The total mixed curvature of open curves in $$E^3$$

The total mixed curvature of a curve in (E^3) is defined as the integral of (sqrt{kappa ^2+tau ^2}), where (kappa ) is the curvature and (tau ) is the torsion. The total mixed curvature is the length of the spherical curve defined by the principal normal vector field. We study the infimum of the total mixed curvature in a family of open curves whose endpoints and principal normal vectors at the endpoints are prescribed. In our previous works, we studied similar problems for the total absolute curvature, which is the length of the spherical curve defined by the unit tangent vector, and for the total absolute torsion, which is the length of the spherical curve defined by the binormal vector.