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A General Formula for Calculating Meridian Arc Length and Its Application to Coordinate Conversion in the Gauss-Krüger Projection 1

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A General Formula for Calculating Meridian Arc Length and Its Application to Coordinate Conversion in the Gauss-Krüger Projection 1 A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection 1 A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection Kazushige KAWASE Abstract The meridian arc length from the equator to arbitrary latitude is of utmost importance in map projection, particularly in the Gauss-Krüger projection. In Japan, the previously used formula for the meridian arc length was a power series with respect to the first eccentricity squared of the earth ellipsoid, despite the fact that a more concise expansion using the third flattening of the earth ellipsoid has been derived. One of the reasons that this more concise formula has rarely been recognized in Japan is that its derivation is difficult to understand. This paper shows an explicit derivation of a general formula in the form of a power series with respect to the third flattening of the earth ellipsoid. Since the derived formula is suitable for implementation as a computer program, it has been applied to the calculation of coordinate conversion in the Gauss-Krüger projection for trial. 1. Introduction cannot be expressed explicitly using a combination of As is well known in geodesy, the meridian arc elementary functions. As a rule, we evaluate this elliptic length S on the earth ellipsoid from the equator to integral as an approximate expression by first expanding the geographic latitude is given by the integrand in a binomial series with respect to e2 , regarding e as a small quantity, then readjusting with a a1 e2 trigonometric function, and finally carrying out termwise S d , (1) 2 2 3 2 0 1 e sin integration. The following example is an approximation of where a and e are the semi-major axis and the first formula (1) obtained by truncating the expansion at order eccentricity of the earth ellipsoid, respectively. e10 : Since formula (1) includes an elliptic integral, it 2 sin 2 sin 4 sin 6 sin8 sin10 S a1 e C1 C2 C3 C4 C5 C6 , 2 4 6 8 10 3 2 45 4 175 6 11025 8 43659 10 C1 1 e e e e e , 4 64 256 16384 65536 (2) 3 15 525 2205 72765 15 105 2205 10395 C e2 e4 e6 e8 e10 , C e4 e6 e8 e10 , 2 4 16 512 2048 65536 3 64 256 4096 16384 35 315 31185 315 3465 693 C e6 e8 e10 , C e8 e10 , C e10. 4 512 2048 131072 5 16384 65536 6 131072 Since the above expression has fractional Despite this fact, formula (2) has long been used coefficients too complicated for most readers, except for generally in Japan to calculate the meridian arc length of those specialized in the given field, to memorize at a the earth ellipsoid. glance, users face a perpetual risk of making mistakes owing to errors in typing the fractional coefficients. 2 Bulletin of the Geospatial Information Authority of Japan, Vol.59 December, 2011 2. Other expressions expressed in formula (3) as 2.1 Bessel’s formula On the other hand, an alternative derivation using a1 n2 1 n S d , (4) 0 2 3 2 yet another quantity n , the third flattening of the earth 1 2ncos2 n ellipsoid defined as expanded the denominator of the integrand in a series 1 1 e2 e2 with respect to n , and then carried out termwise n , (3) 1 1 e2 4 integration. The following formula is a summary of the has been presented by Bessel (Bessel, 1837). expansion result that appeared in Bessel’s original paper: Bessel rewrote formula (1) using the relation S a 1 n 2 1 n 9 2 225 4 3 15 3 175 5 15 2 7 4 35 3 27 5 1 n n n n n sin 2 n n sin 4 n n sin 6 . 4 64 2 8 64 16 4 48 16 (5) 2 The main advantages of this formula over an equivalent value 1 n2 1 n . Helmert then formula (2) are its good convergence, since n is about multiplied the terms in the curly brackets in formula (5) 2 a quarter of the value of e2 , and the reduced number of by 1 n2 , the numerator of the substituted value, terms appearing in the formula despite its almost equal aiming to extract the factor 1 1 n and simplify the precision. fractional coefficients appearing in formula (5) (Helmert, 1880). 