# Recursive process for constructing the refinement rules of new combined subdivision schemes and its extended form

## TL;DR

This paper introduces a recursive method for constructing new combined subdivision schemes with tension parameters, enhancing polynomial properties and allowing flexible control over curve and surface refinement.

## Contribution

It presents a recursive construction approach for (2N+2)-point subdivision schemes with tension parameters, extending to (2N+3)-point schemes and interproximate schemes with variable tension.

## Key findings

- Recursive construction of subdivision schemes from base cases.
- Enhanced polynomial reproduction and generation with increasing N.
- Flexible tension parameters for interpolating and approximating control points.

## Abstract

In this article, we present a new method to construct a family of (2N+2)-point binary subdivision schemes with one tension parameter where N is a non-negative integer. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, the refinement rules of a (2N+2)-point scheme for N=M are recursively obtained from the refinement rules of the (2N+2)-point schemes for N=0,1,2,...,M-1. The complexity, polynomial reproduction and polynomial generation of these schemes are increased by two for the successive values of $N$. Furthermore, we modify this family of schemes to a family of (2N+3)-point schemes with two tension parameters. Moreover, a family of interproximate subdivision schemes with tension parameters is also introduced, which allows a different tension value for each edge and vertex of the initial control polygon. Interproximate schemes generate curves and surfaces such that some initial control points are interpolated and others are approximated.