# The total mixed curvature of open curves in \(E^3\)

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## Abstract

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.

## Keywords

Curve Curvature Torsion## Mathematics Subject Classification

53A04## Notes

### Acknowledgements

The authors are very grateful to the referee for valuable comments, especially for introducing a paper by Ghomi [6].

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