3016974-skmm-08-qibla

Seminar Kebangsaan Matematik & Masyarakat 2008

DETERMINATION OF QIBLA DIRECTION USING MODERN APPROACH

1Shahrul Nizam Ishak And 2Jamaludin Md Ali 1 & 2 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang.

E-mail: 1 [email protected] , 2 [email protected]

Abstract

The problem of determining the direction of qibla is a problem of mathematical geography. The objective of this study is to determine and compare the method to be used for determining qibla direction at Northeast Penang. Thus, this paper describes about the mathematical calculation of qibla by using modern approach. The mathematical methods used were Spherical Trigonometry Method (STM) and Vector Calculus Method (VCM). The discussion is confined to the scientific aspects of the subject and the religion rulings are analyzed only for an underlying scientific assumption and also for knowledge. Keywords : Qibla , Qibla Direction , Mathematical Calculation.

1 Introduction In the Holy Quran (chapter II: verse 144), Muslims are enjoined to face the sacred precincts in Makkah during their prayers. The Kaabah was adapted by the Prophet Muhammad as a physical focus of the new Muslim community and the direction of prayer (qibla) was to serve as the sacred direction in Islam until the present day. Islamic tradition prescribes that certain acts such as burial ceremony, recitation of the Quran, announcing the praying time (azan) and the ritual slaughter of animals for food have to be performed in the qibla direction whereas expectoration and bodily functions should be performed in the perpendicular direction.

However, praying (solah) five times a day as stated in the Holy Quran has been compulsory upon all mature Muslim as an individual obligatory (fardhu ‘ain). Thus, in order to perform the prayer, basically Muslims have to learn how to perform it by learning the law of prayer (fiqh solah) and also understand the philosophy behind the prayer itself. Moreover, the solah must be performed with sincere reverence and humility (khushu‘), otherwise it is considered invalid. Solah is performed by facing to the direction of qibla (towards the Kaabah in Makkah) until the day of the judgment.

According to Halim et al. [1] stated that all of the Four-Imam in Islam community such as Imam Maliki, Imam Hanafi, Imam Hanbali and Imam Syafie had agreed that whoever wants to perform the prayer must direct their face and chest in the direction of the Sacred Mosque (Kaabah) in Makkah. Mughniyah [2] stated that according to Imam as-Syafie has briefly explained that it is compulsory for every Muslim to face towards Kaabah while performing their prayer regardless of their distance from Kaabah. If a person is able to determine the accurate qibla direction by himself then he or she is expected to face towards its, otherwise it is acceptable to determine the qibla direction based on his or her ability.

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Hence, base on concerning to the problem face by the Muslim, an action has to be taken in order to provide right information to the Islamic community. So, to determine accurate qibla is a Muslim responsibility. Thus, in modern eras, the determination of qibla has changed from traditional approach to modern approach. Thus, many of the Muslim scholars have determined ways and methods regarding the qibla direction. Therefore, this paper will focus on the mathematics calculation in determining the qibla direction, which is referring to the location of the cities that depends on its latitude and longitude.

2 Methodology In view of the present situation, it is appropriate to briefly outline the methodology involved in qibla determination. The reasons because not many people understand the basic formulation and occasionally confusion arise. The determination of the direction of qibla involves solution in terms of latitude and longitude parameters. Therefore, for all of the examples given were based on the geographical coordinate that were adapted from Halim et al. [1] where Kaabah, m : (21 25 15.6 , 39 49 29.1 )North East′ ′′ ′ ′′° ° and the geographical coordinate place or point of interest that have been chosen by the writers which is situated at Northeast District of Penang ; Acheh Mosque, p : (5 24 39 , 100 18 40 )North East′ ′′ ′ ′′° ° .

[Calculation done on other location should correspond to the identified place but the coordinate for Makkah (Kaabah) is fixed.] Problem. The problem involves the solution using the spherical triangular formed by the point or place interest, Makkah (Kaabah) and the North Pole. We are interested in only one point of the two angles, which is the direction of Kaabah. Thus, we need to solve for the angle θ . Note that the writers will focus on the Formula given below because it is more accurate and easy to understand. Method I. Spherical Trigonometry Method (STM)

Generally, to perform calculation with spherical triangles it is necessary to use the formulae of spherical trigonometry. There are three different types special trigonometrical formula for usage in spherical astronomy which can be found in Ilyas [3]. This formula was derived from four-part formula. Formula. 1 sintan

sin cot cos cosC

a b a Cθ − ⎡ ⎤= ⎢ ⎥−⎣ ⎦

where θ = the qibla angle from North ( anticlockwise )

C = the different of longitude between p and m a = 90o – the latitude of the point interest

b = 90o – the latitude of the Kaabah ( ),p pφ λ = the geographical coordinate of the point interest, p

( ),m mφ λ = the geographical coordinate of the Kaabah, m

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Proof. Given the four-part formula cos cos sin cot sin cota C a b C B= −Thus,

sin cot sin cot cos cosC B a b a C= − sin cot cos coscot

sina b a CB

C−

=

sintasi

Ba b a

=nn cot cos cos

CC−

1 sintan

sin cot cos cosCB

a b a C− ⎡ ⎤∴ = ⎢ ⎥−⎣ ⎦

Notation. B is the angle of qibla and denoted as θ in the above formula.

