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Solar Energy 93 (2013) 72–79

Brief Note

Determination of time and sun position system

Richard Kittler ⇑, Stanislav Darula 1

Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava, Slovakia

Received 4 June 2012; received in revised form 11 March 2013; accepted 30 March 2013Available online 1 May 2013

Communicated by: Associate Editor David Renne

Abstract

Accurate calculations and recording of time either in local clock time or true solar time in regular daytime measurements of sun andsky radiation and light, in computer programs evaluating solar irradiance or sunlight and skylight illuminance currently apply differentlydefined solar hour angles as well as angularly determined solar altitude and azimuth coordinates. The historical basis of solar geometryused to construct sundials for the measurements of true solar time were gradually replaced during Middle Ages by spherical trigonometryrelations of solar altitude and azimuth angles including solar declination, local geographical latitude and time changes in better precision.However, in different scientific and practical applications like in solar energy or daylight calculations are currently used several systemsand formulae defining sun position and time which need to be unified and standardised in computer programs or measurement evalu-ations. Potential confusion and inconsistent results should be avoided, possible mistakes have to be checked and corrected after the validISO international standard. This paper discusses various approaches considering time and sun coordinates in different systems.� 2013 Elsevier Ltd. All rights reserved.

Keywords: Sun position; Orientation and time; Solar hour angle; Solar altitude and azimuth

1. Introduction

The need to determine orientation in space or locationand in fluent time was felt a long time ago during the eraof food gatherers and hunters and even stronger in thebeginning of civilisation when the first settlements werebuilt. The periodic changes of daytime and night-time withthe typical regular sun-paths logically led to simple sundi-als using the sun shadow of a vertical stick thrown by sun-beams in equatorial regions. Long ago were also noticedthe two extraordinary days when these were divided intotwo exactly same and equal periods with sunrise and sunseton the horizon opposite to each other (Kittler et al., 2012).

Probably in Egypt 5000 years ago a Solar calendar with365 days, i.e. 12 times 30 days plus 5, was used with 24 h aday. Also in old Sumer roughly about 3000 BC the correct

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⇑ Corresponding author. Tel.: +421 2 59309267.E-mail address: usarkit@savba.sk (R. Kittler).

1 ISES member.

spring and autumn equinox dates were identified and thesewere expected to appear in any place also far from theequator and seemed to be a perfect expression of the“heaven–nature justice” given by gods to men. Dungi asthe first king of Ur established since 2650 BC the sexages-imal system for measuring time and space after a moon cal-endar with 12 month, 30 days each and double 12 h in aday. So, the day was divided to 12 h of daytime and 12 hof night-time expressing each the 180� sun-path range withan hourly angular step of 15� (Paturi, 1989). This is an evi-dent remnant of the sexagesimal calculation system devel-oped by Sumerians which is still alive nowadays with the60 min in an hour and 60 s per minute too. The Sumeriansystem of 360� circularly returning periods enabled thedivision possibility using integer numbers. Uncertain wasthe compensation necessary to adjust the calendar to thetrue length of a solar year, i.e. to 365.25 days.

Anyhow, the long experience with sundials and sun-pathobservations resulted in its perfect geometrical interpreta-tion probably stored on clay tablets in the Babylonian main

R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79 73

archives of the Mardok church where the chief priest Ber-ossos has studied the geometrical principle later called ana-lemma by Vitruvius (MS. 13 BC printed 1487). Thedescriptive orthogonal projection of the sun-path diagramswere later used in urban planning, for the orientation ofhouses, churches or pyramids (Rossi, 2007, Kittler andDarula, 2008a,b) and these were also taught in architec-tural schools (Tregenza and Wilson, 2011). However, itseems that recently the analemma rule (Kittler and Darula,2006) is almost forgotten due to the use of spherical trigo-nometry in computer programs (Doerfler, 2007), thus thedescriptive imagination of typical sunpath changes is some-times not realized.

2. Time determination and measurement system

The oldest traditional sundial inherited from Sumerianand Mesopotamian knowledge and graphical descriptionapplied the symmetrical N–S projection of the hemispheresection determined the parallel line to the globe pole andequator plane related to the horizon and zenith point.The local sun shade studies enabled to determine also thesolar declination range which was taken as 1/15th part ofthe hemisphere circle, i.e. d ¼ 360

�=15 ¼ �24

�, (now more

exactly 23.45�).Evidently, the noon sun culmination at 12o0clock true

solar time (TST) in the Northern globe hemisphere hasalways the South direction and dividing the whole circlefor 24 h, the time or hour angle is 15� or s = pH/12 in radi-ans, when H is hour number in TST.

