Using Google Maps and Spherical Quadrilateral Approach Method for Land Area Measurement

Adi Setiawan Eko Sediyono Department of Mathematics Master of Information System Programme Faculty of Science and Mathematics Universitas Kristen Satya Wacana Universitas Kristen Satya Wacana Jl. P. Diponegoro 52-60 Salatiga 50711, Indonesia Jl. P. Diponegoro 52-60 Salatiga 50711, Indonesia Email : [email protected]

Abstract— In this paper, we present how to determine the Sjoberg explained how numerical integration is used in the land area measurement by using spherical quadrilateral determination of the area in fields, spheres and ellipsoid [5]. approach method. The proposed method is compared to Likewise, Pedzich and Kuzma presents a method of calculating quadrilateral approach method with the UTM (Universal the area of geodesic polygons used in the Polish region [6]. Transverse Mercator) coordinate system. The spherical quadrilateral approach method delivers in the calculation of the The polygon method and square method are used in the area of DIY and Central Java provinces 3328.2355 km2 and determination of the plantation area of the subdivision of 34978.488 km2, respectively, while the quadrilateral approach Pendowo TBM 2015 PTPN IX Getas, Semarang Regency by method yielded 3613.1760 km2 and 35207.7300 km2. The results considering the land as un contoured land. It is described in obtained by spherical quadrilateral approach method are bigger Alivah et al. [7]. Devi et al. described how the area of land that than the reference areas but less than 10 %. is Semarang regency can also be done using spherical triangle approach method [8]. In addition, Alivah et al. also explained Keywords— Google Maps, Measurement of Land Area, how to determine the area of contoured land with cone and Quadrilateral Approach Method, Spherical Quadrilateral square cone method and it is applied to Gili Trawangan island Approach Method. and plantation land of Afdeling Tembir sub Sembir subdistrict of Pendowo TBM 2015 PTPN IX Getas, Semarang Regency I. INTRODUCTION [9]. Land area measurement is very important in agriculture, plantation and housing. The border of the land are often In this paper, it will be presented how to determine the area irregular and contoured. Determining the coordinates of land of region by using spherical quadrilateral approach method boundaries by using GPS (Global Position System) requires without considering the contour of the region. The borders of field visits to the site of land will be very costly. Today, land can be obtained by using Google Maps. Google Maps provided by Google Inc. is widely used in GIS In this section, it is explained the distance between 2 points (Geographic Information System) e.g. to determine the latitude on the earth that can be considered as an ellipsoid by using and longitude coordinates of location on the surface of the Vincenty formula [10] and [11]. Suppose we have 2 points on earth such that the coordinate of boundary of a region or the earth with coordinate ( ,L ) and ( ,L ) where  ,  are position on the surface of the earth can be determined. More 1 1 2 2 1 2 latitude and L1 and L2 are longitude. In this case, the earth is information of Google Earth can be found in [1] and [2]. considered as an ellipsoid such that a = 6378137.0 meters i.e. Furthermore, the coordinates of the border of the region can be length of semi-major axis of the ellipsoid (radius at equator), used in land area measurement. b = 6356732.314245 meters i.e. length of semi-minor axis of II. LITERATURE REVIEW the ellipsoid (radius at the poles) and f = 1/298.257223563 i.e. flattening of the ellipsoid. Furthermore, it can be used the First, Various methods of land area measurement can be symbols as follows : L = L2 – L1 i.e. the difference in longitude used if the coordinates of the border of the region are known. of two points, Setiawan et al. presented how the circle approach method can U arctan(1 f )tan( ) be used to determine the area of Gili Air island in Lombok 1 1 and West Nusa Tenggara [3]. Devi et al. also explained how the circle approach method is used to determine the area of flat U 2 arctan(1 f ) tan(2 ) region. It was used to determine the football field area at i.e., reduced latitude (latitude on the auxiliary sphere), 1, 2 Universitas Kristen Satya Wacana Salatiga, Rawa Pening Lake i.e. longitude of the points on the auxiliary sphere, 1, 2 i.e. area in Semarang regency and the are of Salatiga City. These forward azimuths at the points,  i.e. azimuth at the equator, s three areas are located in Central Java, Indonesia. The concern i.e. ellipsoid distance between the two points. Given the regions can be considered as the surface on the earth coordinates of the two points (1,L1) and (2,L2), the invers considered as an ellipsoid [4]. problem proposed by Vincenty want to find the azimuths and the ellipsoidal distance s. It is calculated U1, U2 and L and set or initial value of -L, then iteratively evaluate the following d  (230199.56313373.34)2 (9205585.54 9239273.69)2 equations until converges i.e. 89737.22 meter. That means Vincenty distance is less sin  (cosU sin )2 (cosU sinU sinU cosU cos )2 , 2 1 2 1 2 22.587 meter. cos   sin U sin U  cos U cos U cos  , 1 2 1 2 III. RESEARCH METHODS In this research, latitude and longitude coordinates of  sin     arctan  , border of the region are used to determine the land area  cos   measumerent. The region of concern are the DIY and Central cos U cos U sin  Java provinces. A total of 29 points on the top border and 29 sin   1 2 , points on the bottom boder of DIY province are used to sin  determine the area of the region. For the province of Central cos 2 1 sin 2  , Java, it is used 45 points on the top border and 45 points on the bottom border. The area is determined by using two spherical 2 sin U 1 sin U 2 cos( 2  m )  cos   2 , triangles approach by considering the earth as an ellipsoid. As a cos  comparison, the quadrilateral method by using UTM f coordinate system is also used in determining the area of C  cos 2  [ 4  f (4  3 cos 2  ) ], 16 concern region using the same points. The results are compared 2 to the area based on the information of Central Bureau of D Csin cos(2 )C cos (12cos (2 ))  m m  Statistics (BPS - Badan Pusat Statistik) i.e. 3185.80 km2 for the

