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Curvature Preserving Fractal Interpolation Functions: A Hybrid Geometric Approach

K R Tyada

TL;DR

This work addresses the lack of curvature fidelity in classical fractal interpolation by introducing curvature-preserving cubic fractal interpolation functions (CP-CFIFs) built on a cubic spline reference. A penalty-based optimization targets the curvature difference between the fractal interpolant and the spline, while a Read-Bajraktarević-type contraction guarantees a unique fixed-point interpolant that preserves data values. Theoretical results establish uniform convergence to the spline as the scaling factors vanish and provide curvature deviation bounds and stability with respect to parameter perturbations. Numerical experiments across low-curvature, high-curvature, and noisy datasets demonstrate improved curvature fidelity over standard cubic splines, with a quantified computational trade-off, suggesting strong applicability to CAD/CAM, geometric modeling, and geometry-aware data interpolation.

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.

Curvature Preserving Fractal Interpolation Functions: A Hybrid Geometric Approach

TL;DR

This work addresses the lack of curvature fidelity in classical fractal interpolation by introducing curvature-preserving cubic fractal interpolation functions (CP-CFIFs) built on a cubic spline reference. A penalty-based optimization targets the curvature difference between the fractal interpolant and the spline, while a Read-Bajraktarević-type contraction guarantees a unique fixed-point interpolant that preserves data values. Theoretical results establish uniform convergence to the spline as the scaling factors vanish and provide curvature deviation bounds and stability with respect to parameter perturbations. Numerical experiments across low-curvature, high-curvature, and noisy datasets demonstrate improved curvature fidelity over standard cubic splines, with a quantified computational trade-off, suggesting strong applicability to CAD/CAM, geometric modeling, and geometry-aware data interpolation.

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.
Paper Structure (13 sections, 5 theorems, 36 equations, 4 figures, 2 tables)

This paper contains 13 sections, 5 theorems, 36 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. General Framework of Fractal Interpolation Functions
  3. Construction of Curvature Preserving Cubic Fractal Interpolation Functions (CP-CFIFs)
  4. General Framework of Cubic FIFs
  5. Curvature Preserving Cubic Fractal Interpolation Function
  6. Penalty-Based Optimization Framework
  7. Theoretical Analysis
  8. Stability Analysis
  9. Numerical Results
  10. Test Case 1: Low-Curvature Data
  11. Test Case 2: High-Curvature Data
  12. Test Case 3: Noisy Data
  13. Conclusion

Key Result

proposition 1

(Barnsley Barnsley86) The IFS $\{C, w_i : s \in \Im^*\}$ possesses one and only one attractor $G^*$, which represents the graph of a continuous interpolant $\psi^* : I \to K$ that interpolates all data points $\psi^*(x_t) = u_u$ for $t \in \Im$.

Figures (4)

  • Figure 1: Curvature Comparison and Error for Low-Curvature Data
  • Figure 2: Curvature Comparison and Error for High-Curvature Data
  • Figure 3: Curvature Comparison and Error for Noisy Data
  • Figure 4: Parameter sensitivity analysis showing first derivatives for different scaling factors $\lambda$ across all datasets.

Theorems & Definitions (10)

  • proposition 1
  • theorem 1
  • remark 1
  • remark 2
  • theorem 2
  • proof
  • theorem 3: Curvature-Constrained Scaling Factor Bound
  • proof
  • theorem 4: Curvature Stability
  • proof