In fact in the equatorial belt the noon sun position ispassing from the South to the North but in the Southernglobal hemisphere typical are noon solar azimuth anglesdue North.

So, to be precise a North oriented hour angle s has to beclearly marked and distinguished from the South oriented t

as differently are then determined the solar altitude and azi-muth angles. As already published (Kittler and Mikler,1986a, p. 42) the mutual relations are t ¼ 180

� � s andcos t = �cos s, sin t = sin s, where s ¼ 15

�H , while

t ¼ 15� ð12� HÞ if H 6 12 and t ¼ 15

� ðH� 12Þ if H > 12,or t ¼ 15

� jH� 12j. All old sundials, water, candle and pen-dulum time measuring apparatus were respecting the solarhour angle, i.e. 360�/24 = 15�, but could not care about theminute and second precision. However, even if the 15� hourangle is valid the daily period in number of hours a.m.(ante meridian) or after noon p.m. (post meridian) can beused as applied in Britain. So, various hour angle systemsare presented in Fig. 1.

Early mechanic clocks gravitationally driven by weightwere installed since the 13th Century in many church ortown hall towers. Also later watches with their hour andminute hands rotating over the clock dial covered a 360�clockwise circle twice from hour number H = 0–12 perday. All these mechanical clocks and watches were basedon the circular turning of cogwheels, therefore needed acoordinated step between the hour and minute rotation

which meant for minutes and seconds 360�/60 = 6� whilefor an hour 360�/12 = 30� i.e. twice the hour angle. Thus,if in a day is the division of the circular clock face coversonly 12 h, the 180� turn is precisely 30 min or half an hour.However, this technical solution of the clock dial dividingthe day in two 12 h parts on 360� dials brings the clockhands fictitiously either at noon or midnight to the upperposition both seemingly indicating the North direction.So in fact inherently is applying a solar angle twice whichmeans artificially introduced distortions in the coordinatedsystem of time and space orientation.

It has to be noted that the now accepted Universal TimeCoordination (UTC) and the so called Greenwich MeanTime (GMT) is also based on the globe rotation and iscounted from 0.00 midnight and valid for the referencegeographical longitude 0� (Greenwich, London). Of course,the local clock time (LCT) within defined time zones is usu-ally based on UTC respecting state or national territoriesthat sometimes do not coincide with the 15� longitudescale.

3. Determination of solar position angles

Besides the analemma geometrical construction of solaraltitude and azimuth angles published in several editions ofVitruvius’s books (MS 13 BC, printed 1487), astronomerswere trying to define separately the position of different starsincluding the sun using spherical geometry. Probably thefirst trials by Johannes de Sacrobosco or John of Holywood(1230) to determine the position of stars by their angles fromthe globe pole and local zenith were applied also to sun posi-tion expressing its spherical angles. His imagination ofspherical trigonometry (Paturi, 1989) applying sphericalangles for navigation and calendars made him renown inMedieval years after his book was first printed in 1472 and65 times reprinted in the next 75 years (Daly, 2008).

However, after the old Ptolemy’s “Mathematical Syn-taxis” written about 150 AD and rediscovered in the 12thCentury Medieval Europe in an Arabic translation knownas al-Majisti or Almagest corrected by Jabir ibn Aflah in“Islah al-Majisti” implemented also Arabic knowledge ofspherical trigonometry. Later similarly Regiomontanus(1436–1476) during his studies at the Rudolfina Universityin Vienna tried to translate the original Greek Almagestand introduced with his teacher Georg von Peurbach(1461) trigonometric functions instead of the old angularchord system. In his seminal book on triangles Regiomont-anus (1464, 1533) probably copied Aflah’s spherical trigo-nometry and following Peurbach’s suggestions workedout probably the first tables of sine functions in “Compos-itio tabularum sinum recto”. These later compiled bySantbech et al. (1561) with edition of his summary on pla-nar and spherical trigonometry enabled also the solar alti-tude and azimuth angles to be determined anew. However,only after sine, cosine, tangent and cotangent tables wereworked out by Rheticus and after his death published by

Fig. 1. Alternative systems of defining hour angles and solar azimuth angles.

74 R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79

his pupil Oth (1596), the solar trigonometry could be usedto define the momentary sun position in any location.