2.2 Helmert’s formula More than thirty years after that, Krüger About forty years after the publication of Bessel’s summarized Helmert’s result in his paper published in result, Helmert derived a simpler expression by replacing 1912 (Krüger, 1912) as the following formula: the 1 n 2 1 n factor appearing in formula (5) with a n2 n4 3 n3 15 n4 35 315 S 1 n sin 2 n2 sin 4 n3 sin 6 n4 sin8 . (6) 1 n 4 64 2 8 16 4 48 512 As formula (6) shows, the result derived by (1880) seems to be not only difficult to understand but Helmert has a simpler and more concise expression than also hard to generalize. does formula (2), which was truncated at order e10 , and has almost the same or greater precision despite its 3. Derivation of general formula truncation to no more than order n4 (corresponds to In order to generalize formula (6), we choose e8 ), as reported by Tobita et al. (Tobita et al., 2009). another approach that does not start from formula (4) Nevertheless, the derivation process used by Helmert directly. First, in accordance with a well-known result A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection 3 regarding elliptic integrals and elliptic functions (e.g., 2 2 2 e sin 2 Byrd et al., 1954), we find that the integral in formula (1) S a 1 e sin d . (7) 0 2 1 e2 sin 2 can be regarded as a special case of incomplete elliptic integrals of the third kind. Thus, it can be divided into an incomplete elliptic integral of the second kind and a term Bearing the relation expressed in formula (3) in of elementary functions as* mind and letting 2 , it is not hard to see that we can rewrite formula (7) as a 1 2 2nsin 2 S 1 2ncos n2 d . (8) 1 n 2 0 2 1 2ncos2 n At a glance, the result of formula (8) appears to term in the parentheses in formula (8) (we denote this be a favorable transformation, since the factor a 1 n, term S1 ) can be regarded as a generating function of 1 2 which also appears in formula (6), appears spontaneously. Gegenbauer polynomials Ci cos (e.g., Now, we can begin to examine formula (8) in order to Abramowitz et al., 1965), we can expand S1 in a power generalize Helmert’s result. series with respect to n and rearrange all the terms as First, since the integrand appearing in the first *Since we could find no references describing the derivation of this result in detail, a simple proof is given in the appendix. 4 Bulletin of the Geospatial Information Authority of Japan, Vol.59 December, 2011 2 1 2 S1 1 2ncos n d 2 0 2 1 i 1 2 n Ci cos d 0 2 i0 1 2 i k 1 2 i k 1 2 n i cosi 2k d 0 2 2 i0 k0 k!i k ! 1 2 2 n2 j 2 j k 1 2 2 j k 1 2 cos2 j 2k d 0 2 j0 2 k0 k!2 j k !1 2 2 j1 2 n2 j1 k 1 2 2 j 1 k 1 2 cos2 j 1 2k d 0 2 j0 2 k0 k!2 j 1 k ! 1 2 j 2 2 jk 2 j1 k 2 j 1 3 3 3 n 1 cos2 j 2k 1 1 d 0 2 2r 2r 2s j0 r1 k0 r1 s1 j k 2 j1k 2 3 3 n2 j1 cos2 j 1 2k 1 1 d 0 2r 2s j0 k0 r1 s1 j 2 j jk jk 2 1 3n 3n 3n n cos2k n n d 0 2 2r 2r 2s j0 r1 k1 r11 s j1 jk1 jk 2 3n 3n cos2k 1 n n d 0 2r 2s j0 k1 r11 s j j jk jk j1 jk1 jk 2 sin 4k sin4k 2 r 2k r s 2k 1 r s j0 r1 k1 r1 s1 k1 r11s j 2 j jl1 2 jl 2 2 sin 2l k l r s j0 k1 l 1 r1 s1 2 j 2 j l m sin 2l 1 (9) m . k l j1 m 2 j0 k1 l1 m1 In formula (9), i 3n 2i n , x denotes function. From the above result, for the time being we the Gamma function, and x denotes the floor rewrite formula (8) as 2 j 2 j l a sin 2l 1 m 2nsin 2 S m . (10) k j1 m 2 1 n l 1 2ncos2 n2 j0 k1 l1 m1 Next, we turn to the second term in the d2S d2 1 2 1 1 2ncos n2 d 2 2 parentheses (we denote this term S2 ) in formula (8). d d 2 0 First, we take the following favorable relation between 2nsin 2 S2 . (11) 1 2ncos2 n2 S1 and S2 into account: A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection 5 On the other hand, as we have already seen the form to that of S1 , i.e., final result of formula (9), it is easy to understand that 2 2 S1 can be represented as a trigonometric expansion that d S d S 1 sin 2l 2 2 2 0 l has the form d d l1 4l 2 sin 2l .
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