Catmull-Rom splines are widely used in computer graphics and animation for defining smooth trajectories that pass through given control points. In this study, we investigate a new class of splines derived by modifying the tangent vectors of Catmull-Rom splines using a λ parameter. The λ parameter serves as a geometric control, adjusting the stiffness and turning behavior of the spline. It is numerically and graphically demonstrated that this modification preserves both C¹ and G¹ continuity. Furthermore, Catmull- Rom splines generated with various λ values are compared, and an application to sample path design is presented to illustrate the practical implications of the proposed method.

In this study, we give the structure of ruled surfaces parameterized on a product time scale. Mean and Gaussian curvatures of these surfaces are constructed on distinct time scales. Then, we obtain principal curvatures for ruled surfaces. Finally, we will clarify the issue with some examples.