In mapping the globe, states or countries, in urban andlocal cadastral maps these are oriented from the North car-dinal point identifying places by their geographical latitudeand longitude and with the compass orientation of sitesmeasured clockwise relaying on the magnetic globe poles.In fact these geodetic measurements started a very longtime ago, since 1160 BC in China were used magnetite nee-dle-like pieces inset in straw and flown in water. The mag-netite always turned to align with the N direction of theNorth Star and oppositely to S noon position of the Sun.Later in 1269 AD Pierre de Maricourt studied magnetiteand explained the existence of two equally strong magneticpoles and an Italian Sanuto in 1588 discovered the N and SEarth magnetic poles and assumed that these are placedclose to globe poles. Nowadays geodetic measurementsare using theodolite with the compass adjustment directingthe equipment to the magnetic globe meridian due North.

In fact the main problem was to establish a unified sys-tem binding the time progress, i.e. changes in the hourangle, with the solar altitude and azimuth angular changes.Several orientation systems were used in different scientificand practical regions of measurements as shown in Fig. 1.

In geodetic measurements, in surveying and geographicalmapping several more sophisticated systems of azimuthdetermination are currently based on spherical trigonometry

and solar position observations (Boucher, 1983). Astron-omy calculations used a similar system of 360� clockwiseangles are also used but taken from the South are preferredprobably due to simple apparent angles in spherical trigo-nometry representation. However, in nautical navigationdue to ship routes between the Northern and Southern globehemispheres a dual orientation system was used in practice.

Although these differences were summarised by Vinaccia(1939) he proposed for building purposes to use the solarazimuth taken in morning from South to North anticlock-wise while in the afternoon the clockwise angles both 0–180� were taken in plus or minus angular values. Some-times in solar engineering textbooks as e.g. by Iqbal(1983) this system has still some tradition in spite of thefact that neither astronomy nor geodesy and cartographydo not favour it. Some books in building science and archi-tecture (e.g. Markus and Morris, 1980) also favour the azi-muth zero from the Southern cardinal point with ±180�eastward and westward angles respectively probably dueto similar insolation conditions for buildings with symme-try to the sun noon position. However, currently urban andbuilding designs are based on geodetic maps of settlementswith the orientation of building sites oriented from theNorthern cardinal point with measured angles 0–360�clockwise, thus hour and sun position angles are coordi-nated with the geodetic system. In fact the North orientedsystem is now standardised by ISO, 2003 world-wide.

R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79 75

4. Solar positions expressed by spherical trigonometry

relations

The spherical trigonometry assuming a fictitious spherewith a unity radius can be divided by any three great circlesforming an arbitrary triangle with three sides and sphericalangles. Their arc distances can be divided by the sphereradius to yield radians, now preferred angular units in com-puter programs. Given three elements of the spherical tri-angle it is possible to solve those remaining.

To determine the solar altitude and azimuth the bestchoice is the triangle formed by great circles passing theobserver’s zenith Z, the celestial pole P and the sun posi-tion S in Fig. 2. Thus the three triangle sides are definedby co-declination, co-altitude and co-latitude angles. Theseare identified and measured from the hemisphere centre,i.e.

�

Fig. 2. The orthogonal sketch of the fictitious sky hemisphere with a unityradius in the N–S section and plan showing the sun position in angulardistances from the zenith and the direction of the globe pole thus formingthe spherical triangles.

– a – the co-declination angle or polar angular distance ofthe sun, which is a ¼ 90

� � d, thus cos a ¼ sin d orsin a = cos d,

– b – the co-altitude angle b ¼ 90� � cS equal to solar

zenith distance ZS i.e. b ¼ ZS ¼ 90� � cS , thus

cos b = cos ZS = sin cS or sin b = cos cS = sin ZS,– c – the co-latitude uZ, i.e. the angular distance of the

pole to the zenith of the hemisphere, c ¼ uZ ¼ 90� � u

and cos c = cos uZ = sin u or sin c = cos u where u isthe geographical latitude of the locality.

To define the solar zenith distance ZS or the solar alti-tude cS the cosine basic formula of spherical geometrycan be applied as:

cos b ¼ cos a cos cþ sin a sin c cos b ð1Þi:e: cos ZS ¼ sin cS ¼ sin d sin u� cos d cos u cos s ð2Þ

where s ¼ 15�H

To determine the solar azimuth angle ANS taken from

North clockwise the second cosine formula can be appliedas:

cos a ¼ cos b cos cþ sin b sin c cos a ð3Þ

i:e: sin d ¼ sin cS sin uþ cos cS cos u cos ANS ð4Þ

where from is:

cos ANS ¼

sin d� sin cS sin ucos cS cos u

¼ sin dcos cS cos u

� tan u tan cS ð5Þ

The first relation corresponds with that in Tregenza andWilson (2011).

If in Eq. (5) is inserted the sin cS from formula (2), then:

cosANS ¼

1

coscS cosusind� sinuðsinusind� cosucosdcossÞ½

¼ cosdcoscS

ðcosu tandþ sinucossÞ

ð6Þ

The last Eq. (6) was recommended by Kittler and Mikler(1986a,b) as a best azimuth cosine function for the Northoriented system which was derived by applying the cotan-gent and sine functions leading directly to Eq. (7) for morn-ing hours, i.e. for H 6 12 is:

cos ANS ¼ cot AN

S sin ANS

¼ 1

sin sðcos u tan dþ sin u cos sÞ cos d sin s

cos cS

¼ cos dcos cS

ðcos u tan dþ sin u cos sÞ

¼ 1

cos cScos u sin dþ sin u cos d cosð15

�HÞ

� �ð7Þ

Then afternoon azimuth for H > 12 can be calculatedafter:

76 R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79

cos ANS ¼ 360

�

� 1

cos cScos u sin dþ sin u cos d cos ð15

�HÞ

� �ð8Þ

Because computer calculations are generally performedin radians above formulae are slightly adapted, i.e. forthe solar altitude Eq. (2) is taken in the form:

cos ZS ¼ sin cS ¼ sin d sin u� cos d cos u cospH12

� �ð9Þ

The solar azimuth in radians after Eq. (7) for H 6 12and after Eq. (8) if H > 12 can be written as follows:

cos ANS ¼

1

cos cScos u sin dþ sin u cos d cos

pH12

� �� �ð10Þ

cos ANS ¼ 2p

� 1

cos cScos u sin dþ sin u cos d cos

pH12

� �� �

ð11Þ

Note that if the computer calculations are processedonly in radians then all angles in Eqs. (9)–(11) have to bereduced to their radiant values similarly as the hour angle.

s ¼ 15�pH

180� ¼

pH12¼ 0:2617992H ð12Þ

i.e. any angle for x, d and u has to be taken in radians as:

x ¼ px�

180� ¼ 0:017453x

� ð13Þ

Therefore in some formulae e.g. (ASHREA, 2009) theangles d and u are denoted differently as d and l. Herethe actual geographical latitude of the location u is takenfrom maps with a positive value in the Northern globehemisphere or with a negative value for locations in theSouthern hemisphere. The solar declination d can be takenfrom astronomical almanacs or after Meeus (1998) or Redaand Andreas (2004) in a precise value with respect to theactual year/calendar changes. In the engineering calcula-tions approximate simpler formulae can be used after dif-ferent formulae summarised by Kittler and Mikler(1986b), including e.g. the one by Smith and Wilson(1976) in degree.

d ¼ 23:45�

sin360

� ðJ� 81Þ365

� �ð14Þ

or by Pierpoint (1982) in radians:

d ¼ 0:4093 sin2p ðJ� 81Þ

365

� �ð15Þ

where J is the day number within a year.

5. Comparison of several published azimuth formulae

The advantage when taking the hour angle from theSouthern cardinal point is in its sine positive function

either in the astronomical or in Vinaccias systems that yieldthe same formula, i.e:

sin cS ¼ sin d sin uþ cos d cos u cos t ð16Þ

The same positive equation is recommended for solarengineering purposes by ASHREA (2009), probably fol-lowing Iqbal (1983), thus it is evident that the South ori-ented system for solar position definition is proposed. Incontrary, the North oriented geodetic system is used incadastral maps, in urban planning and architecture, in illu-minating engineering calculations (IESNA, 2000) as well asin daylight science (Kittler et al., 2012), i.e. the coordinatedissue in defining hour angles and solar position angles tothe same basic Northern cardinal point. Thus the hourangle s ¼ 15

�H in degrees or s ¼ pH=12 in radians and

then cos t = �cos s, thus after Eq. (2) is:

sin cS ¼ sin d sin u� cos d cos u cosð15�HÞ ð17Þ

In some publications the basic orientation record ofsolar altitude and azimuth is not provided and the hourangle relationship to true solar time is uncertain. However,although the solar altitude difference is only in the positiveor negative relationship of the formulae two members, evenmore complex occur in various azimuth formulae.

The older Vinaccia (1939) formula is based on the sinefunction and the two-sided South orientation:

sin ASS ¼

cos d sin tcos cS

ð18Þ

which is mentioned by Iqbal (1983) with a note that “thisequation gives improper values when AS

S P 90�

and shouldbe avoided”. Therefore he recommended to apply a cosinefunction similar to that in Eq. (5), but with the two-sidedSouth orientation:

cos ASS ¼

sin cS sin u� sin dcos cS cos u

ð19Þ

However, the ASHREA Handbook (2009) is using boththe sine Eq. (16) as well as a cosine function with the two-sided South orientation as is:

cos ASS ¼

1

cos cSðcos t cos d sin u� sin d cos uÞ ð20Þ

Eq. (20) in the North oriented system, whencos t = �cos s leads to:

cos ANS ¼

1

cos cSð� cos s cos d sin u� sin d cos uÞ ð21Þ

while Vinaccia’s Eq. (18) for morning hours in the Northoriented system is t ¼ 180

� � 15�H and

sin t ¼ sin s ¼ sinð15�HÞ, then:

sin ANS ¼

cos d sin scos cS

ð22Þ

The tangent form can be derived applying Eqs. (22) and(21) as:

R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79 77

tan ANS ¼

sin ANS

cos ANS

¼ �½cos d sinð15�HÞ�= cos cS

�½cos d sin u cosð15�HÞ þ sin d cos u�= cos cS

ð23Þwhich coincides with the IESNA (2000) formula given inradians, i.e.:

tan ANS ¼

�½cos d sinðpH12Þ�

�½cos u sin dþ sin u cos d cosðpH12Þ� ð24Þ

A similar but simpler formula was also derived by Bou-cher (1983) in degrees as:

tan ANS ¼

sin scos u tan d� sin u cos s

ð25Þ

However, it seems that there is a disadvantage in usingtangential function as in 90� or in 270� angles it reachesinfinity. This discontinuity can cause uncertain calculationerrors in the general simulation of fluent changes due todisruptions especially in extreme cases when the wholerange 0–360� or 0–2p has to be modelled for the sunpathsfluently. Therefore the cosine functions in Eqs. (7) and (8)in degree, or in Eqs. (9) and (10) in rad. were also recom-mended by Muneer (1997) for computer algorithms. Treg-enza and Sharples (1993) used both the sine and cosinefunctions of the solar altitude after Eq. (5) which inherentlyinclude also the two tangent functions as shown in secondpart of this equation. Thus misunderstandings can becaused in calculations for equinox days at globe poleswhere zero azimuths for the whole days would be indi-cated. More sophisticated calculation methods for accuratesolar positions were suggested recently by Blanc and Wald(2012) recommended for astronomy purposes and satellitedata evaluations but their use has to expect rather tediousand complex algorithms.

Markus and Morris (1980) in their sunpath diagrams wit-tily introduced the possibility to use the same diagrams for

Table 1Solar altitudes and azimuths in degrees in an equinox day 21st March.

Hour in LCT Quito Savannahu = 0�; k = �80� Ref(k) = �75� u = 32�; k = �cS AS AN

S cS A

6 – – – –7 7.20 89.94 89.94 6.138 22.00 89.94 89.94 18.729 37.20 89.93 89.93 30.88

10 52.20 89.91 89.91 42.1111 67.20 89.86 89.86 51.4712 82.20 89.59 89.59 57.2213 82.80 �89.56 270.44 57.34 �14 67.80 �89.85 270.15 51.78 �15 52.80 �89.91 270.09 42.53 �16 37.80 �89.93 270.07 31.35 �17 22.80 �89.94 270.06 19.22 �18 7.80 �89.94 270.06 6.6419 – – – –

the Northern globe hemisphere (with N on the upper side)as well as for the Southern hemisphere localities by turningthe diagrams 180� around. So in fact both the third andfourth azimuth systems on Fig. 1 were applied. A similar pos-sibility of using Eq. (22) is recommended by Martin andGoswami (2005, p. 2), when defining the solar azimuth angle:

ASS ¼ arcsin

cos d sin tcos cS

� �ð26Þ

where the hour angle t is defined as 15� times hours fromlocal solar noon, while the solar azimuth angle is deter-mined as “the angle between the projection of the earth–sun line on the horizontal plane and the due South direc-tion (Northern hemisphere) or due North (Southernhemisphere)”.