C

North

θm

pQibla

a b

E

quator

( ),p pφ λ ( ),m m φ λ

Figure 1. Qibla Determination using STM Example. Let ( ),p pφ λ = ( 5.4108 , 100.3111 ) ° °

( ),m mφ λ = ( , ) 21.4210° 39.8247°

Then, C = ( p mλ λ− ) = 60.4864°

a = 90o – pφ = 84.5892 ° b = 90o – mφ = 68.5790°

Thus, by substituting into the formula, we will get :

1 sin 60.4864tan(sin84.5892 cot 68.5790) (cos 68.5790 cos 60.4864)

θ − ⎛ ⎞= ⎜ ⎟× − ×⎝ ⎠

= 2568° ′ 29′′

∴ Azimuth Qibla = 360 θ°− = 34291° ′ 31′′

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Method II. Vector Calculus Method (VCM) Abdali [4] said that if a plane cuts a sphere, the curve of intersection on the sphere is always a circle. If the plane cutting the sphere also passes through the sphere’s center, then the sphere is cut into two equal parts (hemispheres), and the circle of intersection is called a great circle. A great circle has the same center and radius as the sphere itself, and is the largest circle that can be drawn on the spherical surface. These concepts are illustrated in Figure 2.

Figure 2 ABCD is a Great Circle

Assumption. Consider the following assumption of this model: the earth is sphere in shape but in reality earth is not a true sphere, actually geoids. According to Durani [5] the earth can be considered as an APPROXIMATE SPHERE. The variation between the actual shape of the earth (oblate ellipsoid or spheroid) and the approximating sphere is not more than few degrees in azimuth and is not very significant for the purpose of determination of the direction of Kaabah. This VCM was developed to improve on the method for the determination of qibla direction. Firstly, we have to determine the coordinates p and m (refer Figure 4). Thus, this problem can be most easily solved by using spherical coordinates on the earth. If we assume that the earth to be a sphere of radius 6378.137kmρ = , then each point on the earth has spherical coordinate of the form ( 6378.137, θ , φ ) where

, .latitude longitudeφ θ= = The transformation from spherical coordinates to normal rectangular coordinates where R is the radius of the earth. Let denote the sphere of radius R centered at the origin in RRS 3. In rectangular coordinates ( ){ }2 2 2 2, , |R x y z x y z RS = + + = and in spherical coordinates ( ){ }cos( )sin( ), sin( )sin( ), cos( ) | 0 2 ; 0R R R RS θ φ θ φ φ θ π φ= ≤ π≤ ≤ ≤

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X

Y

Z

0 z

θ

φP ( ρ , θ , φ )

ρ

Q

Figure 3 Spherical Coordinate

Theorem 1. [Spherical Phytagorean Theorem] For a right triangle ABCΔ on a sphere of radius R with a right angle at vertex C and sides of length a, b and c, then

cos cos cosc a bR R R

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

B

A C

c

γ

b

a

β α

Figure 4 Spherical Right Triangle

Proof. Rotate the sphere so that the point A has coordinates (R,0,0) and the point C

lies in the xy-plane. This will make β and 2π the angles determining the point B.

Here β is the central angle determined by side AC, α is the angle determined by AC and γ is the angle determined by AB. This gives us the following spherical coordinates for the vertices of the triangle:

( )( )( )

,0,0cos( )cos( ), cos( )sin( ), sin( )cos( ),0, sin( )

A RB R R RC R R

α β α β

β β

α

=

=

=

Using what we know about the dot product in R3 w can find the cosine of the angle , eγ . According to Anton et al. [6] the dot product u v• is defined by :

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Y

p m

O X

v u

Z

Therefore, based on the Figure 4, it shows that the vector u is a unit vector that directs to the North and also tangent for the point of interest, p. Vector v is a vector that has direction to the Kaabah, m.

