Continued Fractions, Intermediate Fractions and Their Relation to the Best Approximations

Milinkovic, Luka; Malešević, Branko; Banjac, Bojan

doi:10.46939/j.sci.arts-20.3-a05

Cited by 1 publication

(3 citation statements)

References 12 publications

(18 reference statements)

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“…By the position of the arrow mark (2) (γ) at the zodiac (14), the local mean sidereal time can also be determined on the same division (3) or on the equator circle (8) scale. The ring (12) is the celestial Tropic of Capricorn and represents the inner border of the dial. The position of the Sun (4) on the ecliptic circle (14) indicates the current date of the year.…”

Section: The Face Of the Astronomical Clockmentioning

confidence: 99%

“…Since the number of gear teeth is an integer, the gear ratios must be approximated by rational numbers. The proper gear ratios of the mechanism gear trains are determined by the method of continued fractions since this calculation technique was mathematically proven [11] to produce the best possible rational approximation of a real number [12]. (Given a number x, its best rational approximations of the first kind are those fractions p/q such that x is closer to p/q than to any fraction with a smaller denominator.…”

Section: Approximations Of Astronomical Datamentioning

confidence: 99%

“…The proper gear ratio of the particular gear train must be determined by the method of continued fractions. This method generates the set of convergents or semi-convergents of the continued fraction representation of the real ratio [12,13].…”

Section: Approximations Of Astronomical Datamentioning

confidence: 99%

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The Design and Simulation of an Astronomical Clock

et al. 2021

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This paper describes and explains the synthesis of an astronomical clock mechanism which displays the mean position of the Sun, the Moon, the lunar node and zodiac circle as well as the Moon phases and their motion during the year as seen from the Earth. The clock face represents the stereographic projection of the celestial equator, celestial tropics, zodiac circle (ecliptic) and horizon for the latitude of Belgrade from the north celestial pole to the equator plane. The observed motions of celestial objects are realized by a set of clock gear trains with properly calculated gear ratios. The method of continued fraction is applied in the computation of proper and practically applicable gear ratios of the clock gear trains. The fully operational 3D model of the astronomical clock is created and the motion study of its operation is accomplished by using the SolidWorks 2016 application. The simulation results are compared with the ephemeris data and the detected differences are used to evaluate the long-term accuracy of the astronomical clock operation. The presented methods of the clock mechanism synthesis can be useful for the design, maintenance and conservation of large-scale city astronomical clocks since these clocks represent a precious historical and cultural heritage of European civilization.

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Section: The Face Of the Astronomical Clockmentioning

confidence: 99%

Section: Approximations Of Astronomical Datamentioning

confidence: 99%