---

A cross in Table 3 indicates the required r-value that shall be known.

NOTE In Table 3, blank cells indicate directions that should not be used for calculation, therefore the knowledge of r of these directions is not relevant in this standard.

5.3.2 Linear interpolation in the r-table

When a value of r is required for values of tan £ and p lying between those given in the r-table, the linear interpolation shall be retained.

The mathematical procedure is similar to that described for the /-table in 5.2.2 with tan £ replacing C half plane angles and p replacing у angles.

Again, in these formulae, interpolation can be first carried out in the tan £ values and then in the p half planes. If desired this procedure can be reversed (that is the interpolation is first carried out in the p half planes followed by tan £ values) and the same result obtained.

6 Calculation of ЦС, y)

To determine the luminous intensity from a luminaire to a point it is necessary to find the vertical photometric angle (y) and photometric azimuth (C) of the light path to the point. To do this, account shall be taken of the tilt in application in relation to the tilt during measurement, the orientation, and rotation of the luminaire. For this purpose it is necessary to establish mathematical sign conventions for measuring distances on the road and for rotations about axes. The system used is a right-handed Cartesian coordinate system. The corrections for turning movements do not allow for any change in the luminous flux of the light source due to turning movements.

A (x, y) rectangular coordinate system is used (Figure 5). The abscissa is aligned with the reference direction, which, for a straight road, lies in the longitudinal direction. Then:

*lp " ^- (9)

Лр=Л-Л (¹⁰)

where

(x ,y ) ^are coordinates of the calculation point;

(x_L,y_L) are the coordinates of the luminaire.

Key

1. edge of carriageway

2. calculation point

3. luminaire

Figure 5 — (x,y) coordinate system for locating luminaire in plan

NOTE In order to obtain positive x and у coordinates for all grid points, it is advisable to place the origin in the low left corner of the calculation field, (see Figure A.l).

6.3 Mathematical conventions for rotations

Figure 6 shows the axes of rotation in relation to the (x у z) right-handed coordinate system. In this system rotation angles are positive when pointing the right thumb along the third axis in the positive direction, the fingers curl in the direction leading from the first axis toward the second one (right hand rule).

Axis I is fixed in space, axis II and axis Ill can be turned about axis I.

Key

1. axis Ill

2. longitudinal direction

3. axis II

4. axis 1: first photometric axis

6. — Axes of rotation in relation to the (x,y) coordinate system

7. shows the relation of tilt for calculation to tilt during measurement and tilt in application. From this it is evident that:

6

(И)

- 0f - 6_m where

6 is the tilt in degree for calculation;

9, is the tilt in degree in application;

в_т is the tilt in degree during measurement.

Key

6 tilt for calculation

0f tilt in application

0m tilt during measurement

1 horizontal

Figure 7 — Tilt during measurement, tilt in application, tilt for calculation

6.4 Calculation of C and у

NOTE These can be determined in four stages:

x' = x(cos v cos 0 - sin v sin 6 sin 0) + y(sin v cost/) + cos v sin 6 sin г/)} + H cos 5 sin i// (12)

y' = -x sin v cos 6 +y cos v cos S - H sin 6 (13)

H' = -x(sin v sin 6 cos ip + cosy sin 0) -y(sin v sin </» - cos v sin 6 cos i/i) + H cos 6 cos ip (14)

where

xandy are the longitudinal and transverse distances between the calculation point and the nadir of the luminaire in Figure 5;

H is the height of the luminaire above the calculation point;

v, 8 and i/) are the orientation, tilt for calculation, and rotation.

NOTE x',y' and Н' are used in the calculation of C and у when the luminaire has been turned through v, 6, and They correspond to x,y and H in the unturned coordinate system and for calculation purposes may be regarded as intermediate variables (see Figure 6).

Caution shall be paid in Formulae (12), (13) and (14) to the value of H which is currently the mounting height of the luminaire to the road surface for horizontal or hemispherical illuminance and road luminance evaluations.

For the calculation of veiling luminance in /ті 1,5 (m) stands by default for the height of the eyes of the observer. Similarly in vertical and semicylindrical illuminance evaluations, the calculation points considered are conventionally located at 1,5 m high from the ground. In that case H - 1,5 shall be substituted to H in Formulae (12), (13) and (14) to define correctly the direction of luminous intensity interpolated in the /-table.

Evaluation of arctan — gives:

X

-

(15)

90° < arctan — < 90°

The angular quadrant in which <p lies is determined by:

where

(p is the installation azimuth in degree;

v is the orientation in degree (Figure 6), obtained from the formulae in 6.4, x' andy' being used in place ofx and у respectively.

