---

This annex includes the logical flow charts introducing /ті, and Rei calculations but with the general mathematical conventions and limitations given in Clause 4.

The variables and arrays names correspond mainly to three types of "actors" in a lit scene:

P, the current point of calculation;

L, the current luminaire;

Obs, the current observer position.

The road surface is defined for luminance calculation thanks to the present CIE r-tables files (see CIE 144:2001 mentioned in Bibliography).

A certain number of auxiliary variables and arrays need to be created for the sake of computer algorithms and cumulative variables used in lighting calculations. Programmers are advised to find in the last column of Table A.l the suggested symbols of parameters, variables and indices of the logical flow charts codified in ASCII.

The system of coordinates of the calculation program can be seen in Figure A.l.

1 lane axis

2 birds eye view of section of road

3 current« P » grid point (Xp,y_p, z_p)

Figure A.l — System of coordinate: example of road with two lanes

Table A.1 — Symbols and corresponding designations of variables, tables and parameters
used in the logical flowchart of the calculation program (in alphabetic order)

NOTE It is advised not to confuse the codification of arrangement in this table with the key numbers of Figure 10 in 7.1.4. In this figure, number 2 is not a current layout and can be dealt with the proposed logical flow chart as two single sided installations, one by carriageway, changing simply the overhang of luminaires.

As stipulated in Clause 4, all calculation results are presented with a required number of significant digits and decimal places. The objective is not to express the real accuracy of measured values dealt with in EN 13201-4, but to comply to performance requirements of the tables of EN 13201-2 with an allowed rounding for presentation.

A.2 Linear interpolation in the tables

When the required luminous intensity (or the reduced luminance coefficient) lies between measured values, an interpolation is necessary.A value z(x,y) can be found for the needed direction (x,y) as shown in Figure A.2.

Be: z(xl,yl), z(x2,yl), z(xl,y2), z(x2,y2)

four values in the table corresponding to four directions defined by the table entries:

(xl,yl), (x2,yl), (xl,y2), (x2,y2)

closest to and surrounding the direction (x,y).

Figure A.2 — Linear interpolation in the tables

There are three equivalent methods to obtain z(x,y):

1. A linear interpolation between z(xl, yl) and z(x2, yl) finding for the direction (x, yl) an intermediate value z(x, yl), followed by a second interpolation between z(xl, y2), and z(x2, y2) producing a z(x,y2) value. A third interpolation between z(x,yl) and z(x,y2) gives the searched z(x, y) value for the direction (x,y).

2. A linear interpolation between z(xl, yl) and z(xl, y2) finding for the direction (xl, y) an intermediate value z(xl, y), followed by a second interpolation between z(x2, yl), and z(x2, y2) producing a z(x2,y) value. A third interpolation between z(xl,y) and z(x2,y) gives the searched z(x, y) value for the direction (x,y).

3. A linear regression directly between z(xl, yl), z(x2, yl), z(xl, y2) and z(x2, y2) in a single interpolation.

Method 1:

For computing purposes the general calculation formulae are:

z(x, у) = z(x, yl) + ^{У У} [z(x, y2) - z(x, yl)] (A. 1)

у 2 - yl

which gives:

z(x, yl) = z(x1, yl) + -^{X x1}[z(x2, yl) - z(x1, yl)] (A.2)

x2 - xi

and:

z(x, y2) = z(x1, y2) + ^{x x1}[z(x2, y2) - z(x1, y2)] (A.3)

xZ - xi

Method 2:

Similarly and alternatively to the latter case, producing the same result is:

z(x, y) = z(x1, y) + ^x1 [z(x2, y) - z(x1, у)] (A.4)

xz - xi

which gives:

z(xl,y) = z(xl,yl) +

y-yl
—
y2-yl

(A.5)

z

(A.6)

(x2, y) = z(x2, yl) + ^У [z(z2, y2) - z(x2, yl)]
y2-y1

Method 3:

In the case of linear interpolation, the recourse to linear regression is also possible directly between the four measured values that make a cell. The general formula using Lagrange polynomial interpolation can be written:

z(x,y) = P_n xz(xl,yl) +P₂₁ xz(x2,yl) +P₁₂xz(xl,y2) + P₂₂xz(x2,y2) (A.7)

Where Pij > 0 are such that:

p₂₂₌^L. х-Я (a.s)

x2-x1 y2-y1 x2-xl y2-yl x2-x1 y2-y1 x2-xl y2-yl

The choice is given to programmers to create a subroutine "interpolation" among these three equivalent methods. These subroutines are usable for both /-tables or r-tables with respectively (C, y) or (tan c, 0) for the defined direction (x,y).

Taking account of standard r-tables format defined in Table 2, complementary programming tests are needed to avoid using the blank cells. By default most of programming languages assign 0 values to these blank cells. If there is no test to exclude the calculation when these cells are involved at the border of filled cells, the ensuing interpolation gives an incorrect rvalue (instead of nothing).

A.3 Information Technology requirements

Regarding logical flow charts, programmers are advised to use common shared variables and re­dimensioned arrays all along the different phases which are detailed in the different flow charts, and also in the subroutines. It is anticipated that these flow charts use global variables whose meaning and values do not change during the calculation program. That means all input or calculated data remain available in RAM (Random Access Memory) refreshed in real time as soon as the calculation program is launched. Apart from the edition of input data and calculation results on the computer screen, on a printer or on a plotter, they should be saved in the computer either in a specific format readable by the software or in a file in a portable document format to keep a track of the lighting design

.

Input road surface reflection properties: Rie names of r-tables (dry and wet) in CIE format

I

Input light distribution: File name of l-table (identifiable format)

I

Display of all input data

Are input data correct ?

лхр= 10

nxp = 1 + int (S/3)

S/3 = int(S/3)

r?Xp — S/3

S>30

•|

лхр = 2 лхр I

Arrangement # 3

Arrangement = 1

I crow

dx -Sin*?

xLmax = 12H * S

I xlmin = -5H

Arrangement «3

xLmax = 12H + 2S

( Start 1

Input geometric data (see mathematical conventions) including :
Carriageway width, W_K -Number of lanes,
Width of central reservation (nil when not existing), W_a
Observer distance to the first luminaire before the grid points,
Mean height of observers' eyes / road surface (default value: 1,5 m). г^,
Age of observer (default value: 23 y), A

Luminaires: Arrangement code: 1 to 5 (see conventions)
Mounting height, H - Spacing, S - Tilting, T

Overhang versus carriageway edges, CLfalgebraic value)
Assigned lamp flux of the luminaires,
Global maintenance factor, f_u

Compute grid point numbers and distances:
Transversally to die road: ny_P = 3. rij^,; dy = W_RI ny?

Along the road axis: such drat:

ntbef_fiald = int (I xtmln | IS)
nLafter_field = int (| xLmax | IS)
nt - nibef fleld + ntafter field + 1

irow = 0

iraw

I nL TI(lrow) = lnt(5O0 / S) I

xminL(irow) = -ni.bef_field«S

■| xminLfl) = xminLfl) ♦ S |

Arrangement о 3

ТІ calculation?

Limited number of luminaires?

Input nl_TI(irow)

Save input data
Save grid points data :
(nxp, d, ny_P and D)
Save luminaires’data:
ntbef_field, ntafter_field, nrow
nt(1 to nrow), nL_TI(1 to nrow)
xminL(iraw= 1 to nrow))

irow = nrow

Figure A.3 — Standard program: data input (editor)

Figure A.4 — Calculation process of point luminances, illuminances and minimum point
luminance

Figure A.5 — Calculation process of average initial illuminance, overall uniformity, average
initial luminance