# Variations in 2D and 3D models by a new family of subdivision schemes and algorithms for its analysis

## TL;DR

This paper introduces a new family of subdivision schemes with a tension parameter that interpolates between approximating and interpolatory schemes, providing flexible design of limit curves and surfaces, along with analytical algorithms for their properties.

## Contribution

A novel family of subdivision schemes with a tension parameter is proposed, along with algorithms for analyzing their properties and efficiency.

## Key findings

- Limit curves vary between approximating and interpolatory schemes.
- Algorithms effectively analyze the properties and time complexity of the schemes.
- Graphical analysis demonstrates the flexibility of the proposed schemes.

## Abstract

A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision schemes provide the flexibility in designing the limit curves and surfaces. Therefore, the limit curves generated by the proposed subdivision schemes variate in between or around the approximating and interpolatory curves. We also design few analytical algorithms to study the properties of the proposed schemes theoretically. The efficiency of these algorithms are analyzed by calculating their time complexity. The graphical representations and graphical properties of the proposed schemes are also analyzed.