  L  (1  C ) f sin  D . DIY province and 32548 km2 for the Central Java province. When has converged to the desired degrees of accuracy, evaluate the following : IV. RESULTS AND DISCUSSION a 2  b 2 The proposed method is applied to determine the area of u 2  cos 2  , b 2 Central Java province and can be described as follows. Table I 2 presents 5 coordinates of top border and 5 coordinates of u A 1 4096  u 2 [  768  u 2 (320  175 u 2 )], bottom border respectively A, B, C, D, E and A, B, C, D, E. 16384 Based on Table I, the given coordinates can be conversed in 1 E  B [ cos (2 ) (3  4 sin 2  ) (3  4 cos 2 (2 ))], UTM coordinates as presented in Table II. To obtain the UTM 6 m m coordinate system from longitude and latitude coordinate system can be used web based program package. 1 2 F  B [ cos  (  1  2 cos ( 2 m ))  E ], 4

  B sin  ( cos( 2 m )  F ), s b A(  ),  cos U sin    2   1  arctan  ,  cos U 1 sin U 2  sin U 1 cos U 2 cos    cos U sin    1   2  arctan  .   sin U 1 cos U 2  cos U 1 sin U 2 cos   Later, Vincenty formula is modified by using formula : k 2 1  1 4  3 2  A  , B  k1 1  k1  1  k 1  8  where 1  u 2  1 k 1  . Fig 1. Map of Central Java. 1  u 2  1

To illustrate the use of the Vincenty formula, suppose point A has coordinates (-7,180433, 108,557056) and point B has The area of the spherical quadrilateral region is presented in coordinates (-6,879217, 109,310992). After 5 iterations,  will Fig. 1 with the vertices B, C, B and C that are located on the converge such that Vincenty distance between A and B is surface of the earth by considering the earth as an ellipsoid. 89714.636 meter. In UTM coordinate system, point A and B Using the Vincenty formula, the distance c = BC, d = CB, has coordinate (230199.56, 9205585.54) and (313373.34, 9239273.69), respectively. By using UTM coordinate, the e = BC, g = CC are spherical quadrilateral sides where distance between A and B can be found by using Euclide c = 75183.014934, d = 102616.794342, e = 81188.155580, distance : g = 130577.299289 (in meters). Furthermore, it also can be obtained side h = BC = 130577.299289 meters. 2 2 d  ( x A  x B )  ( y A  y B )  d  c h   1 1 1  Table I. Latitude and Longitude Coordinate of Border of Central Java.  1  arc cos   0.90212495 ,  c2 h2  No. Location Latitude Longitude  h1  c1d1  1 A -7.180433 108.557056   arc cos  1.62709448.  c d  2 B -6.879217 109.310992  2 2  3 C -6.929661 109.989398 In a similar way, 2, , 2 in spherical triangle BCB can also 4 D -6.480940 110.698016 be determined by cosines rule in spherical triangle i.e. 5 E -6.639848 111.560239 cos h cos g cose  sin g sin e cos ,

