A certified classification of first-order controlled coaxial telescopes

TL;DR

This paper provides a geometrical classification of three-mirror telescopes using algebraic and topological methods, identifying their connected components and invariants under first order approximations.

Contribution

It introduces a novel intrinsic geometrical classification framework for three-mirror telescopes based on semi-algebraic set analysis and topological invariants.

Findings

Semi-algebraic description of telescope configurations

Explicit count of connected components

Exact topological invariant for three-mirror systems

Abstract

This paper is devoted to an intrinsic geometrical classification of three-mirror telescopes. The problem is formulated as the study of the connected components of a semi-algebraic set. Under first order approximation, we give the general expression of the transfer matrix of a reflexive optical system. Thanks to this representation, we express the semi-algebraic set for focal telescopes and afocal telescopes as the set of non-degenerate real solutions of first order optical conditions. Then, in order to study the topology of these sets, we address the problem of counting and describe their connected components. In a same time, we introduce a topological invariant which encodes the topological features of the solutions. For systems composed of three mirrors, we give the semi-algebraic description of the connected components of the set and show that the topological invariant is exact.