Internal illumination prediction based on a simplified radiosity algorithm

Solar Energy 80 (2006) 260–267

www.elsevier.com/locate/solener

Internal illumination prediction based on a simplifiedradiosity algorithm

Darren Robinson *, Andrew Stone

BDSP Partnership Ltd., Summit House, 27 Sale Place, London W2 1YR, UK

Received 19 August 2004; received in revised form 10 February 2005; accepted 25 February 2005

Available online 5 April 2005

Communicated by: Associate Editor Jean Rosenfeld

Abstract

This paper describes a new model for predicting natural interior illumination in the urban context. A simplified radi-

osity algorithm is used to define the external luminous environment and this model structure is used to predict internal

illumination from sky and external obstructions for the range of sky conditions. This illumination is predicted with

comparable accuracy to the ray tracing program RADIANCE, but at a computational cost several orders of magnitude

lower. A standard diffusing sphere approximation (the BRS split flux equation) is currently used to solve internally

reflected light. Comparisons suggest that the accuracy of this approach is adequate for the purpose of predicting the

energy implications of photoresponsive lighting control. The model is in a form readily amenable for inclusion into

other programs that require internal illuminance as an input, to improve the accuracy of their predictions without

recourse to external simulation programs.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Daylight; Radiosity; Model; Theory; Validation

1. Introduction

At the scale of the urban neighbourhood—when

accumulated resource supply and demand profiles have

a relatively high and stable base—the financial viability

of integrating renewable and high efficiency technologies

(e.g., wind and solar energy conversion technologies,

district co-generation) improves. It is also at this scale

0038-092X/$ - see front matter � 2005 Elsevier Ltd. All rights reserv

doi:10.1016/j.solener.2005.02.016

* Corresponding author. Address: Solar Energy and Building

Physics Laboratory, Swiss Federal Institute of Technology

(EPFL), CH-1015 Lausanne, Switzerland. Tel.: +41 21 693

4543; fax: +41 21 693 2722.

E-mail address: [email protected] (D. Robinson).

that important decisions influencing solar/daylight ac-

cess and the external microclimate, and thereby pedes-

trian quality of life, are made. For this reason

increasing emphasis is being placed on using environ-

mental modelling tools to help make rational urban de-

sign decisions. It is in this context that the European

Commission is funding Project SUNtool (Sustainable

Urban Neighbourhood modelling tool) to develop a

new integrated resource flow modelling and associated

educational tool (Robinson et al., 2003). At the core of

the modelling tool is a thermal model and this has as

an input a daylight model (amongst several others), so

that heat gains and electrical energy demand from artifi-

cial lighting is considered. In this it is important that

daylight predictions are sensitive to urban obstructions,

ed.

mailto:[email protected]

Nomenclature

A surface area, m2

E illuminance, lm m�2 (Lux)

F luminous flux, lm

I irradiance, W m�2

L luminance, lm m�2 Sr�1

U solid angle Sr

h angular displacement, rad

r,u,x view factor (sky, self and external obstruc-

tions respectively)

n angle of incidence, rad

s transmittance

c altitude, rad

a azimuth, rad

g luminous efficacy, lm W�1

q reflectance

D. Robinson, A. Stone / Solar Energy 80 (2006) 260–267 261

that they are reasonably accurate and not too computa-

tionally demanding. To satisfy these needs a new gener-

alised model has been developed and tested.

The following section describes the approach to the

calculation of sky, externally reflected and internally re-

flected light. Comparisons with a truth model are dis-

cussed in the subsequent section, together with

suggestions for further improvements to predictive accu-

racy in handling internally reflected light.

1 Note that external surfaces are assigned a mean reflectance,

so that the transmission of light to the external environment

through glazing from rooms on the other side of external

surfaces is ignored.

2. Theory

2.1. Sky and externally reflected light

In order to calculate the interior illumination due to

the sky and external obstructions, we must first define

their luminance. To characterise the sky, we may simply

use one of the numerous luminance distribution models

that are available. Of these we use the all weather model

due to Perez et al. (1993). Determining the luminance of

obstructions however is a little more involved, since we

must calculate contributions from the sky, other external

obstructions and the sun. Considering the first contribu-

tion, if we discretise the sky vault into some set of p sky

patches, each of which subtends a solid angle / (Sr) and

has luminance L (lm m�2 Sr�1) then, given the mean an-

gle of incidence n (radians) between the patch and our

calculation point together with the proportion of the

patch that can be seen r(0 6 r 6 1), we have a conve-

nient general solution for direct sky illuminance

(lm m�2) (1).

