Fractal and convexity analysis of the quaternary four-point scheme and its applications

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme (Q(4)-scheme) with a shape parameter and then find the precise range of the shape parameter of the Q(4)-scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures.