on the dial curve where we wish the shadow to intersect it at time t, and the di- ... of the same radius as the dial, and the dial is circumscribed on the cylinder.
SUNDIALS AND MATHEMATICAL SURFACES Keijo Ruohonen Institute of Mathematics, Tampere University of Technology P.O. Box 692, 33101 Tampere, Finland
1. Introduction The ancient art of sundial designing and making has produced an astonishing variety of designs, many of them ingenious and works of art. The history and a fair sample of various designs can be found in [Ro] or [Pe,Sch,SP]. The principles of working are in many cases forgotten and understood only by a very tedious reverse-engineering, often made much more difficult by wellmeant but incompetent repair work. This is the case e.g. for the famous Schissler bowl sundial, made in 1578, see [S]. Nowadays designing sundials might not be considered an important undertaking. Nevertheless, the art is not dead. Some of the more advanced mean solar time showing sundials are of relatively recent origin, see [Ro,L]. The “missing” latitude-independent sundial was first described in [F]. (Any three of the four variables altitude, declination, azimuth — related to Sun’s position in the sky — and latitude determine the fourth, so in principle only three of them are needed to set up a sundial.) The goal of the present paper is to show that use of mathematics programs can be a tool in invigorating the art of sundial designing. The central principle here is to use throughout the usual equal-hour 24 h clock face. This has the consequence that the shadow must be created by a rather complex surface. Generating pictures of these surfaces or computing files of data describing them requires much numerical processing. This is where mathematics programs show their value. The program MAPLE® is used here throughout, but there are many other programs, equally useful (such as MATLAB® or MATHEMATICA ® ). As theoretical tool, vector-matrix calculus was used. It is strongly felt that the more usual method of spherical trigonometry has had its time. Separate sections are devoted to the usual solar time sundials, the more complicated mean solar time sundials, and shifted-time sundials. A more mathematically oriented treatment can be found in [Ru]. Earth’s orbital parameters and dynamics, among many other things, are excellently explained in [A] and [Se].
®
MAPLE is trademark of Waterloo Maple Software. MATLAB is trademark of The Mathworks Inc.. MATHEMATICA is trademark of Wolfram Research Inc..
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2. General Ideas For the purposes of this paper, a sundial consists of a curve where a scale is fixed, and a shadow caster. (The usual thing is, of course, to choose the perimeter of a circular sundial as the curve, a 24 h equal-hour scale, and the gnomon stick as the shadow caster, as in the common armillary spehere.) To determine what we want our sundial to show we need two things: the point P on the dial curve where we wish the shadow to intersect it at time t, and the direction of Sun’s rays at that time (and current date). There is naturally considerable freedom in the choice of the curve and the scaling. A curve is mathematically defined by a parametrisation, i.e., an expression P( p) which gives the point P = P( p) on the curve corresponding to parameter value p. (For our purposes it is only important that a reasonably well-behaved mathematical expression P( p) can be obtained.) The direction of Sun’s rays depends on two parameters, Earth’s rotational position and (via declination) its orbital position. These parameters can be given as angles τ and φ, respectively. It is assumed that τ = 0° at 12 o’clock noon and φ = 0° at perigeum. For mathematical purposes, once P( p) is given, it is allowed that p depends on both τ and φ, symbolically we write this as p = T ( τ , φ) (and assume again that a well-behaved mathematical expression for this dependence can be obtained). The time shown by our dial for given values of τ and φ then corresponds to the point P = P(T ( τ , φ)) on the dial curve. Note that dependence on φ is required e.g. if the mean solar time (more or less the time in our watches) is needed. What remains is to choose the shadow caster. This will be mathematically a 3-dimensional surface, which in some cases degenerates to a space curve. Using the fact that Sun’s rays showing time on our sundial must touch the surface but not intersect it, it is possible to find a mathematical expression for the surface, and to draw it using computer. (It would also be possible to generate a file of numerical data to drive numerically controlled machining devices.) These mathematical expressions, when opened, can be enormous. Fortunately, it is not necessary to expand the expressions, if a so-called vectormatrix representation is used. Modern mathematical programs can use this representation directly. Without going into details, it can be said that the technique outlined above is extremely general, and can be used to produce an endless variety of sundial designs. In what follows we will consider only the case where the curve is (part of) the perimeter of a circle, and an equal scaling along it. Nearly
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any other type of curve could be used, and any other scaling. (The shadow caster surfaces will then, however, often be quite complicated and difficult to analyse.) A circle is naturally parametrised by sector angle θ, once a point corresponding to θ = 0° is fixed. This point is chosen in the sequel as the northernmost point on the dial, determined by the intersection of the dial and the meridian plane through the center of the dial.
3. Solar Time Sundials To obtain sundials based on solar time, we choose a T ( τ , φ) which does not depend on φ, this is denoted by writing θ = T ( τ) . It is a good idea to check the ideas described in the previous section against some well-known cases, where the dial curve is a circle parallel to Earth’s equator. Here are four test cases Case 1. Choosing
T ( τ) = τ
we get the familiar solar time equatorial sundial. The shadow caster surface shrinks into a line at center of the dial and at right angles to it. Case 2. Choosing
T ( τ) = 0°
we get a sundial which shows the constant time 12.00. The shadow caster surface again shrinks into a line, but this time it is at the dial curve at P(0°) . This design is of course quite useless, and here only for testing purposes. Case 3. Choosing
T ( τ) = τ + 90°
we get the usual “barrel sundial”. The shadow caster surface is now a cylinder of the same radius as the dial, and the dial is circumscribed on the cylinder. The same design is of course obtained also by choosing T ( τ) = τ − 90° . These two are simple examples of time-shifted designs, more about them in Section 5.
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Case 4. Choosing T ( τ ) = 2τ we get a solar time equatorial sundial with a 12-hour scale. Comparing with Case 1, the shadow now moves at twice the speed. Again the shadow caster surface degenerates to a line. The line is at the dial curve at P(180°) . The design is depicted in Figure 1. Note that actually two circular dials are needed to show the time at all hours.
Figure 1.
Let us then move on to more interesting cases. Case 5. A horizontal sundial is one where the dial face is parallel to Earth’s surface. This means that the axis of the dial is tilted by an angle δ = 90° − local latitude. Mathematically this is easily achieved by properly defining P = P(θ) . Choosing now δ = 45° and
Figure 2.
T ( τ) = τ we get the design depicted in Figure 2. Shadow caster surface is here an astroidal cylinder, cross section of the cylinder, shown in Figure 3, is a curve known as the astroid. The axis of the cylinder is parallel to Earth’s axis. (The radius of the cylinder is
(1 − cos δ) ×
dial radius ,
so, for δ = 0° (Case 1) it will shrink into a line.) It is obvious that this design is not useful as it is since the four lobes of
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Figure 3.
t h e astroidal cylinder obstruct each other. To obtain a practical design, the dial shoul be dissected into four parts and then these parts together with their corresponding lobes should be rearranged preserving their orientation. There are numerous possible arrangements for the four parts, one such is shown in Figure 4. For 12.00–18.00 the lower left quarter is used, for 18.00–24.00 it is the upper left one, for 0.00–6.00 the upper right one, and for 6.00–12.00 the lower right quarter. An interesting choice for δ is δ = 90° , i.e., a horizontal dial on the equator. This design is shown in Figure 5. Note that now the dial is totally inside the astroidal cylinder. We leave it to the reader to figure out how the dial should be dissected and assembled to obtain a practicable design.
Figure 4.
Case 6. To get a vertical southfacing sundial, we should choose the tilt angle δ = − local latitude. (A negative angle simply means tilting in opposite direction.) For δ = −45° the design is the same as in Figure 2 (Case 5), only the dial is now in another position. Figure 5. Case 7. Is it possible to have an equatorial dial where the shadow moves, not clockwise, but counterclockwise? Indeed, it is! To get this design we choose T ( τ) = − τ . The resulting design is depicted in Figure 6, seen from the direction of Earth’s
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axis. The shadow caster surface is again an astroidal cylinder, as in Cases 5 and 6 except that the radius of the cylinder is twice that of the dial, and the dial is totally inside the cylinder. A possible way of dissecting and reassembling the dial is given in Figure 7. There is the additional difficulty that, for certain hours, the dial is between Sun and the shadow caster. This happens for instance at 12.00. In such cases the corresponding part of the dial should be rotated by 180°. Fortunately here the dial design is already sufficiently symmetric so that this is not needed. The dial must however be dissected into eight parts. If we number these octants clockwise from I to VIII, and then rearrange them preserving orientations as in Figure 7, then the octants are used in the following order: IV: 12.00–15.00, VII: 15.00–18.00, VI: 18.00–21.00, I: 21.00–24.00, VIII: 0.00–3.00, III: 3.00–6.00, II: 6.00–9.00, V: 9.00–12.00.
Figure 6.
Figure 7.
Case 8. Choosing T ( τ) = τ 2 we get a semicircular equatorial sundial. The shadow caster surface is again a cylinder but not an astroidal one, see Figure 8. (The cross section is a curve belonging to the cycloid family of curves.)
