On the Design of Reflecting Systems with Virtual Sources Open Access
Pittman, Dylanger (Summer 2023)
Abstract
We consider the problem of the determination of a system of reflecting surfaces jointly transforming a given radiance distribution from a point source into an irradiance distribution appearing to an observer as produced by some virtual sources. Our work continues the work by Kochengin et al. [13] which dealt with the case when the required reflector is a single surface. Here, the reflector is allowed to consist of several disjoint surfaces.
Table of Contents
Contents
1 Introduction 1
1.1 Virtual source Reflector Problem . . . . . . . . . . . . . . . . . . . . 5
1.2 Dissertation Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Notation and Terminology 9
3 Convex Weak Solutions to the Virtual Source Reflector Problem 11
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Hyperboloids of Revolution . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Convex Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Weak Solutions in the Discrete Case . . . . . . . . . . . . . . . . . . 25
3.6 Weak Solutions in the General Case . . . . . . . . . . . . . . . . . . . 29
3.7 Rotationally Symmetric Convex Refractors on the Surface of a Right
Circular Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Generalized Weak Solutions to the Virtual Source Reflector Prob-
lem 40
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Generalized Weak Solutions Constructed from Hyperboloids . . . . . 41
4.3 Generalized Refractors . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 The Rotationally Symmetric Case . . . . . . . . . . . . . . . . . . . . 53
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Appendix A Formulation of the PDE for the Virtual Source Reflector
Problem 61
Appendix B No Set of Points Satisfies Both Hypotheses H1 and H2 63
Appendix C Blaschke’s Selection Theorem 67
Appendix D Reidemeister’s Theorem About Singular Points on Con-
vex Sets 68
Appendix E A Constructive Proof of Lemma 4.3.2 70
E.1 Nonatomic Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
E.2 Generalization of Hall’s Matching Theorem . . . . . . . . . . . . . . . 71
E.3 Proof of Theorem E.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
References 79
About this Dissertation
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