Formulas for second curvature, third curvature, normal curvature, first geodesic curvature and first geodesic torsion of implicit curve in n-dimensions

We present closed formulas for computing the differential geometry properties of the intersection curve of n−1n−1 implicit hypersurfaces in RnRn. These closed formulas include second curvature, third curvature, normal curvature, first geodesic curvature and first geodesic torsion for implicit curves in n-dimensions.

► We obtain the compact formulas of (κ2,κ3)(κ2,κ3). ► We obtain the normal curvature, the first geodesic: curvature and torsion. ► These properties are for the intersection curve of n−1n−1 implicit hypersurfaces in RnRn. ► We obtain the Darboux frame by using the Gram–Schmidt orthogonalization process.