The reflection and refraction of light are the most basic methods for the application of geometric optics. In the processing of modern precision optical elements, the different expression methods can provide different solutions for light tracing, prism error analysis as well as prism assembly and adjustment. The traditional form expression methods of the law of reflection and the law of refraction of light were introduced, and the expression methods of vector, matrix and quaternion were derived. Through the simulation calculation assisted by Matlab, the reflected rays of incident light in two different areas of the Schmidt prism inspection optical path was symmetrically distributed on both sides of the reflected rays on the front surface in the horizontal direction, which was consistent with the practical application. The application of expression methods of vector, matrix and quaternion in Schmidt prism inspection optical path was realized. The three expression methods can provide scientific and practical solutions for light tracing, prism error analysis as well as prism assembly and adjustment.