6 A -7.325493 108.558280 cos e  cos g cos h  sin g sin h cos  2 , 7 B -7.689379 109.256835 cos g  cos h cos e  sin h sin e cos  2 . 8 C -7.857403 109.973341 It can be found that 2 = 0.74554123, 2 = 0.6602806 and 9 D -7.799129 110.711686  = 1.73585942 such that 10 E -7.183703 111.554887  = 1 + 2 = 0.90212495 + 0.6602806 = 1.5624056 Table II. UTM Coordinate of Border of Central Java. and

 =  +  = 0.61246812 + 0.74554123 No. Location Easting Northing 1 2 = 1.3580094. 1 A 230199.56 9205585.54 Thus, the sum of the angles in spherical quadrilateral is more 2 B 313373.34 9239273.69 than 2 i.e. 3 C 388355.68 9233908.05  +  +  +  4 D 466609.91 9283620.87 = 1.5624056 + 1.6270945 + 1.3580094 + 1.7358942 5 E 561926.17 9266029.38 = 6.2834037. 6 A 230421.23 9189536.01 Without rounding, it is found 7 B 307733.25 9149650.10  +  +  +  - 2 8 C 386818.18 9131327.78 equals to 1.18347929 × 10-4. Finally, area of the spherical 9 D 468212.54 9137897.79 quadrilateral region can be BCCB can be found as 10 E 561264.49 9205906.23 Area  ab (         2 ) = 6378137 ( 6356752.314245 ) 1.18347929 × 10-4 2 2 The spherical quadrilateral region of BCCB can be i.e. 7439027858 m or 7439.027858 km . Similarly, it can be determined by the formula found the area of spherical quadrilateral ABBA, CDDC and Area  ab(     2 ) DEED equal to 4241,646306, 9906,37276 dan 9702,569173 2 where  =  +  and  =  +  . In this case, (in km ), respectively such that the area of Central Java 1 2 1 2 approximates to 31290.080613 km2.  =  CBC,  =  CBB,  =  BCC,  =  BBC, 1 2 In UTM coordinate system, the distance between two  =  BBC and  =  BCB. The formula is known as 1 2 points can be found by using Euclide distance such that the Girard Theorem. The angles 1, , 1 located within the area of triangle can be found by using Heron formula. The spherical triangle BCC can be determined using the cosine area of triangles ABB, AAB, BCC, BBC, CDD, CCD, rule of the spherical triangle DEE dan DDE are found 3632.1643, 615.99, 3849.9792, cos h cos ccos d  sin csin d cos , 3595.607, 5741.5593, 4169.677, 2871.1769 dan 6834.405 (in km2), respectively, such that the area of Central Java equal to cos d cosccosh sincsinh cos1 , 31310.560 km2. cos c coshcosd  sin hsin d cos 1 . If the spherical rectangular method is used and by using 58 2 Practically, 1, , 1 (in radians) can be determined by border points , the total area of DIY is 3328.2355 km so that more 4.47% of the reference area. When used UTM c1  cos ( c ( /180 ) /( 2r / 360 ))  0.99993037 , d  cos ( d ( /180 ) /( 2r / 360 ))  0.99987029 , coordinate system obtained the calculation of the area of the 1 province of Yogyakarta equals to 3613.1760 km2 or 13.42% h1  cos ( h ( /180 ) /( 2r / 360 ))  0.99978997 , more than the reference area. If the spherical quadrilateral

