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Understanding and Constructing Sundials with a Hypothetical

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Understanding and Constructing Sundials with a Hypothetical K.P. Cheung, Department of Architecture, The University of Hong Kong, Pokfulam Road, Hong Kong This page started in July 1997 UNDERSTANDING AND CONSTRUCTING SUNDIALS WITH A HYPOTHETICAL CYLINDRICAL SKY VAULT OF SUNPATHS Abstract Sundials were used for thousands of years and some are still used nowadays. Yet there is a need for an explicit full field graphical illustration of how sunlight direction at all possible combinations of day, time, and latitude are related to the gnomon and its shadow cast on the marking surface of the sundial. Starting with conceiving the sunpath traversing an imaginary cylindrical sky vault, this paper attempts to offer this explicit graphical illustration as presented on a horizontal sundial for Hong Kong. Nomenclature AST Apparent Solar Time, or true solar time. d the declination angle of the sun with respect to the centre of the earth, d=+23.44 deg at Summer Solstice and d=-23.44 deg at Winter Solstice. Different declination angles correspond to the different days of the year [Table 1]. GMT Greenwich Mean Time L geographical latitude of a place at Northern hemisphere t hour angle, t=0 deg for solar noon (i.e. AST, apparent solar time = l2:00 noon). For one hour, the hour angle elapsed is l5 deg., t is positive for AST p.m. and negative for AST a.m. UT Universal Time Introduction There are numerous ways of presenting the relationship of apparent solar time, day of year, solar azimuth angle, solar altitude angle for a defined latitude, as observed from a fixed point contained within a horizontal plane at that latitude. The commonly used methods of presentation are based on the annual sun path appearing on an imaginary hemispherical sky-vault [Olgyay 1957a, Rohr 1965, Waugh 1973a, Fig 1]. In reality, the sun never traverses part of a perfect sphere relative to the earth. With the hemispherical sun-path sky-vault presentation, it is extremely inconvenient, if not impossible, to make the full field illustration, because projections involving spherical vault surface involve a very large number of curves for full field projection. This perhaps has hindered authors so far from producing the explicit full field illustrations. Although there were many possible graphical approaches mentioned in a recent publication (Waugh 1973b), it quoted a graphical method published in Paris in 1790 by Dom Francois Bedos de Celles for laying out a horizontal sundial. In this method, only the AST hour lines for the selected latitude were shown. The day curves were not shown. This method, and the other graphical approaches did not give full field illustrations. In an attempt to make the full field graphical illustration for sundials, it is considered that projections involving an imaginary cylindrical sky vault for the sunpath would result in the least possible number of projected lines and curves, thus providing explicit and clear illustrations. In this imaginary cylindrical sky vault approach (Cheung 1997), the sun path is conceived as part of an imaginary cylindrical surface, with the line of AST noon and the cylinder axis lying in the meridian plane which contains the observer, or the sundial. Also the axis of the imaginary cylindrical surface is inclined at an angle equal to the latitude of the location of the observer, or the sundial, [Fig 2] to a horizontal plane containing the observer, or the sundial. With this approach, it will be demonstrated that clear and explicit graphical http://www.arch.hku.hk/~kpcheung/sundial/hypoth.htm (1 of 3) [1/11/2004 2:51:06 PM] relationships linking the various sundial parameters, the sundial gnomon, and the sundial surface can be shown in a full field manner. The basic assumption The sun travels with respect to an observer 0 (or the tip of the gnomon of a horizontal sundial) on the surface of the earth (i.e. topocentric observation) from d=+23.44 deg to -23.44 deg from Summer Solstice to Winter Solstice and vice versa, year by year [Olgyay 1957b, Fig 2]. Because the distance between the sun and the earth is more than 2000 times the radius of the earth, for general purposes including sundial design, it is assumed that there is practically no difference between topocentric observation and geocentric observation (i.e. observing the sun fictionally at the centre of the earth) in relating the variables of concern of the sundials. Thus the observer point 0, and hence the shadow of the gnomon falling on the sundial surface, are interchangeable freely between a location on the surface of the earth, and fictionally the centre of the earth, provided that the parallelity of the sundial components are kept at both locations. That is, if a sundial were placed fictionally at the centre of the earth in this way, it would have the same reading as placed on the earth surface. This is most prominently illustrated in universal sundials. The contracted imaginary cylindrical sky vault Daily the sun can be considered to move about the cylinder axis passing through 0 [Fig 2] on a circular path at a speed of l hour AST equal to l5 deg from East to West, relative to the stationary observer 0, or any point along the gnomon of the sundial provided that the gnomon is parallel to the axis of the earth. In one year, the sun could appear to traverse an imaginary cylindrical surface MQNQM, having an axis passing through 0, this axis being parallel to the earth axis. Each line on the imaginary cylindrical surface represents a particular moment of AST. All AST lines are inclined at an angle L to the reference horizontal plane passing through 0 which is at latitude L. Different days of the year on the same AST line [Fig 2] correspond to the various solar declination angles [Table 1] with respect to the stationary observer 0, or the location of the sundial. In actual fact, the sun never traverses a cylindrical path in the sky as described, nor does it ever truely traverse a path occupying part of a sphere in the sky. The transposition of the position of the sun relative to the earth onto either part of an imaginary spherical surface or an imaginary cylindrical surface are both acceptable for relating the directional relationship of the solar altitude angle and solar azimuth angle with the different times of the day, the different days of the year (i.e. the solar declination angle) and the latitude(s) in which the sundial is designed for. In the conventional spherical sky-vault transposition the sun appears to traverse an arc for a particular apparent solar time [Fig 1], moving from one day to the next. However in this proposed cylindrical transposition, the sun appears to traverse a straight line such as line MQN [Fig 2], under the same conditions and assumptions of topocentric and geocentric observations. The directional relationship of sunlight with the gnomon of the sundial is identical whether the sun is interpreted to traverse a hemispherical sky vault, or a cylindrical sky vault, because the same practically parallel sunlight exists for both interpretations. Based on the latter interpretation, a proposed method of constructing a horizontal sundial is discussed [Fig 3]. In this method, the sky vault containing the sunpath is imaginarily contracted into a small cylinder occupying a small area in a piece of paper. The essential components of a sundial The essential components of a sundial is the gnomon and the sundial surface. Sunlight shines on the gnomon, casting a shadow or a bright spot on the sundial surface for reading the day/time. There are a variety of gnomon and sundial surface. For a horizontal sundial illustrated [Fig. 3], the sundial surface is a horizontal surface containing the AST hour lines and day curves, and the gnomon is a vertical pointer stemming from the dial surface. The virtual line joining the tip of the pointer to the converging point of the AST hour lines makes an angle with the AST noon line, this angle being qual to the angle of the latitude at which the sundial is designed for. A proposed method of constructing a horizontal sundial Starting with the imaginary contracted cylindrical sky vault for the sun path, the construction [Fig 3] of the horizontal sundial is self-explanatory to a person having a background of basic technical drawing. The construction [Fig 3] starts with the direction of sunlight defined for the day/time combinations. It also enables the days for any shadow point on the dial surface such as the point E to be obtained (i.e. March 14 and September 30, with d = -2.73 deg = angle QOb and line ab is parallel to line OQ, point a cutting 17:00 line) by direct reverse projection if its time (i.e. 17:00) is known. Furthermore, it enables the day(s) and the time for any shadow point on the dial surface such as the point S' to be obtained (i.e. 15:30 and Feb 18 / Oct 24) by reverse projection if both the day(s) and time of the shadow point are not known initially. These are elaborated in the following steps of constructing and understanding a horizontal sundial [Fig 3] : [Click here for downloading Fig 3] 1. Set line RG'F'V, such that Point H coincides with Point O, line HG' is at 90 deg to line RG'F'V, line HF' coincides with the axis of the cylinder, and length HG' = (tan L) x length G'F'. http://www.arch.hku.hk/~kpcheung/sundial/hypoth.htm (2 of 3) [1/11/2004 2:51:06 PM] 2.
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