Figure 8.
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4. Mean Solar Time Sundials Imagine a plane through (the center of) Sun and Earth’s axis. During a year this plane turns 360°. Thus Earth “loses” one turn in a year. Indeed, the mean sidereal day is about four minutes shorter that the mean solar day. The rate of turning of the plane is not constant, however. The precise formulation of this rate is rather complicated, but a quite sufficiently accurate formulation can be obtained starting from Kepler’s laws. It is then an easy exercise in vector calculus to obtain a formula for the instantaneous angular turning speed α (in degrees per year) of the plane in terms of Earth’s orbital position: α = A(φ) . This speed corresponds to solar time. The speed corresponding to mean solar time is the (artificial) constant speed 360° year . The speed at which the difference between these two is accumulated is then 360° year − A(φ) . From this speed the accumulated error is easily obtained, once a reference time is fixed. The resulting error (in minutes of time), the so-called Equation of T i m e , is depicted in Figure 9. It is denoted by E(φ) . (Note that in many texts it is the negative of this error that is called Equation of Time.) To obtain an equatorial mean solar time sundial we need to choose then
14 12 10 min 8 6 4 2 1
2
months from New Year 3 4 5 6 7 8 9 10 11 12
0 -2 -4 -6 -8 -10 -12 -14 -16
T ( τ , φ ) = τ + E( φ ) .
Figure 9.
For any fixed time of day (that is, a fixed value of τ) Sun’s rays through P( τ + E(φ)) form a conelike surface depicted in Figure 10. The ubiquitous lopsided figure of eight, or the
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analemmic figure, is seen here clearly. It can be seen from Figure 10 that the shadow caster surface is self-intersecting. In fact, investigating the mathematical equation of the surface, one can see the surface should be divided in two: one part for spring and the other for autumn. The turning points are the times of solstices. It can also be seen that these two surfaces are surfaces of revolution and they both taper to a point. But examining the mathematical expression of the surface a serious drawback is revealed: the surfaces have singularities at the times of solstices. This means, in this case, that the surfaces are infinite, or rather that not all parts of the surFigure 10. faces are between the dial and Sun. In Figures 11 and 12 the spring surface and the autumn surface, respectively, are depicted both in relation to the dial and enlarged. These are narrow spindlelike surfaces, and, as can be seen, they are not the same. Some 14 h was removed from each end of the time intervals, nevertheless the effect of the singularities can be seen by the beginning “flaring” of the surfaces. Even more prominently this is seen in Figure 13 where only some 80 minutes was removed from the ends of the time interval (the autumn surface). To dispose of the flaring parts of the surfaces at least some 40 h should be removed from each end of the time intervals, resulting in a total time of un-
Figure 11. 8
Figure 12. availability of at least one week. As two of the removed intervals correspond to times of “busiest usage” of the sundial this could be considered as unacceptable. Again investigating the mathematical expression for the shadow caster surface one can see that the reason for the singularities is that zero values of the Equation of Time do not occur at the solstices but slightly aside of them. Once a mathematiFigure 13. cal expression for the Equation of Time is obtained, it is easy to stretch and translate it slightly in such way that the resulting approximate equation of time is zero exactly at the times of the solstices. By this operation the singularities are removed. Moreover it is seen then that, although the spring and autumn surfaces are not the same, they are quite similar. Thus an opportunity to replace them with one surface offers itself: One can take the average of the two surfaces. This results in an approximate equation of time which is zero at the times of the solstices and is symmetric, see Figure 14. Of course the price to pay for this convenience is inaccuracy. It turns out, as might be expected, that the worst error occurs at the times of the solstices, it is however only about ±100 seconds of time. The resulting single shadow caster surface is depicted in Figure 15 in its entirety. A natural question is what do the shadow caster surfaces look like for a horizontal mean solar time sundial. In view of the previous construct we 9
16 14 12 10 min 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -14 -16
1
2
3 4 5 6 7 8 9 10 11 12 months from New Year
Figure 14.
Figure 15.
would expect that two surfaces are needed, each of them divided into four “lobes”. On the other hand, it would not be surprising if the singularities would then turn out to be of a more virulent kind. Indeed, this is what happens. The surfaces are very complicated and it would probably require ray-tracing to check whether they obstruct themselves. Even so, a large period of unavailability would be unavoidable since using an approximate equation of time, zero at the times of the solstices, only works for equatorial sundials.