In Australia is frequent only the fourth azimuth systemin Fig. 1 taking azimuth angles from North, which wasused by Roy et al. (2007) in the computer programMAM for calculating sky luminance within any windowsolid angle after the 15 ISO standard skies. To documentand compare azimuth angles and sunpath changes in typi-cal localities in Tables 1–3 were chosen:

– Roy’s Australian hometown Perth, u ¼ �32�S;

k ¼ 116�E.

– Opposite in the Northern hemisphere Savannah, USA.u ¼ 32

�N ; k ¼ 81

�W .

– Equatorial locale Quito in Ecuador u ¼ 0�; k ¼ 80

�W .

In all these locales are traced sunpaths in the equinoxday 21st March in Table 1 as well as in the extreme sol-stices, i.e. 21st June (in Table 2) and 21st December (inTable 3) in hourly steps in local clock time (LCT). In alltables are as AS given azimuth values in the Australian sys-tem preferred by Roy, while in the next column are AN

S azi-muth values after the standard ISO North oriented system.

Of course the seasonal solar altitudes indicate the shiftof the summertime to December in high sunpaths in the

Perth81� Ref(k) = �75� u = �31.9�; k = 116� Ref(k) = 120�

S ANS cS AS AN

S

– –93.78 93.78 7.70 85.11 85.11

102.15 102.15 20.27 76.64 76.64111.86 111.86 32.35 66.70 66.70124.29 124.29 43.42 53.80 53.80141.53 141.53 52.44 35.78 35.78165.49 165.49 57.56 11.02 11.02166.57 193.43 56.90 �16.87 343.13142.35 217.65 51.75 �40.22 319.78124.87 235.13 41.16 �56.93 303.07112.29 247.71 29.81 �69.03 290.97102.51 257.49 17.58 �78.55 281.45�94.10 265.90 4.96 �86.84 273.16

– –

Table 2Solar altitudes and azimuths in degrees in a solstice day 21st June.

Hour in LCT Quito Savannah Perthu = 0�; k = �80� Ref(k)=�75� u = 32�; k = �81� Ref(k)=�75� u = �31.9�; k = 116� Ref(k) = 120�

cS AS ANS cS AS AN

S cS AS ANS

6 – – 7.13 66.76 66.76 – –7 7.89 66.32 66.32 19.10 73.74 73.74 – –8 21.56 64.68 64.68 31.49 80.37 80.37 7.20 56.57 56.579 34.92 60.98 60.98 44.13 87.29 87.29 17.21 46.90 46.90

10 47.61 53.84 53.84 56.84 95.67 95.67 25.59 35.16 35.1611 58.68 40.08 40.08 69.27 109.02 109.02 31.61 21.05 21.0512 65.75 14.41 14.41 79.74 144.99 144.99 34.50 4.99 4.9913 65.11 �19.04 340.96 78.55 �136.24 223.76 33.75 �11.63 348.3714 57.22 �42.73 317.27 67.48 �106.41 253.59 29.52 �27.02 332.9815 45.80 �55.21 304.79 54.97 �94.26 265.74 22.45 �40.17 319.8316 32.98 �61.69 298.31 42.26 �86.22 273.78 13.34 �51.01 308.9917 19.55 �65.03 294.97 29.64 �79.39 280.61 2.84 �60.00 300.0018 5.86 �66.43 293.57 17.30 �72.75 287.25 – –19 – – 5.41 �65.67 294.33 – –

Table 3Solar altitudes and azimuths in degrees in a solstice day 21st December.

Hour in LCT Quito Savannah Perthu = 0�; k = �80� Ref(k) = �75� u = 32�; k = �81� Ref(k)=�75� u = �31.9�; k = 116� Ref(k) = 120�

cS AS ANS cS AS AN

S cS AS ANS

6 – – – – – 9.33 111.91 111.917 8.73 113.77 113.77 – – 21.41 105.07 105.078 22.38 115.52 115.52 6.42 122.89 122.89 33.87 98.47 98.479 35.71 119.38 119.38 16.50 132.45 132.45 46.54 91.45 91.45

