Bertrand Surface Family Pairs Preserving Common Characteristic Curves

Bertrand curve pairs share the same principal normals, creating a geometric symmetry useful in design and modeling. The geometry of a surface can be characterized and studied through three types of characteristic curves: geodesics, curvature lines, and asymptotic curves. We introduce a method to construct corresponding surface family pairs from a Bertrand curve pair, ensuring that both curves serve as the same type of characteristic curve on each surface family, thereby extending curve symmetry to surface symmetry. We build surface pairs by linearly combining the Frenet frame of a Bertrand curve, with coefficients acting as shape functions. We establish necessary and sufficient conditions that these functions must satisfy to guarantee that the Bertrand curves become the same characteristic type on both surface families. This provides flexible control over surface geometry and curve type. We further derive conditions for developable surface pairs, proving that no developable pair can contain a twisted Bertrand curve as a curvature line or asymptotic curve. To illustrate this, we construct surface pairs from a circular helix, and the resulting surfaces exhibit an aesthetically pleasing symmetry, demonstrating the flexibility and interactivity of our framework.