Es ¼Xpi¼1

LUr cos nð Þi ð1Þ

Using the Perez model, the sky luminance within the

ith patch is given by (2), where ‘vi is the relative lumi-

nance of an arbitrary sky point (a function of the zenith

angle of the considered sky point Z and the angular dis-

placement of this point from the sun h :‘v = f(Z,h)) andthis is hemispherically normalised using diffuse horizon-

tal irradiance Id in conjunction with a diffuse luminous

efficacy gd (e.g. using the model due to Perez et al.,

1990).

Li ¼‘viIdgdP2p

j¼1 ‘vU cos nð Þjð2Þ

The sky vault discretisation scheme proposed by

Tregenza and Sharples (1993) has found widespread fa-

vour, in which the vault is split into 145 patches that

each subtend a similar solid angle—defined by the patch

azimuthal range Da and altitudinal c limits (3).

Ui ¼ Dai sin ci;max � sin ci;min

� �ð3Þ

Finally, views to the i th sky patch may be diminished

by both self-obstructions from our external plane u and

external obstructions x, so that our sky view factor is gi-

ven by ri = 1 � ui � xi.

The illuminance due to external obstructions Eo is gi-

ven by a similar expression to that for the sky, but this

time summating to 290 (or 2p) patches for two hemi-

spheres, one above and one below the horizontal plane

(4)

Eo ¼X290i¼1

L�Ux cos nð Þi ð4Þ

where xi = 1 � ri � ui. In solving for (4) we make the

simplification that the luminance of obstructions occlud-

ing a given patch can be represented by that of the dom-

inant obstruction L*.

The total external surface luminance L is then given

by (5) in which Ibn is the incident beam irradiance, gbis the corresponding beam luminous efficacy and q is

the surface reflectance.1

262 D. Robinson, A. Stone / Solar Energy 80 (2006) 260–267

L ¼ Ibngb þX145i¼1

LUr cos nð Þi þX290j¼1

L�Ux cos nð Þj

!q

,p

ð5Þ

The terms in brackets correspond to contributions

to external surface illuminance from the sun, sky and

external obstructions (assuming Lambertian reflectance

characteristics) respectively. A set of simultaneous

equations describing total surface illuminance for an

entire external scene can be formulated into a matrix

and solved by inversion for infinite reflections (Appen-

dix A) or by iteration (e.g. Gauss-Seidel) for finite

reflections. For further details and application of this

Simplified Radiosity Algorithm (SRA) to solve for

external solar radiation prediction, see Robinson and

Stone (2004).

Note that Robinson and Stone (2004) also describe a

basic procedure for deriving the sky, self obstruction

and external obstruction view factors. In brief this in-

volved discretising each sky patch into a regular (102)

grid of cells and, by performing intersection tests

within these cells, determining a view factor incidence

angle product due to the n occluded cells (e.g.

x cos n ¼ 10�2Pnj¼1 cos nj); the dominant external sur-

face (*) is simply that which is responsible for the largest

contribution to x cos n. A more efficient solution in-

volves rendering the urban scene from a given view

point, each surface having been assigned a unique col-

our.2 Each pixel is translated into angular coordinates

to identify the corresponding patch as well as the angle

of incidence. For our view factors then, Ux cos n are

treated as single quantities obtained by numerical inte-

gration of cos n � dU across each patch. This technique

is considerably less computationally demanding but, if

the extent of the rendered scene is small, may suffer from

aliasing errors due to pixel size.

With this pre-processing of the luminance of the exter-

nal scene complete, we may proceed to determine the cor-

responding contributions to interior illumination (6).3

Es ¼X145i¼1

LUr cos nsð Þi ð6aÞ

Eo ¼X290i¼1

L�Ux cos nsð Þi ð6bÞ

2 Starting at one end of the r, g, b scale for the first surface

and cycling through towards other the end of this scale, so that

excepting the background colour, up to 2563 � 1 unique surface

identifiers are available—considerably more than is likely to be

needed for our purposes.3 For calculation points aligned with the floor surface normal,

we may summate to just 145 for Eo in (6).

