Exact Finite Element Formulation of Frenet Formula of Curve in Geospatial Database

Abstract. Directly using the difference form of Frenet formula will cause the three basic vectors losing their orthogonal features rapidly. As the analytic form is exact for infinite short length region, for finite length segregation, the omitted items should be retrieved to get high precision. Based on the unit orthogonal transformation in geometrical field theory, the Frenet formula is reformed for finite length region. Then, for given triple at the initial end of curve, using the curve parameters of curvature and torsion, the exact finite element formulation of Frenet formula is obtained to get the triples at any length position until the end point of curve. This method can be used as a high-precision technology for the measure, store, and retrieve of complicated curve.