Curvature Preserving Fractal Interpolation Functions: A Hybrid Geometric Approach

TL;DR

This paper introduces a novel curvature-preserving fractal interpolation method that combines fractal functions with classical splines to improve geometric fidelity in data modeling.

Contribution

A new curvature-aware fractal interpolation framework based on cubic splines, optimizing parameters to better preserve shape features compared to traditional methods.

Findings

Achieves improved curvature preservation over standard splines

Provides theoretical conditions for curvature approximation

Demonstrates effectiveness through multiple examples

Abstract

Fractal interpolation functions (FIFs) generated using iterated function systems (IFS) provide a powerful framework for modeling self-similar and irregular data, yet traditional constructions often neglect geometric fidelity such as curvature. In this paper, we introduce a curvature-preserving variant of FIFs built upon a classical cubic spline interpolant. We define a curvature-aware iterated function system (IFS) with parameters optimized via a penalty-based approach to minimize deviation from the curvature of the classical spline. Theoretical conditions for interpolation and curvature approximation are derived. We compare the curvature of the proposed FIF with that of the classical cubic spline and discrete data curvature across multiple examples. Our method achieves both data interpolation and shape fidelity, preserving curvature more accurately than standard splines. The approach has potential applications in geometric modeling, computer graphics, and scientific data interpolation.