For this we require additional renderings from each

internal calculation point, from which we simply process

the information within the visible extremities of external

openings to define the sky and external surface view fac-

tors and for the latter to identify the dominant

obstructions.

Note that whilst solar illumination may also contrib-

ute to internal and external surface luminance, it is as-

sumed within this standalone model that users will

interact with blinds if there is a risk of direct insolation

in the vicinity of our internal calculation points. How-

ever, within the overall SUNtool modelling environ-

ment, a stochastic lighting and blind model is called to

predict the probability of and outcome from human

interaction with blinds.

2.2. Internally reflected light

By approximating the reflection characteristics of a

room to that of a perfectly diffusing sphere, the standard

solution to internally reflected light takes the form of the

following infinite geometric series (7):

Ei ¼ F q=A 1� �qð Þ ð7Þ

for incoming flux F reflected initially from surfaces of

reflectance q and subsequently throughout the room of

surface area A and area weighted mean reflectance �q.In splitting the treatment of the numerator in (7) into

flux entering the room in the downwards direction FD to

be modified by the mean reflectance of the lower half of

the room qL and that entering in the upwards direction

FU to be modified by the mean reflectance of the upper

half of the room qU, we have the BRS split flux equation

(8) (Hopkinson et al., 1966).

Ei ¼ F DqL þ F UqUð Þ=A 1� �qð Þ ð8Þ

This simply requires that we summate the luminous

fluxes at the mid-point of each of the n windows of glaz-

ing area Ag from our hemispheres, the upper hemisphere

for FD and the lower inverted hemisphere for FU (9).

F D ¼Xni¼1

Ag;i IbngbsþX145j¼1

LUrs cos nð Þj

þX145k¼1

L�Uxs cos nð Þk

!i

ð9aÞ

F U ¼Xni¼1

Ag;i

X290j¼146

L�Uxs cos nð Þj

!i

ð9bÞ

For this we may re-use information from renderings

executed at the relevant external surface.

Fig. 1. Proportions of hypothetical room, showing internal

virtual photocells.


3. Comparisons

For the purposes of comparing predictions from this

SRA-based model with a reference model, a simple cu-

boid room with a single window wall has been adopted.4

The room has height H, depth 2H and width 3H. The

window wall is fully glazed above sill height S, so that

the glazed fraction is (H � S)/H. Calculation points

coinciding with the sill height are positioned within the

front and rear halves of the room (e.g. 3H/2 from the

sides and H/2 and 3H/2 respectively from the window

wall), so representing two photoresponsive control

zones5 (Fig. 1).

Predictions from the SRA-based model for this hypo-

thetical room are compared with those from a reference

model under two scenarios. In the first instance we

examine an unobstructed scene. In the latter, we exam-

ine a scene which is obstructed by a nearby plane that

has been composed to diminish both horizon and cir-

cumsolar brightening regions of the sky throughout

the year. This obstructing plane is 20 m distant from

our room, which forms part of an identical facade

(which in turn obstructs sky views to our obstructing

facade). Between these two parallel surfaces is a ground

plane, which is splayed at its ends (Fig. 2).

In order to verify the SRA-based model, a reference

model should consider the range of sky luminance distri-

butions, preferably using an identical basis to that with-

in the SRA-based model, in modelling scenes of similar

geometric complexity with due regard for diffuse reflec-

tion. This is principal leaves two options: physical mod-

elling under a controllable discretised sky dome or a

simulation program. Physical modelling of this type

introduces several sources of error (Ashmore and Ri-

chens, 2001; Mardaljevic, 2002), amongst them (i) model

construction errors, e.g. due to scaling inaccuracies, (ii)

parallax errors due to discrete source representation of

the sky luminance distribution, (iii) errors in the mea-

surement of material optical properties (reflectance and

transmittance) as well as introduction of surface rough-

ness and specularity (iv) measurement errors due to

internal obstruction from the photocell, (v) potential cal-

ibration errors due to differences in size of measurement

surface. These errors may be eliminated when using sim-

ulation programs, but numerical errors are then intro-

4 But note that equations (6) and (8) are applicable to rooms

of arbitrary shape and glazing design (concerning number,

position and optical properties).5 Based on the work of Hoch (1988), glazing can be

represented within this same room by adjusting the glass

transmittance, so that we have a design-independent represen-

tation of the magnitude of glass. This is appropriate for urban

masterplanning studies, for which the design details are

unavailable either to support specific glazing representations

or indeed room layouts.

