Abstract

We present an analytic solution for the irradiance at a point due to a polygonal Lambertian emitter with radiant exitance that varies with position according to a polynomial of arbitrary degree. This is a basic problem that arises naturally in radiative transfer and more specifically in global illumination, a subfield of computer graphics. Our solution is closed form except for a single nonalgebraic special function known as the Clausen integral. We begin by deriving several useful formulas for high-order tensor analogs of irradiance, which are natural generalizations of the radiation pressure tensor. We apply the resulting tensor formulas to linearly varying emitters, obtaining a solution that exhibits the general structure of higher-degree cases, including the dependence on the Clausen integral. We then generalize to higher-degree polynomials with a recurrence formula that combines solutions for lower-degree polynomials; the result is a generalization of Lambert’s formula for homogeneous diffuse emitters, a well-known formula with many applications in radiative transfer and computer graphics. Similar techniques have been used previously to derive closed-form solutions for the irradiance due to homogeneous polygonal emitters with directionally varying radiance. The present work extends this previous result to include inhomogeneous emitters, which proves to be significantly more challenging to solve in closed form. We verify our theoretical results with numerical approximations and briefly discuss their potential applications.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Figures (7)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Equations (143)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Metrics

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription