# On the Design of Reflecting Systems with Virtual Sources Open Access

## Pittman, Dylanger (Summer 2023)

Permanent URL: https://etd.library.emory.edu/concern/etds/vx021g60h?locale=en
Published

## 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.

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