Revisiting Spherical Trigonometry with Orthogonal Projectors

Para ulama sepakat bahwa menghadap kiblat menjadi syarat sahnya salat, tetapi mereka tidak sepakat bahwa orang yang salat wajib menghadap ke bangunan Ka’bah atau ke arah Ka’bah. Untuk kesempurnaan ibadah, diperlukan usaha mencari arah kiblat yang tepat. Teori yang biasa digunakan untuk menghitung arah kiblat adalah teori ilmu ukur segitiga bola (spherical trigonometry). Sekarang, teori geodesi dengan rumus vincenty menjadi teori yang lebih akurat digunakan untuk menghitung arah kiblat daripada teori spherical trigonometry. Tulisan ini berusaha menggambarkan penggunaan teori geodesi dalam perhitungan arah kiblat untuk kota Banjarnegara, Purbalingga, Banyumas, Cilacap dan Kebumen. Teori geodesi menghasilkan arah kiblat untuk kota Banjarnegara 294°40’01.042”, Purbalingga 294°44’11.376”, Banyumas 294°45’25,582”, Cilacap 294°54’21.568”, dan Kebumen 294°44’16.752”. Jika arah kiblat tersebut dibandingkan dengan arah kiblat yang selama ini dipedomani, maka terdapat selisih rata-rata 0°7’32.74”. Dengan adanya hasil perhitungan yang mempunyai tingkat akurasi lebih tinggi, tentu akan menambah keyakinan dalam beribadah, walaupun arah kiblat yang selama ini menjadi pedoman juga masih dalam lingkup menghadap kiblat.

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Mathematical models of Gauss – Kruger projection for calculation of meridians conversion on the plane and scale of the image

Geodesy and Cartography ◽

10.22389/0016-7126-2018-934-4-2-7 ◽

2018 ◽

Vol 934 (4) ◽

pp. 2-7

Author(s):

P.A. Medvedev ◽

M.V. Novgorodskaya

Keyword(s):

Mathematical Models ◽

Analytical Methods ◽

Theoretical Studies ◽

Spherical Trigonometry ◽

Cylindrical Projection ◽

Rectangular Coordinates ◽

In Series ◽

Observational Errors ◽

The Difference

This work contains continued research carried out on improving mathematical models of the Gauss-Krueger projection in accordance with the parameters of any ellipsoid with the removal of points from the axial meridian to l ≤ 6° . In terms of formulae earlier derived by the authors with improved convergence for the calculation of planar rectangular coordinates by geodesic coordinates, the algorithms for determining the convergence of meridians on the plane and the scale of the image are obtained. The improvement of the formulae represented in the form of series in powers of the difference in longitudes was accomplished by separating spherical terms in series and then replacing their approximate sums by exact expressions using the formulae of spherical trigonometry. As in previous works published in this journal [7, 8], determining the sums of the spherical terms was carried out according to the laws of the transverse-cylindrical projection of the sphere on the plane. Theoretical studies are given and formulae are proposed for estimating the observational errors in the results of the derived algorithms. The maximum of observational errors of convergence of meridians and scale, proceeding from the specified accuracy of the determined quantities was established through analytical methods.

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