duced. Of the numerous lighting simulation programs

available, RADIANCE (Ward Larsen and Shakespeare,

1997) has been extensively validated (Mardaljevic, 1995)

and repeatedly surpassed competing programs in terms

both of functionality and accuracy. For these reasons

we have chosen RADIANCE as our reference model,

which using a companion module gendaylit, provides

the ability to represent the sky luminance distribution

used by our SRA-based model.

3.1. Unobstructed absorbing room

As noted earlier, further renderings are in principal

required from each calculation point within the rooms

to be modelled, to determine the three view factors

(for the sky and obstructions—self and external)—with

internal surfaces also having a unique colour. However,

to reduce this pre-processing cost for this simple room

we have rendered for only one of our internal points

and for this we have removed the window wall. In this

way, with our knowledge of window geometry and

assuming that the angular position of external obstruc-

tions remains essentially unchanged, we are able to

translate the view information from the rear to the front

calculation point6—each time processing only the infor-

mation within the extremities of the projected window

plane. In stereographic projection, the resultant image

corresponds to Fig. 3.

Adopting this approach we have the following results

(Fig. 4) for a south facing room, using climate data for

Kew UK (1967), based on the use of view factors from

the rear calculation point. The agreement with RADI-

ANCE is rather good, with over predictions of just ca.

0.7% and 0.8% at the front and rear calculation points

respectively. The small discrepancy is a combination of

errors in calculating view factors and discretising the

sky vault within the SRA-based model and numerical er-

rors within RADIANCE—particularly at the rear where

6 This is useful because the same room may be modelled

behind several thousand external surfaces in a masterplanning

study, within the SUNtool modelling environment.

y = 1.0069xR2 = 0.9999

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

RADIANCE Calculated Illuminance (lux)

SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

y = 1.0084xR2 = 0.9997

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

(a)

(b)

Fig. 4. Sky illuminance within the front (a) and rear (b) halves

of a south facing room.

Fig. 2. Obscured sky patches (a) from which a corresponding plane (b) was produced, with ground plane between which is splayed at

its edge to maximise reflection interaction (c).

Fig. 3. Illustration of masking of view due to self obstructions

from opaque internal surfaces at the front (a) and rear (b)

calculation points.


the window subtends a relatively small solid angle

(�0.26Sr).

The annual RADIANCE dataset, using moderately

accurate settings, for these two calculation points, takes

approximately five orders of magnitude longer to pro-

duce than that using the SRA-based model. This relative

difference is increased when solving for multiple rooms,

since the SRA-based model solves for all obstruction

luminances simultaneously.

3.2. Obstructed absorbing room

The method of accounting for self obstructions de-

scribed above (translating view information from one

point to another) assumes that the angular coordinates

of external obstructions are insensitive to position within

a room, which may not be appropriate for nearby

obstructions. For either the front or rear of the room

good agreement is expected between models when the

calculation position coincides with view factor calcula-

tion point (with divergence in the direction of over-

prediction as our mask moves towards the front and

under-prediction towards the rear). The question is

whether to base our calculated view factors at the front

or the rear. Since light switching probability within our

hypothetical room is likely to be more sensitive to pre-

dictions at the rear of the room, where illuminances will

more frequently be close to some switching threshold,

we base our predictions on view factors calculated at

the rear of the room and accept the corresponding error

with distance from this point in the interests of compu-

tational efficiency. On this basis we have the following

results for a view to the sky which is obstructed by the

configuration described in Fig. 2 (Fig. 5).

