A generalization of surface family with common line of curvature

• We deduce the condition for the given curve to be the line of curvature on surface when the marching-scale functions are in more general expression.
• Two functions θθ(s) and λλ(s) control the shape of the surface.
• We classify the conditions by the expression of θθ(s).

We can represent the surface with a linear combination of the components of Frenet–Serret frame. Based on this representation, in the work of Li et al. [C.-Y. Li, R.-H. Wang, C.-G. Zhu, Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Des. 43 (9) (2011) 1110–1117], we derive the necessary and sufficient condition on the marching-scale functions for which the given curve is a line of curvature of the resulting surface. For convenience, we assumed the marching-scale functions can be decomposed into two factors. In this paper, we derive the sufficient condition for the given curve as a line of curvature of the surface when the marching-scale functions are in more general expressions. Finally, we give some representative examples to illustrate the convenience and efficiency of this method.