10 48.33 126.81 126.81 24.99 144.04 144.04 59.25 82.61 82.6111 59.24 141.15 141.15 31.17 157.98 157.98 71.57 67.43 67.4312 65.93 167.61 167.61 34.28 173.92 173.92 81.01 21.35 21.3513 64.77 �159.14 200.86 33.78 �169.48 190.52 76.84 �53.05 306.9514 56.56 �136.29 223.71 29.77 �153.98 206.02 65.20 �76.80 283.2015 45.03 �124.31 235.69 22.89 �140.68 219.32 52.58 �87.63 272.3716 32.16 �118.07 241.93 13.91 �129.66 230.34 39.86 �95.25 264.7517 18.71 �114.87 245.13 3.50 �120.58 239.42 27.28 �101.94 258.0618 5.02 �113.57 246.43 – – 15.00 �108.60 251.4019 – – – – – – 3.21 �115.79 244.21

78 R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79

Southern globe hemisphere in contrary to wintertime in theNorthern one which is associated with cold dull weatherand sunshine deprivation.

Bearing in mind the comparison of the ISO recom-mended orientation with that used in Australia the solarazimuth in any locality is different only in afternoonhours, i.e. then AN

S ¼ 360� � jAS j which means a trivial

correction of AS in Tables 1–3. However, if a particularcomputer tool is to be utilised further, e.g. needs anyinput of building, window or solar collector orientations,these have to be defined in the specific orientation systemof such computer program. So, for instance in use ofMAM the window orientation input should be in azimuthAS, i.e. respecting minus azimuth for western windowsand fac�ades.

6. Conclusions

The main purpose of this article was not only to reviewthe historically very long development trial to determinesolar coordinates influenced by the specific localities, theirgeographical latitude and longitude, with seasonal solardeclination and daily hourly changes. The ancient graphi-cal determination of sunpaths for sundials known as theAnalemma rule preserved in the Vitruvius manuscript weregradually later replaced by trigonometric functions andspherical images of angular relations that nowadays serveeffectively in computer algorithms. However, differentapproaches and ways in this development search broughtseveral relations using hour angles resulting in differentexpressions for solar azimuth angles.

R. Kittler, S. Darula / Solar Energy 93 (2013) 72–79 79

At the same time the separate long history of geographicand geodetic measurements resulted in global, country orstate maps, as well as in town and village plans and cadas-tral division of ownership and real estate documentationwhich have to be respected in urban planning or architec-tural and building design. All professions linked with envi-ronmental and climatic aspects of these designs have to usein their calculations and computer programs a unified sys-tem of time and orientation mutually understandableworldwide. Such a unified and coherent system of timeand orientation is now standardised by ISO. It is necessaryto respect it at least in the computer programs for buildingphysics, solar engineering and illuminating engineering cal-culations when specifying the true solar time, solar hourangle as well as solar altitude and azimuth in calculationsor measurements. Therefore, to comply with the ISO19115 international standard, solar azimuth AN

S with theNorth zero turning clockwise around to 360� after Eqs.(10) and (11) should be recommended in all solar energyand illuminating engineering calculations as well as in com-puter programs. To avoid misunderstanding and errors insolving practical problems by computer programs azimuthcoordinates in other coordinate systems have to be checkedand corrected if necessary as indicated and demonstrated inthis paper.

Acknowledgement

This paper was written under the financial support ofthe Slovak Grants APVV-0177-10 and VEGA 2/0029/11.

References

ASHREA Handbook. 2009. Fundamentals. In: Climatic Design Infor-mation. American Society of Heating, Refrigerating and Air-Condi-tioning Engineers, Inc., Atlanta. (Chapter 14)

Blanc, Ph., Wald, L., 2012. The SG2 algorithm for a fast and accuratecomputation of the position of the sun for multi-decadal time period.Sol. Energy 86 (10), 3072–3083.

Boucher, P., 1983. Azimuth determination by solar observation: newperspectives on an old problem. Surveying and mapping 43 (3), 307–314.

Doerfler, R., 2007. The analemmas of Vitruvius and Ptolemy. Deadreckonings. In: Lost Art in the Mathematical Sciences. <http://www.myreckonings.com/wordpress>.

Daly, J.F., 2008. Sacrobosco, Johannes de (or John of Holywood). In:Complete Dictionary of Scientific Biography. <http://www.encyclope-dia.com/topic/Johannes_de_Sacrobosco.aspx>.

Holywood, J., (Johannes de Sacrobosco), MS 1230, first printed 1472. DeSphaera. Tractatus de Sphaera, Algorismus, de Arte Numerandi.Ferrara.