The small increase in over-prediction at the front of

the room is due to an unrealistically low angle

of obstruction skyline elevation arising from the re-use

of view information from the rear of the room.7 The

7 But note that this is avoidable, by performing additional

renderings.

y = 1.0808xR2 = 0.9984

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

y = 1.277xR2 = 0.9798

0

1000

2000

3000

4000

5000

6000

7000

0 1000 2000 3000 4000 5000 6000 7000


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

(a)

(b)

Fig. 6. Sky, externally reflected and internally reflected illumi-

nance within the front (a) and rear (b) halves of a south facing

room.

y = 1.0409xR2 = 0.9998

0

2000

4000

6000

8000

10000

12000

14000

0 2000 4000 6000 8000 10000 12000 14000


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

y = 0.9984xR2 = 0.9994

0

200

400

600

800

1000

1200

1400

1600

1800

0 200 400 600 800 1000 1200 1400 1600 1800


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

(a)

(b)

Fig. 5. Sky and externally reflected illuminance within the front

(a) and rear (b) halves of a south facing room, based on view

factors determined from the rear calculation point.


change to under predictions at the rear of the room is

principally due to an under prediction in the mean lumi-

nance of the obstructing wall, which is split into hori-

zontal strips of uniform luminance calculated at the

strip centroid. However, this accounts for only a 0.1%

change in predictive accuracy.

3.3. Obstructed reflecting room

The inclusion of internally reflected light into our

problem introduces a considerable source of error, with

over predictions of ca. 8% and 28% for the front and

rear calculation points respectively (Fig. 6).

There are several reasons for this poor agreement be-

tween the SRA-based model when using the split flux

equation and RADIANCE. For example

(i) When the source of above horizontal illumination

is strongly anisotropic (e.g. due to sunlight), the

incoming energy is incorrectly weighted by the

mean reflectance of the lower half of the room.

Similar though less significant errors occur for

the upwards contribution.

(ii) Although the incoming energy is weighted by the

mean reflectance of the relevant half of the room,

the room is nevertheless assumed to be completely

isotropic. The results are therefore insensitive

either to receptor direction or position. The results

also neglect the effects of anisotropy following

subsequent reflections (e.g. brightening of a ceiling

due to reflection of a floor sun patch).

Within the current context however, our key objec-

tive is to inform light switching/dimming control. For

this we need reasonable accuracy around the thresholds

of this control, say in the range 0–1 kLux. Within this

range, our over predictions are more than halved (Fig.

7). This is predominantly due to the relative isotropy

of the sources of illumination (i.e., skylight/externally re-

flected skylight are dominant).

At the front of the room the sky is not clear either

when the sun is within the field of view (there are no

internally reflected sun patches) or when its angle of inci-

dence on obstructions is small (there is little externally

reflected sunlight). At the rear of the room, where views

to the sky are relatively restricted, the situation is a little

more complex. A strand of points is evident along a

unity slope line, suggesting that we have both isotropic

sources and anisotropic sources—for example when

the sun is incident on our window plane at an oblique

y = 1.0362xR

2 = 0.9999

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

y = 1.1292xR2 = 0.9975

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000


SRA

Cal

cula

ted

Illum

inan

ce (l

ux)

(a)

(b)

Fig. 7. Sky, externally reflected and internally reflected illumi-

nance within the front (a) and rear (b) halves of a south facing

room: values 61 kLux only.


azimuth angle, but the solar intensity is reasonably low

and the circumsolar region is not visible at the rear of

the room. For both cases, the accuracy of predictions

improves with reduced control threshold. Notwithstand-

ing this, for the purposes of urban masterplanning to

which this paper relates, these errors are likely to be

within the uncertainty band due to room configuration

(internal walls and furnishings, glazing shape and posi-

tion, surface reflectances etc).

Nevertheless work is ongoing to improve the resolu-

tion of predictions of internally reflected light within the

framework of this SRA-based model. Possibilities in-

clude more appropriate allocation of the incoming en-

ergy using the split flux equation (e.g., a three term

split flux, for energy incident on walls, floor and ceiling),

a radiosity solution or perhaps some hybrid—explicitly

solving for the illuminance due to the first reflection

and assuming that the room is diffuse for subsequent

reflections (e.g., the majority of room anisotropy is de-

scribed by the first reflection). It is anticipated that the

results from this work will be reported in a future paper.

