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Derivation of formulas in spherical trigonometry based on rotation matrix

Hsieh T. H., Wang S. Z., Liu W., Zhao J. S., Chen C. L.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2019

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Vol. 13, no. 3

553--558

EN

The formulas of spherical triangle, which are widely used to solve various navigation problems, are the important basic knowledge of nautical mathematics. Because the sine rules and the cosine rules for the sides are the fundamental formulas to derive the other spherical triangle formulas, they are also called the genetic codes of the spherical triangle formulas. In the teaching process, teachers usually use the geometric method to derive and prove these fundamental formulas. However, the derivation of geometric methods is complicated and difficult to understand. To improve the teaching process, this paper proposes the three-dimensional rotation method, which is based on conversion of two cartesian coordinate frames using the rotation matrices. This method can easily and simultaneously derive the sine rules, the cosine rules for the sides, and the five-part formulas (I), and is also helpful to solve different kinds of spherical navigation problems.

2

Some New Dimensions in Sextant-Based Celestial Navigation Aspects of Position Solution Reliability with Multiple Sights

Zevering K. H.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2008

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Vol. 2, no. 3

271--278

EN

The traditional approach relying on sight reduction tables, a non-programmatic location of the position fix and an inadequate allowance for observation errors is still widely pursued and advocated. In the late 1970s the programmatic Least Squares method (LSQ) was introduced which determines a random error fix (FixQ) for any multiple sights combination. B.D Yallop & C.Y Hohenkerk (1985) expanded LSQ to incorporate the computation of the random error margin of a fix. Several marketed PDA-based programs apply LSQ, but none have fully incorporated the random error margin as a guide for the navigator. All existing LSQ applications have two drawbacks. One is, all observation error is attributed to random sources, whereas the possibility of systematic error has in fact a long theoretical and practical background in celestial navigation. Systematic error represents a bias in statistical random error theory and can and should be allowed for. A major drawback is that existing LSQ program applications incorporate the running fix technique (RFT) traditionally applied in coastal navigation. It has no general validity in celestial navigation. The position circle of an earlier celestial sight can only be mathematically correctly transferred when its Geometric Position (GP) is transferred for the run data. A final aspect of reliability is the strategy adopted at the sight planning stage. At least during twilight observations, navigators should aim at getting three or four sights with a total azimuth angle >180o, with three successive subsights on each body. In such configurations FixQ and FixS will be relatively close together, generally obviating the need to process the sights for possible systematic error.

3

MAAPCNIA: A Boost to Authentic MET Instruction

Baylon A.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2008

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Vol. 2, no. 3

285--290

EN

This paper presents an innovation in teaching strategy in one major subject in the Maritime Education and Training (MET) program of the Maritime Academy of Asia and the Pacific (MAAP). This strategy makes use of an instructional aid called MAAPCNIA (MAAP Celestial Navigation Instructional Aid), conceptualized and designed by Capt. Daniel S. Torres, Jr. MAAPCNIA encompasses color-coded spheres that depict navigational triangle. This triangle is composed of a blue celestial sphere representing horizon system of coordinates and a white sphere enclosing a globe characterizing time diagram and celestial equator and terrestrial systems of coordinates. With this instrument, celestial spheres can be easily visualized compared when using one-dimensional drawing. Hence, understanding of orthographic projection, which is essentially the core of appreciation and mastery of this tough subject, is enhanced. This paper focuses on the development and impact of MAAPCNIA since its conception in 2004, towards authentic teaching-learning process in Celestial Navigation.

4

Estimation of Altitude Accuracy of Punctual Celestial Bodies Measured with Help of Digital Still Camera

Bobkiewicz P.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2008

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Vol. 2, no. 3

279--284

EN

Measurement of altitude traditionally made with sextant may be done with help of digital still camera. Factors influencing accuracy of this measurement done with help of such a device are described in the paper. Values of errors introduced by each of these factors are estimated basing on example technical data of typical digital camera. This analysis shows, which factors are the most important and if accuracy of altitude is sufficient for purposes of celestial navigation.