(21)

7 Calculation of photometric quantities

The luminance at a point shall be determined by applying the following formula or a mathematically equivalent formula:

V ⁷k ^rk(tan£,^)

k=l

ні

(22)

where

is the maintained luminance in candelas per square metre;

is the index of current luminaire in the summation;

ni_u

is the number of luminaires involved in the calculation;

Ж/)

is the luminous intensity in candela of the k^th luminaire being Ck and yk calculated as indicated in 6.4;

is the overall maintenance factor, depending on light source lumen maintenance factor and luminaire maintenance factor;

rkftan £, P)

is the reduced luminance coefficient for the current incident light path with angular coordinates (fk, )?k), in reciprocal steradians (see 7.1.1.2 and Figure 4);

Hk

is the mounting height of k^th luminaire above the surface of the road, in metres.

In Formula (22) tan e and /? are the entries of the r-table r_k(tan e; fi)

tan e and ft are evaluated for each observer position and each luminaire.

From Figure 4 we can calculate:

_tPT tans- = —

H

^(xp-jq.)² +(у_р-уі)²

H

(23)

where

PT is the distance on the ground of the observed point P(x_p; _y_p) to the geometrical projection of the optical centre of the luminaire to the ground T(x_L;y_L);

H is the mounting height of the luminaire.

Similarly from Figure 4 ft is evaluated from the oriented angle between 2 vectors in the horizontal plane of the ground:

/? = arccos

(o_hP, PTj = arccos

(Жоь)ЖМФр~ЖЬЖр)
^(^xP~^xOh) +(3^/P^-J^/0h) 'д/(^хЕ~^хр)²⁺(А “-УР)

(24)

where

Oh (xohjyoh) is the projection of the observer eye position on the road surface.

NOTE 1 When the P point lies on the vertical through the luminaire, cos p is indeterminate. In this case /? can take any value (see the first line of any r-table where the r value should be the same for all fi angles).

To take account of the mirror symmetry due to the assumed isotropy of the road surface the r-table is limited to p varying between 0° and 180° Using another method than the previous formula, p could be in symmetrical quadrants such that:

-180° <p<0° (25)

In that case, a test is needed to reach the field of definition of p thanks to the sign changed in Formula (26):

If-180° < p< 0° then p= -p (26)

NOTE 2 The interest to use the inverse cosine function with the algebraic scalar product is to evaluate the p angle directly in the field of definition of the r-table: 0° to 180°.

7.1.2 Field of calculation for luminance

In the longitudinal direction of the relevant area, the field of calculation shall enclose two luminaires in the same row (see Figure 8). When there is more than one row of luminaires and the spacing of the luminaires differs between rows, the field of calculation shall lie between two luminaires in the row with the larger or largest spacing.

T

5 first luminaire in field of calculation

6 observer

7 observation direction

his last procedure may not give accurate luminances for the whole installation as luminances will differ in the different spans between adjacent luminaires. As calculations are carried out to comply with the requirements of EN 13201-2, the field of calculation that gives the worse results shall be chosen among the possible fields of calculation in the relevant area.

Key

1. edge of relevant area

2. field of calculation

3. width of relevant area W_r

4. last luminaire in field of calculation

Figure 8 — Information for luminance calculations;
field of luminance calculations for the relevant area

NOTE Relevant area is defined in CEN/TR 13201-1:2014, 3.17.

7.1.3 Position of calculation points

The calculation points shall be evenly spaced in the field of calculation as shown in Figure 9.

The first and last transverse rows of calculation points are spaced at one half the longitudinal spacing between points from the boundaries of the calculation field.

NOTE This grid is similar to the grid used for illuminance calculations as regards the positioning of the first and last row of calculation points in the transverse direction (see Figure 14).

Key

1 edge of lane

2 last luminaire in calculation field

3 field of calculation

4 centre-line of lane

5 first luminaire in calculation field

6 observation direction

7 observer’s longitudinal position

X denotes lines of calculation points in the transverse and longitudinal directions.

Figure 9 — Information for luminance calculations; position of calculation points in a driving
lane

The spacing of the points in the longitudinal and transverse directions shall be determined as follows: a) In the longitudinal direction

(27)

where

D is the spacing between points in the longitudinal direction, in metres;

S is the spacing between luminaires in the same row, in metres;

N is the number of calculation points in the longitudinal direction with the following values:

for5< 30 m, W= 10;

for S > 30 m, the smallest integer giving D < 3 m. The first transverse row of calculation points is spaced at a distance D/2 beyond the first luminaire (remote from the observer).

b) In the transverse direction

The spacing (d) in the transverse direction is determined from the formula:

d = ^- (28)

where

d is the spacing between points in the transverse direction, in metres;

IVi is the width of the lane, in metres.

The outermost calculation points are spaced d/2 from the edges of the lane.

Where there is a hard shoulder and luminance information is required, the number and spacing of the calculation points shall be the same as for a driving lane.

When illuminance calculations are provided together with luminance calculations for the same relevant area on a carriageway the definition of calculation points used for both calculations shall respect the definition of calculation points detailed previously in this paragraph.

7.1.4 Position of observer

For luminance calculations the observer’s eye is 1,5 m above the road level and at 60 m ahead the calculation field of the relevant area.