4. Discussion and conclusions

The external luminous environment may be repre-

sented by two hemispheres, each discretised into patches

of known solid angle. Occlusions to these patches may

be combined and represented by some patch fraction

for which the luminous characteristics are defined by

the dominant occlusion. By solving for luminous ex-

changes between each surface in an external scene and

these patches, we have a simplified radiosity algorithm

(SRA). This SRA has been shown to produce accurate

predictions of both external illuminance and shortwave

irradiance. By extension, this detailed characterisation

of the external luminous environment may be used to

solve for illumination within arbitrarily complex rooms

composed of Lambertian surfaces, to support energy

prediction due to artificial lighting. Results from this

model, for a simple cuboid room, have been compared

to the ray tracing program RADIANCE. From this

we conclude that:

• The SRA-based daylight model produces predictions

of sky and externally reflected light within rooms that

are of comparable accuracy to the ray tracing pro-

gram RADIANCE, but at a computational cost sev-

eral order of magnitude lower.

• The split flux equation for internally reflected light

performs poorly when exterior luminances are

strongly anisotropic. However, at normal light

switching thresholds this tends not to be the case,

so that the model may be considered adequate for

supporting energy predictions.

• In producing fast and accurate predictions of sky and

externally reflected light as well as improved resolu-

tion of the incoming flux for internally reflected light,

this SRA-based daylight model represents a consider-

able improvement over simplified (e.g. daylight factor

based) models typically found within current energy

modelling software.

• Furthermore, work is in progress to improve the res-

olution of predictions of internally reflected light

from this SRA-based model. This will be reported

in a future paper.

Finally, this SRA-based daylight model has been

embedded within a prototype of the integrated resource

flow modelling tool, being developed within the context

of Project SUNtool. The model (and subsequent

improvement to it) is also readily amenable for inclusion

into other programs that require indoor illumination

(and indeed external surface irradiance) as an input.

Acknowledgements

The funding for this work by the European Commis-

sion�s Directorate General for Transport and Energy is

gratefully acknowledged. Thanks are also due to review-

ers of a draft of this paper for their helpful and construc-

tive remarks.


Appendix A. Solution by matrix inversion

An alternative approach to iterative solution of sur-

face illuminance is to formulate the problem as a matrix

equation and solve by inversion. This would give

Ed = AEg + BL; where Eg = Ed + Eb, is a vector listing

the global irradiance on each surface, and L a vector giv-

ing the luminance of each sky patch.

Rearranging:

Ed ¼ ðI � AÞ�1ðAEb þ BLÞ ðA:1Þ

The matrix A is square and describes how the direct

component of illuminance on each surface is eventually

distributed around the n surfaces in the scene (entry (i, j)

in the array describes the proportion of direct insolation

on surface j that is reflected to surface i):

A ¼

q1k1;1p

q2k1;2p . . .

qnk1;np

q1k2;1p

. .. ..

.

..

. . .. ..

.

q1kn;1p

q2kn;2p � � � qnkn;n

p

26666664

37777775

ðA:2Þ

where qi is the reflectance of surface i, and ki,j describes a

scaling factor for the effect of the energy reflected from

surface j to surface i. If surface j obstructs m sky patches

when viewed from surface i, denoted by x1,x2, . . . ,xm,then: ki;j ¼

Pmk¼1Ui;xk ð1� ri;xk � xi;xk Þ cos ni;xk , where

ri;xk is the view factor from surface i to sky patch xk

and Ui;xk is the solid angle of sky patch xk from surface i.

Matrix B describes the contribution from each sky

patch (of unit luminance) to the illuminance received

by each surface within the scene:

B ¼

U1;1r1;1 cos n1;1 U1;2r1;2 cos n1;2 � � � U1;pr1;p cos n1;p

U2;1r2;1 cos n2;1. .. ..

.

..

. . .. ..

.

Un;1rn;1 cos nn;1 Un;2rn;2 cos nn;2 � � � Un;prn;p cos nn;p

26666664

37777775

ðA:3Þ

The matrices (I � A)�1A and (I � A)�1B need only

be computed once for any given geometry. Therefore

at each time step once the direct component of illumi-

nance has been solved for, only two matrix multiplica-

tions and an addition are required to solve for the

diffuse component.

Note that by replacing Ex by Ix and L by R in (A.1)

we have predictions of surface irradiance.

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Internal illumination prediction based on a simplified radiosity algorithm

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