcos（ α+θ) rsin( α+θ)] ， The coordinate of point P in the coordinate system ox’y’z’is [x rcosα rsinα] ， Then the above coordinate system transformation can be expressed by matrix operation as:

Matrix M is the transformation matrix from coordinate system Oxyz to coordinate system Ox’y’z’: