# The Design and Simulation of an Astronomical Clock
## 4 The Face of the Astronomical Clock  
The positions of the celestial objects on the celestial sphere are determined by angular coordinates (altitude--azimuth, declination--right ascension or ecliptic latitude and longitude). Since the stereographic projection is conformal and thus preserves angles, it can  
Figure 4: The clock of Stara Bystrica, Czech Republic [5]. be used for the design of the astronomical dial. The construction of the astronomical dial for the latitude of Belgrade (44.80 N) will be presented and explained in this paper in all necessary detail. Since the celestial sphere rotates around the celestial axis \(NS\), the dial of this astronomical clock can be geometrically constructed by the stereographic projection of characteristic circles on the celestial sphere from the celestial pole \(N\) or \(S\) to the plane of the celestial equator \(e\). Since Belgrade is located in the northern hemisphere, the stereographic projection is obtained from the North Pole \(N\). As is shown on Figure 5, the zodiac circle (circle of ecliptic) \(z\), the celestial Tropic of Capricorn \(t_{S}\), the celestial Tropic of Cancer \(t_{N}\) and the horizon \(h\) for the latitude of Belgrade are stereographically projected from the pole \(N\) to the plane of the celestial equator \(e\). The circles of ecliptic \(z\) and celestial equator \(e\) intersect at two points: one of them is the first point of Aries or the cusp of Aries \(\gamma\) and the second one is the first point of Libra \(\Omega\). As usual, the astronomical dial is rimmed by the celestial Tropic of Cancer \(t_{N}\) since the Sun and the Moon, as observed from the Earth, never exceed this circle. The zodiac circle (the circle of ecliptic) \(z\) just touches both celestial tropics and intersects equator \(e\) and horizon h at the same pair of points--\(\gamma\) and \(\Omega\). Due to the fact that the positions of the Sun and the Moon are determined on this astronomical dial by the ecliptic longitude, the representation of the ecliptic circle (zodiac) is a crucial and inevitable part of its face. Thus, the zodiac circle must be geometrically constructed as well. The stereographic projections of the zodiac circle and its division into  
Figure 5: The stereographic projection of the celestial sphere. 12 equal segments from the pole \(N\) to the plane of the celestial equator are given in Figure 6. The zodiac circle is stereographically divided into 12 equal segments each of which is 30 degrees of ecliptic longitude wide. These segments are named after the well-known astrological signs: Aries P, Taurus O, Gemini R, Cancer S, Leo T, Virgo U, Libra V, Scorpio V, Sagittarius V, Capricorn V, Aquarius \(\widetilde{\text{\tiny{\text{\circlecircle}}}}\), and Pisces O. All segments are equally graduated with six divisions of five degrees each. It is important to emphasize what can be easily seen on the zodiac circle, namely that stereographic projections of these divisions are unevenly distributed since the stereographic projection is not affine but central. Figure 7 shows the model of an astronomical dial created by the SolidWorks application. The Sun (4) and the Moon (16) symbols (icons) are attached to the clock hands which rotate around the clock axis (13). The zodiac (ecliptic) circle (14) also rotates around the axis (13) and supports the Sun and the Moon icons by the two circular grooves. By the position of the Sun pointer, the local mean solar time can be read on the 24 numeral divisions placed at the edge of the celestial Tropic of Cancer (3). By the position of the arrow mark (2) (\(\gamma\)) 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)  
Figure 6: The stereographic projection of the zodiac and its division into 12 equal segments. indicates the current date of the year.