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An original classification of obscuration-free telescopes designs unfolded in two dimensions

Benjamin Aymard, Andrea Delahaye, Audric Drogoul

TL;DR

The paper presents an exact, geometry-driven framework to classify unobscured, two-dimensional unfolded telescope designs with up to four mirrors by treating design spaces as path-connected components of semialgebraic sets defined via an obscuration cost. It introduces the GOFFA nomenclature to encode intrinsic topological and manufacturability properties, and proves exhaustiveness for two, three, and four-mirror configurations under planar symmetry. Using Cylindrical Algebraic Decomposition (CAD) and a linear-path heuristic, it computes representative solutions and enumerates the component counts: $|\mathrm{CC}(\widetilde{\mathcal{A}}_2)|=6$, $|\mathrm{CC}(\widetilde{\mathcal{A}}_3)|=32$, and $|\mathrm{CC}(\widetilde{\mathcal{A}}_4)|=288$ (with 144 classes for the 4-mirror case considering symmetry). The approach yields a practical starting point set for optimization under the no-obscuration constraint and provides a scalable framework for classifying large families of optical designs, with potential extensions to fully three-dimensional configurations.

Abstract

In this article we propose an original classification method for unobscured imaging systems unfolded in two dimensions. This classification is based on a study of off-axis properties, and relies on topology and algorithm of real algebraic geometry to find at least one instance by connected component of a semialgebraic set. Our corresponding nomenclature provides intrinsic information about the system, in terms of geometry and manufacturability. The proposed systems for each name of the nomenclature, can be used as starting points for parallel optimizations, allowing for a much more comprehensive search of an unobscured solution, given a set of specifications. We exemplify our method on three and four mirrors imaging systems.

An original classification of obscuration-free telescopes designs unfolded in two dimensions

TL;DR

The paper presents an exact, geometry-driven framework to classify unobscured, two-dimensional unfolded telescope designs with up to four mirrors by treating design spaces as path-connected components of semialgebraic sets defined via an obscuration cost. It introduces the GOFFA nomenclature to encode intrinsic topological and manufacturability properties, and proves exhaustiveness for two, three, and four-mirror configurations under planar symmetry. Using Cylindrical Algebraic Decomposition (CAD) and a linear-path heuristic, it computes representative solutions and enumerates the component counts: , , and (with 144 classes for the 4-mirror case considering symmetry). The approach yields a practical starting point set for optimization under the no-obscuration constraint and provides a scalable framework for classifying large families of optical designs, with potential extensions to fully three-dimensional configurations.

Abstract

In this article we propose an original classification method for unobscured imaging systems unfolded in two dimensions. This classification is based on a study of off-axis properties, and relies on topology and algorithm of real algebraic geometry to find at least one instance by connected component of a semialgebraic set. Our corresponding nomenclature provides intrinsic information about the system, in terms of geometry and manufacturability. The proposed systems for each name of the nomenclature, can be used as starting points for parallel optimizations, allowing for a much more comprehensive search of an unobscured solution, given a set of specifications. We exemplify our method on three and four mirrors imaging systems.
Paper Structure (16 sections, 12 theorems, 40 equations, 11 figures)

This paper contains 16 sections, 12 theorems, 40 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Topological invariants
  4. GOFFA nomenclature
  5. Telescopes classification
  6. Application.
  7. Materials and Methods
  8. Cylindrical Algebraic Decomposition
  9. Linear Path Heuristic on instances found solutions
  10. Intermediate points for four mirrors
  11. Conclusion
  12. Proofs
  13. Used Theorems and Lemmas
  14. Winding number
  15. Unique Reduced Path Theorem
  16. ...and 1 more sections

Key Result

Proposition 1

For $n\in\mathbb{N}$, $\mathcal{A}_n,\mathcal{B}_n$ are dense open semialgebraic subsets of $(\mathbb{R}^2)^{n+2}$, with $\mathcal{B}_n \subset \mathcal{A}_n$ and $\mathcal{B}_n$Zariski openCox:15. More precisely, $\mathcal{B}_n$ and $\mathcal{A}_n$ write as : with, by denoting $a_i=\left( x_i, y_i\right)$:

Figures (11)

  • Figure 1: Non admissible cases according to $\mathcal{J}$\ref{['eq:obscRazCost']}. With (a) for obscuration and (b) for grazing.
  • Figure 2: Three systems composed of the source $S=a_0=b_0=c_0=(0,0)$ , a mirror at $a_1$ (resp. $b_1$, $c_1$) and the image at $a_2$ (resp. $b_2$, $c_2$). The third system is the mean of the first and second system ie. $(c_0,c_1,c_2)=\frac{1}{2}(a_0,a_1,a_2)+\frac{1}{2}(b_0,b_1,b_2)$ Yet it does not share the same connected component (as shown by the sense of rotation, cf. Proposition \ref{['propo:sense_rota']}.)
  • Figure 3: figure \ref{['fig:is_inter_fig']} (resp. figure \ref{['fig:sign_inter_fig']}) represents parts of two systems that are, by Proposition \ref{['propo:is_inter']} (resp. Propostion \ref{['propo:sign_inter']}). not in the same path connected component of $\mathcal{A}_n$.
  • Figure 4: Two systems with 4 mirrors that are, by Proposition \ref{['propo:winding_num']}. not in the same path connected component of $\mathcal{A}_4$.
  • Figure 5: Off axis classification: all topologies of 3 mirror telescopes
  • ...and 6 more figures

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof : Proof of Proposition \ref{['propo:AB_open_alge']}
  • Proposition 2
  • proof : Proof of Proposition \ref{['propo:AB_ineq']}
  • Definition 3
  • Proposition 3
  • proof : Proof of Proposition \ref{['propo:sense_rota']}
  • Proposition 4
  • ...and 15 more