Parametric representation of a surface pencil with a common line of curvature

Line of curvature on a surface plays an important role in practical applications. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. By utilizing the Frenet frame, the surface pencil can be expressed as a linear combination of the components of the local frame. With this parametric representation, we derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface. Moreover, the necessary and sufficient condition for the given curve to satisfy the line of curvature and the geodesic requirements is also analyzed.

► We deduce the condition for the given curve to be the line of curvature on a surface.
► Two functions θ(s)θ(s) and λ(s)λ(s) control the shape of the surface.
► We classify the necessary and sufficient conditions by the expression of θ(s)θ(s).
► Condition for the given curve to be a line of curvature and also a geodesic is analyzed.