IESNA Lighting Handbook., 2000. Illuminating Engineering Society ofNorth America, NY.

Iqbal, M., 1983. An Introduction to Solar Radiation. Academic Press,NY.

ISO 19115, 2003. Geographic Information – Metadata. Correction 2006.International Organisation for Standardisation, Geneva.

Kittler, R., Mikler, J., 1986a. Podklady pre urcovanie svetelnej klımy vstavebnych suboroch. In: Slovak Basis for the Determination of LightClimate in Building Sets. Research Report of the Institute ofConstruction and Architecture, SAS Bratislava.

Kittler, R., Mikler, J., 1986b. Zaklady vyuzıvania slnecneho ziarenia. In:Slovak Basis of the Utilization of Solar Radiation. Veda Publisher,Bratislava.

Kittler, R., Darula, S., 2006. Analemma: the ancient sketch of fictitioussun path geometry – sun, time and history of mathematics. Architec-tural Science Review 42 (2), 141–144.

Kittler, R., Darula, S., 2008a. Applying solar geometry to understand thefoundation rituals of´Old Kingdom Egyptian pyramids. ArchitecturalScience Review 51 (4), 407–412.

Kittler, R., Darula, S., 2008b. Historicka dolezitost solarnej geometrie preorientaciu v case, priestore a v architecture. (In Slovak: Historicalimportance of solar geometry for the orientation in time and space andin architecture), Architektura & urbanizmus, 42, 3–4, 159–165.

Kittler, R., Kocifaj, M., Darula, S., 2012. Daylight Science andDaylighting Technology. Springer, NY.

Markus, T.A., Morris, E.N., 1980. Buildings, Climate and Energy. PitmanPubl., Ltd., London.

Martin, Ch.L., Goswami, D.Y., 2005. Solar Energy Pocket Reference.International Solar Energy Society, Freiburg.

Meeus, J., 1998. Astronomical Algoriths. Willmann-Bell Inc., Richmond,USA.

Muneer, T., 1997. Solar Radiation & Daylight Models for the EnergyEfficient Design of Buildings. Architectural Press, Oxford.

Paturi, F.R., 1989 Chronik der Technik. Slovak translation. In: KronikaTechniky. Fortuna Print. Chronik-Verlag, Dortmunt. Bratislava, 1993,p. 89).

Peurbach, G., 1461. Epytoma in Almagesti Ptolemei de sinubus et chrodis.MS, Vienna.

Pierpoint, W., 1982. Recommended practice for the calculation of daylightavailability. US IES Daylight Guide Draft.

Reda, I., Andreas, R., 2004. Solar position algorithm for solar radiationapplications. Solar Energy 76 (5), 577–589.

Regiomontanus, J., (Muller, J. von Konigsberg), 1464, 1533. De Trian-gulis omnimodus, In: de Triangulis planis et sphaericis libri quinque. J.Schoner Verlag, Nurnberg.

Rheticus, G. J., Oth, V., 1596. Opus Palatinum de Triangularis. J.Schoner, Nurnberg.

Rossi, C., 2007. Architecture and Mathematics in Ancient Egypt.Cambridge Univ. Press, UK.

Roy, G., Kittler, R., Darula, S., 2007. An implementation of the Methodof Aperture Meridians for the ISO/CIE Standard General Sky. LightResearch and Technology 39 (3), 253–264.

Santbech, D., Regiomontanus, J., Peurbach, G., 1561. Triangulis planis etsphaericis libri quinque. Compositio tabularum sinus. Problematumastronomicorum et geometricorum sectiones septem. H.Petri & P.Perna, Basel.

Smith, F., Wilson, C.B., 1976. The shading of ground by buildings.Building & Environments 11 (3), 187–195.

Tregenza, P., Sharples, S., 1993. Daylight Algorithms. Report ETSU.University of Sheffield.

Tregenza, P., Wilson, M., 2011. Daylighting, Architecture and LightingDesign. Routledge, London.

Vinaccia, G., 1939. Il corso del sole in urbanistica ed edilizia. Hoepli Ed.,Milano.

Vitruvius, M.P., MS 13 BC, printed 1487. De architectura libri decem.Rome. English translation by Morgan, M.H.. Vitruvius. In: The TenBooks on Architecture. Cambridge, Mass. Harvard Univ. Press, 1914and Dover. NY, 1960, also by Granger, F. In: Vitruvius of Architec-ture. Heinemann, London, 1970.