In the transverse direction the observer shall be positioned in the centre of each lane in turn. Average luminance (see 8.2), overall uniformity of luminance (see 8.3) and threshold increment (see 8.5) shall be calculated for the entire carriageway for each position of the observer. Longitudinal uniformity of luminance (see 8.4) shall be calculated for each centre-line. The operative values of average luminance, overall uniformity of luminance, and longitudinal uniformity of luminance shall be the lowest in each case; the operative value of threshold increment shall be the highest value.

Figure 10 gives examples of the observer position in relation to the field of calculation.

1

2

3

4

5

6

7

8

x 9

□ 10

Key

1. six lane road with central reservation and twin central luminaire arrangement

2. six lane road with central reservation and single side luminaire arrangement

3. three lane road with single side luminaire arrangement

4. three lane road with double side luminaire arrangement

5. three lane road with staggered luminaire arrangement

6. two lane road with single side luminaire arrangement

7. two lane road with double side luminaire arrangement

8. two lane road with staggered luminaire arrangement

9. observer position

10. calculation field

Figure 10 — Examples of positions of observation points in relation to the field of calculation

7.1.5 Luminaires included in calculation

The boundary of the area for locating luminaires to be included in calculating the luminance at a point is determined as follows (see Figure 11):

1. boundary on either side of the observer: at least five times the mounting height H on either side of the calculation points;

2. boundary furthest from the observer: at least 12/7 from the calculation point in the direction remote from the observer;

3. boundary nearest to the observer: at least 5/7 from the calculation point in the direction towards the observer.

NOTE The extent of these boundaries is governed by the area covered on the road by the r-table. If the mounting height of luminaires is less or equal to 2 m, a distance of 20 times the mounting height around the calculation points for all azimuth angles is necessary. See informative Annex В about the extended r-table format needed.

Key

1. calculation point

2. boundary of field of calculation

3. boundary of area for location of luminaires

4. observation direction

Figure 11 — Boundary of area in which luminaires are located for luminance calculation

7.2 Illuminance

In this standard any of four different types of illuminance might need to be calculated, depending on the design criteria chosen from EN 13201-2. These might be:

1. horizontal illuminance;

2. hemispherical illuminance;

3. semi-cylindrical illuminance;

Calculation points shall be located on a plane at ground level in the relevant area.

The horizontal illuminance at a point shall be calculated from the formula or a mathematically equivalent formula:

у 4(C/)7м-cos³£_k

"

k=l

where

Eh к

Пій

/м

Ek

H_k

Hk

is the maintained horizontal illuminance at the point (lx);

is the index of current luminaire in the summation;

is the number of luminaires involved in the calculation;

is the luminous intensity in candela of the fc^th luminaire being C and у calculated as indicated in 6.4;

is the overall maintenance factor, the product of the light source lumen maintenance factor and the luminaire maintenance factor;

is the angle of incidence of light at the point [°];

is the mounting height of k^th luminaire (m).

NOTE It is advised not to include lamp survival factor in the overall maintenance factor in road lighting if all failed light sources will be spot replaced.

Calculation points shall be located on a plane at ground level in the relevant area.

The hemispherical illuminance at a point shall be calculated from the formula or a mathematically equivalent formula:

"w 4 (C, /) • 4. • Г cos³ r_k + cos² £-_k 1

₌ ў kt M L * у ₍₃₀₎

^hs £ 4H_k^{1 J}

where

Ehs is the maintained hemispherical illuminance at the point (lx);

к is the index of current luminaire in the summation;

niu is the number of luminaires involved in the calculation;

/k(C, y) is the luminous intensity in candela of the k^th luminaire being C and у calculated as indicated in 6.4;

/м is the overall maintenance factor, the product of the light source lumen maintenance factor and the luminaire maintenance factor;

Ek is the angle of incidence of the light at the point (°);

Як is the mounting height of k^th luminaire (m).

NOTE It is advised not to include lamp survival factor in the overall maintenance factor in road lighting if all failed light sources will be spot replaced..

Calculation points shall be located on a plane 1,5 m above the surface in the relevant area.

Semi-cylindrical illuminance varies with the direction of interest. The vertical plane in Figure 12, at right angles to the rear flat surface, shall be oriented parallel to the main directions of pedestrian movement, which for a road are usually longitudinal.

The semi-cylindrical illuminance at a point shall be calculated from the formula or a mathematically equivalent formula:

E (1 + cos^)

*-1 ⁷¹ ' ^LkR²

where

Esc is the maintained semi-cylindrical illuminance at the point, in lux;

к is the index of current luminaire in the summation;

niu is the number of luminaires involved in the calculation;

/k(C, y) is the luminous intensity in candela of the /c^th luminaire being C and у calculated as indicated in 6.4 (cd);

/м is the overall maintenance factor, the product of the light source lumen maintenance factor and the luminaire maintenance factor;

«к is the angle between the vertical plane containing the incident light path and the vertical plane at right-angles to the flat surface of the semi-cylinder, as shown in Figure 12 (°);