Abstract
This study introduces a 4-point interpolatory quinary refinement scheme with two shape parameters and investigates the scheme’s convergence and smoothness. The scheme’s fundamental properties, such as approximation order, polynomial reproduction, and polynomial generation, are also examined. To the best of our knowledge, this is the first quinary interpolatory scheme shown to generate fractal curves with endpoint interpolation, and its versatility is demonstrated by its ability to generate fractal curves within a specific range of the shape parameters. A modified 4-point quinary scheme for endpoint interpolation is also presented to enhance the scheme’s modeling capability. This research offers valuable insights into the behavior and potential applications of the scheme in engineering and industry.