Weak solutions to the near-field reflector problem with spatial restrictions approached with generalized reflectors constructed from ellipsoids

TL;DR

This paper introduces a new variant of the near-field reflector problem with spatial restrictions, focusing on constructing reflectors within a bounded set to achieve desired irradiance from an anisotropic point source.

Contribution

It formulates a novel near-field reflector problem with spatial constraints and constructs solutions using generalized reflectors made from ellipsoids.

Findings

Existence of weak solutions for the problem.

Construction of reflectors from generalized ellipsoids.

Application to finite target sets.

Abstract

We motivate then formulate a novel variant of the near-field reflector problem and call it the near-field reflector problem with spatial restrictions. Let $O$ be an anisotropic point source of light and assume that we are given a bounded open set $U$. Suppose that the light emitted from the source at $O$ in directions defined by the aperture $D\subseteq S^2$, of radiance $g(m)$ for $m\in D$, is reflected off $R\subset \overline{U}$, creating the irradiance $f(x)$ for $x\in T$. The inverse problem consists of constructing the reflector $R\subseteq \overline{U}$ from the given position of the source $O$, the input aperture $D$, radiance $g$, `target' set $T$, and irradiance $f$. We focus entirely on the case where the target set $T$ is finite.