Properties of great circle chart.

Expansion formulae about geodesic line on chart of Mercator projection are derived. In section 2, lengths of tangent and normal to projected geodesic, length of rectilinear chord, and angle between rectilinear chord and tangent to the projected geodesic are expanded in power series of true length S of the geodesic ( eqs. ( 8)～(11）)． In section 3, curvature σ of projected geodesic 〔eq.(22）) on Mercator projection and its derivatives σ', σ",σ''' are given 〔eqs.(27）～（30）〕 with their respective values at the beginning point of the geodesic〔eqs.(31）～（37）〕. Similar quantities for scale k are also obtained 〔eqs.(39）～〔42) and (43〕～（49〕〕． In section 4, some examples of applications of quantities obtained in section 3 are shown. ( i ) For given length m of projected geodesic on chart, minimum limit of scale by which great circle can be approximated by straight line and that by which geodesic can be approximated by great circle are derived for standard parallel〔eqs.(57) and (58）〕. Both of these limits hold for enough larger scales. In TABLES 1 (for m = 1 metre〕and 2 (for 111=0.1 metre〕examples of these scale limits are shown. 〔ii〕 Correction formulae to obtain the terminal of geodesicfrom a point, which is plotted on chart by employing directly given azimuth T of the geodesic and its true length S, are derived〔eqs.(59）～（62）〕 and their auxiliary quantities are given in TABLES 3 and 4. (iii) Maximum displacement of rectilinear chord from the projected geodesic occurs at about mid-point of the projected geodesic when the geodesic is not too large. Formulae to evaluate this mid-point displacement h 1/2 and its corresponding distance on the earth surface H 1/2 are also derived 〔eqs.(65）～（67）〕 and examples are shown in TABLE 5 for latitudes 25° to 45°.

Bibliographic Details

Main Authors:	Sinzi, M. Akira, Kubo, Yoshio
Format:	Journal Contribution biblioteca
Language:	Japanese
Published:	1967
Online Access:	http://hdl.handle.net/1834/16434
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