Definition. If and u v are nonzero vectors in 2-space or 3-space,

and if θ is the angle between them, then

cosu v u v θ• =

Figure 4 Qibla Determination using VCM Coordinates point of interest, p :

Xp = Rcos ( lalitude point of interest )

Yp = 0

Zp = Rsin ( lalitude point of interest )

Coordinates of Kaabah, m :

Xm = Rcos ( lalitude of Kaabah ) cos ( longitude difference between

point of interest and Kaabah )

Ym = Rcos ( lalitude of Kaabah ) sin ( longitude difference between

point of interest and Kaabah )

Zm = Rsin ( lalitude of Kaabah )

Hence, we determined the vector u as

u = ,0 ,p pZ X−

Then, the normal vector has to be determined. The normal vector is obtained N Nby cross product, OM and it is orthogonalOP× to the great circle that connects between p and m.

N = , ,p m p m p m p mZ Y Z X X Z X Y− −

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= , ,a b c

Thus, vector is obtained from cross product OPv N× and it is also tangent to the point of interest. = v , ,p p p pZ b Z a X c X b− − After that, apply the formula for dot product, by substituting all of the values obtained

and solve for theta,θ .

cos u vu v

θ⎛ ⎞•

= ⎜ ⎟⎝ ⎠

Finally, we will get the qibla direction by subtracting the theta with 3600. ∴ Azimuth Qibla = 360 θ°−

Example. Let Xp = cos ( 5.4108 ) ; X° m = cos ( ) cos ( ) 21.4210° 60.4864°

Yp = 0 ; Ym = cos ( ) sin ( 60.4864 ) 21.4210° °

Zp = sin ( 5.4108 ) ; Z° m = sin ( ) 21.4210°

Hence, after calculating the vector u and the vectorv , apply the formula for dot

product :

1cosθ = 0.0943,0,0.9955 0.0302, 0.8101, 0.3189

(1)(0.8711)− • − −

1θ = 111.5738°

Thus , θ = 1(180 )θ°−

∴ Azimuth Qibla = 360 θ°− = 291° 34′ 31′′

3 Numerical Evaluation There are thirty-two samples of data that has been obtained from Halim et al. [1] and it is not suitable to be shown here. The data taken indicates the geographical coordinate of all the mosques around Northeast Penang in terms of latitude and longitude. Thus, based on the data adapted, the values of the qibla direction are generated using ‘VCalQLator’ system which is constructed by the authors. For comparison, from the result obtained, it was analyzed by using the measurement of variability. Hence, Table 1 illustrated the results as follows:

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Method Mean Variance Std. Deviation

STM 0.0137 0.0004 0.0193

VCM 0.0136 0.0003 0.0187

Table 1 Result Analysis of Mean deviation, Variance and Standard Deviation of data According to Table 1, we judge based on the smallest standard deviation between both methods. Hence, we found that the standard deviation for both methods is slightly same but the standard deviation for VCM is 0.0187 which is smaller than STM that is 0.0193. Therefore, comparing the two methods, the numerical results achieved using VCM is found to be better and acceptable for the purpose of determination of the qibla direction from any location. 4 Conclusion In this paper we have presented two methods. STM is the previous method used and VCM is the suggested method for current and future use for determining the qibla direction. Based on the comparison made between STM and VCM, we can conclude that the suggested method can be used to determine the azimuth of qibla from any location in the world. Therefore, from this work we hope that it will give some benefit especially to the Muslim communities regarding their routine worship. To put into a nutshell, the authors recommended that further investigation can be extended using this method of VCM with other point of view. 5 Acknowledgements The authors would like to extend our gratitude and greatly indebted to the school of Mathematical Sciences, Universiti Sains Malaysia for supporting this work under its Short-term Research Grant Account No. 304/PMATHS/637057. References [1] Halim A., Saad G., Aziz A., Zainal B., Jaafar H. & Sadali H.M. (2006), Ar-

Risalah Fi Tayin al-Qiblah, Pulau Pinang : Jabatan Mufti Negeri Pulau Pinang. [2] Mughniyah M. J. (1996), Fiqh Lima Mazhab, Afif Muhammad (Translations),

Jakarta: Penerbit Lentera. [3] Ilyas M. (1984), A Modern Guide to Astronomical Calculation of Islamic

Calendar, Times & Qibla, Kuala Lumpur, Berita Publishing. [4] Abdali S.K., The Correct Qibla, http://www.patriot.net/users/abdali/ftp/qibla.pdf. [5] Durani N.M. (1994), Direction For Kabah (Mathematical) – From Anywhere,

Islamic Society of North America(ISNA), New York. [6] Anton H., Biven I. & Davis S. (2002), Calculus, John Wiley & Sons, 7th Edition.

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http://www.patriot.net/users/abdali/ftp/qibla.pdf

3016974-skmm-08-qibla

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