Bending Fields for Dual Curves

This paper provides several new characterizations of the infinitesimal bending of dual curves, which is defined as an infinitesimal deformation preserving dual arc length (with appropriate precision). The main goal is to consider the infinitesimal deformations of ruled surfaces through the corresponding deformations of dual curves. Some useful properties of the infinitesimal bending of dual curves are obtained, and dual bending fields are determined. The Vekua-type characterization of the infinitesimal bending of dual curves is formulated in terms of the derivative of the dual arc length. Explicit formulas for dual infinitesimal bending fields of dual spherical curves are obtained using the Blaschke frame, considering both an arbitrary real parameter and the dual arc length. A necessary and sufficient condition for the infinitesimal bending of the dual curve to lie on the dual unit sphere is presented in terms of Blaschke and Frenet invariants. Several examples are illustrated graphically using our own software tool.