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Flexible Foil Mesh Generation for Spatial Focal-Body Modeling of a Spherical Mirror

Netzer Moriya

TL;DR

This work addresses the challenge of accurately modeling highly concave focal regions generated by spherical mirrors by applying Flexible Foil Mesh Generation (FFMG). The FB is defined as the 3D region formed by the envelope of reflected rays from an infinitely distant source, and FFMG is used to iteratively refine a convex-hull initial surface into a closed, concave mesh through force-balanced deformation, smoothing, and snapping. The approach combines ray-tracing of optical reflections with a physically-based mesh evolution that biases toward minimal surface configurations while preserving enclosure and topology. Results demonstrate that the evolving mesh captures the concave geometry of the focal region and remains computationally feasible, with insights into complexity and optimization. The work also outlines extensions to include wave-optics effects and adaptive time-stepping, highlighting potential applications in optical design, medical imaging, and other areas requiring accurate concave-surface reconstructions.

Abstract

We present a novel application of the Flexible Foil Mesh Generation (FFMG) method to model the $3D$ Focal Body generated by a spherical mirror collecting light from an infinitely distant source on its optical axis. The study addresses the challenge of accurately representing highly concave structures formed by the focusing effect. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of the FFMG method in capturing the intricate geometry of the Focal Body, with implications for computational geometry, $3D$ reconstruction, and optical system modeling.

Flexible Foil Mesh Generation for Spatial Focal-Body Modeling of a Spherical Mirror

TL;DR

This work addresses the challenge of accurately modeling highly concave focal regions generated by spherical mirrors by applying Flexible Foil Mesh Generation (FFMG). The FB is defined as the 3D region formed by the envelope of reflected rays from an infinitely distant source, and FFMG is used to iteratively refine a convex-hull initial surface into a closed, concave mesh through force-balanced deformation, smoothing, and snapping. The approach combines ray-tracing of optical reflections with a physically-based mesh evolution that biases toward minimal surface configurations while preserving enclosure and topology. Results demonstrate that the evolving mesh captures the concave geometry of the focal region and remains computationally feasible, with insights into complexity and optimization. The work also outlines extensions to include wave-optics effects and adaptive time-stepping, highlighting potential applications in optical design, medical imaging, and other areas requiring accurate concave-surface reconstructions.

Abstract

We present a novel application of the Flexible Foil Mesh Generation (FFMG) method to model the Focal Body generated by a spherical mirror collecting light from an infinitely distant source on its optical axis. The study addresses the challenge of accurately representing highly concave structures formed by the focusing effect. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of the FFMG method in capturing the intricate geometry of the Focal Body, with implications for computational geometry, reconstruction, and optical system modeling.

Paper Structure

This paper contains 41 sections, 30 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Objective
  4. Related Work
  5. Theoretical Background
  6. A Brief Analysis of Light Distribution Reflected by a Spherical Mirror
  7. Problem Setup
  8. Ray Behavior Near the Focal Point
  9. Formation of the Caustic Surface
  10. Spherical Aberration Effects
  11. Flexible Foil Mesh Generation (FFMG)
  12. Minimal Surface Approximation
  13. Methodology
  14. Problem Definition
  15. Ray Tracing Computational Implementation
  16. ...and 26 more sections

Figures (9)

  • Figure 1: A typical simulation setup for spherical mirror with an equal-axis view (left) and non-equal view (right).
  • Figure 2: Initial point cloud representing the focal body.
  • Figure 3: Mesh initialization including Convex Hull vertices and refined mesh in their proximity.
  • Figure 4: Mesh deformation progression at 50 (upper-left), 100 (upper-right), 150 (lower-left), and 199 (lower-right) iterations.
  • Figure 5: Pressure force components acting on a representative test point during the deformation process.
  • ...and 4 more figures