c2  sin ( c ( /180 ) /( 2r / 360 ))  0.011800545 , method and by using 90 border points is applied in Central Java province then the area is 34978.4880 km2 or 7.47% more d 2  sin( d ( /180 ) /( 2r / 360 ))  0.016106162 , than the reference area. By using UTM coordinate system, the h  sin ( h ( /180 ) /(2r / 360 ))  0.020494139 . 2 area will be 35207.3 km2 or 8,17% more than the reference Furthermore, it can be found area.  c  d h  The spherical quadrilateral approach method assumes that   arc cos  1 1 1   0.1246812 , 1   the four points on the boundaries of the region can be viewed  d 2 h2  as spherical quadrilateral on a surface of an ellipsoid. An area of region can be viewed as the sum of spherical quadrilaterals REFERENCES such that it can be used to calculate the area (compare to Riemman Integral [12]). Differences obtained by reference by [1] Huff, T. ,2014, Google Earth : Low-Investment GIS for Extension reference, probably due to poorly sampled results when Proffesionals, Journal of Extension, Vol. 52 No. 4. clicking Google Maps to obtain longitude coordinates and [2] Google , 2012, Google Earth Features latitude coordinates. Likewise possibly due to changes in the http://www.google.com/earth/media/features.html results caused by the rotation of satellites to the earth and the [3] Setiawan, A., Sediyono, E. & Alivah, E. 2016. The Use of Google Maps rotation of the earth and the satellite around the sun. The use and Circle Approach Method in Land Area Measurement. have been presented in International Conference On Theoritical and Applied of UTM coordinate system makes calculating the distance Statistics by ITS Surabaya, 19-20 October 2016 . Surabaya: Institut between 2 points easier by simply using the Euclid distance Teknologi Sepuluh Nopember. compared to the Vincenty distance. However, the spherical [4] Devi, Setiawan, A. & Sediyono, E. 2016. Penentuan luas lahan quadrangle approach method using the Vincenty distance is menggunakan metode pendekatan segitiga sferik (Teorema Girard) relatively closer to the reference. Other methods also used in dengan bantuan Google Maps. Prosiding Seminar Nasional Pendidikan Matematika Ahmad Dahlan 31 Desember 2016, Program Studi measurement of irregular land area are numerical integration Pendidikan Matematika Universitas Ahmad Dahlan Yogyakarta. [5] and geodesic polygons [6]. [5] Sjoberg, L. E. (2006) Determination of Area on the plane, sphere and ellipsoid. Survey Review. 38 (301):583-593. [6] Pedzich, P. & M. Kuzma (2013) Application of methods for area V. CONCLUSION calculation of geodesic polygons on Polish administrative units. In this paper, we have presented how to determine the area [7] Alivah, E.N., Setiawan, A. & Sediyono, E. (2016). Penentuan luas lahan of DIY and Central Java provinces using Google Maps and dengan bantuan Google Earth. Prosiding, Seminar Nasional 3rd CGISE dan FIT ISI yang diselenggarakan oleh FTek UGM, tanggal 27 Oktober spherical quadrilateral approach method. The proposed method 2016. Yogyakarta: Universitas Gadjah Mada. delivers the measurement area of DIY and Central Java [8] Devi, Setiawan, A. & Sediyono, E. (2017). Penentuan luas lahan datar provinces, respectively 3328.2355 km2 and 34978.4880 km2 dengan metode pendekatan lingkaran berbasis Google Earth/Google while the quadrilateral approach method yielded 3613.1760 Maps. Prosiding Seminar Nasional Matematika dan Pendidikan 2 2 Matematika Surakarta 16 November 2016, Program Magister km and 35207.7300 km . The results obtained by spherical Pendidikan Matematika Universitas Sebelas Maret. Hal 910-920. ISBN : quadrilateral approach method are bigger than the reference 978-602-6122-20-9. areas but less than 10%. In the future research, it can be done in [9] Alivah, E.N., Setiawan, A. & Sediyono, E. (2017). Penerapan Metode other regions by using both methods and by taking into account Kerucut Terpancung dan Bujur Sangkar dalam Perhitungan Luas Lahan the contours of the region. Berkontur menggunakan Bantuan Media Informasi Google Earth/Google Maps. Prosiding Seminar Nasional Matematika dan Pendidikan Matematika Surakarta 16 November 2016, Program Magister Pendidikan Matematika Universitas Sebelas Maret. Hal 910- 920. ISBN : 978-602-6122-20-9. ACKNOWLEDGMENT [10] Vincenty, T., (1975), Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations, Survey Review, 23 (176), The authors would like to thank to Directorate General of 88-93. Higher Education, Indonesia for research funding with scheme [11] Wikipedia : https://en.wikipedia.org/wiki/Vincenty’s formulae . of Hibah Kompetensi fiscal year 2017. [12] Anton, H., Bivens, I. , Davis, S. (2012) Calculus 10th edition, John Wiley & Sons River Street.