5. Time-Shifts It may be necessary to show a shifted time to compensate for the difference between local time or, say, daylight savings time. Suppose a time-shift of ζ is required (measured in angles between −180° and 180°, corresponding to time from –12 h to 12 h). There are three methods of achieving this: (1) Rescale the dial for a shifted reference time. (This is very easy for a 24 h circular equal-hour scale, just rotate the dial counterclockwise through the angle ζ.)
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(2) Rotate the whole assembly (the face of the sundial + the shadow caster surface) around an axis parallel to Earth’s axis through the angle ζ. (3) Construct a new shadow caster surface corresponding to w = v + ζ. I.e., choose T ( τ) = τ + ζ (for solar time sundials) or T ( τ , φ) = τ + ζ + E(φ) (for mean solar time sundials). Methods 1 and 2 are very easily implemented, so there would not appear to be much need for both of them or for Method 3. However, one can apply one of the methods and then compensate the effect using the other methods. In this way a lot of new sundial designs are obtained. In what follows, Method 3 is used (and the correction is made by Method 1, say). First, the solar time sundials of Section 3 are considered. For an equatorial solar time sundial the effect of choosing T ( τ) = τ + ζ is easily described: One simply replaces the ordinary gnomon stick with a thicker one, in fact one with radius ±sin ζ × dial radius . For instance, for ζ = ±90° we get the ordinary barrel sundial (Case 3 in Section 3). For a horizontal sundial the situation is more interesting. It was noted in Section 3 that the shadow caster surface consisted of four lobes which were separated from each other. It turns out that for a sufficiently large time-shift this may not be necessary. Case 9. In Figure 1 6 is the shadow caster for ζ = 90° (a 6 h timeshift) and δ = 45° . (We recall that δ = 90° − local latitude.) The surface is not an elliptical cylinder, though very close to one. In fact, the dial circle touches the cylinder at four points and is otherwise outside it. Although everything looks nice, a closer examination reveals that, in fact, the shadow caster is between Sun and the dial only at 18.00–24.00 and 6.00–12.00. Whenever the shadow caster surface is not between Sun and the dial we
Figure 16.
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have to rotate the dial by 180°. Fortunately the surface is symmetric and the rotated parts are already there, but all this means that we still have to divide the dial into four parts since we have to go through the quadrants of the dial in the order III, II, I, IV. Thus it is not necessary to dissect the surface but the quadrants of the dial need to be permuted. The situation of Case 9 is not 80 exactly what we want. A mathematical investigation shows that in order 60 zeta for the shadow caster surface to be be40 tween Sun and the dial, only certain 20 choices for δ and ζ are allowed. Similarly, in order for the shadow caster -60-40-20 0 20 40 60 delta -20 surface to be useful without dissection, again only certain values for δ -40 and ζ are allowed. Pairs δ, ζ which -60 satisfy both these restrictions are the -80 ones we want. In Figure 17 the allowed pairs δ, ζ are those inside the Figure 17. curve. For instance the pair δ = 45° , ζ = 60° (Case 10 below) is allowed while the pair δ = 45° , ζ = 90° (Case 9) is not. The largest value of δ for which there is a nice time-shift is δ = 58° 42' and the time-shift is then ζ = 71° 34' (4 hours 46 minutes). Case 10. The shadow caster surface for ζ = 60° (a 4 h time-shift) and δ = 45° is in Figure 18. The surface need not be dissected and it is always in between Sun and the dial. As for the equatorial solar time sundials, adding time-shifts to the equatorial mean solar time sundials in Section 4 also means replacing the thin shadow caster surfaces by thicker ones. The singularities at the times of the solstices are still there, and, if Figure 18. anything, even worse than in the case of no time-shift. The spring surface and the autumn surface for a time-shift of one hour (ζ = 15° ) are depicted in Figures 19 and 20, respectively. Altogether some 18 days was removed from the ends of the time intervals to remove the flaring parts of the surfaces. There is
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thus a long period of unavailability. It should be mentioned that unfortunately the trick of using an approximate equation of time, zero at the times of the solstices, does not work when time-shifts are used. The only advantage offered by the designs here is that the surfaces do not contain sharp points and are thus somewhat easier to make.
Figure 19.
Figure 20.
There is however a hypothetical situation where a time-shift is useful for equatorial sundials. In order to use an approximate equation of time, it is not necessary that it is zero at the times of the solstices, only that it has the same value, say ε, at these times. A time-shift ζ = − ε will then force the showed time θ = T ( τ , φ ) = τ − ε + E( φ ) to be τ at the solstices. As it happens, at the present time it is possible to choose ε = 0 with negligible extra error. Indeed, the mean of the values of the Equation of Time at the solstices is only a few seconds of time. This might be considered as fortunate. It is even more fortunate that Earth’s axis is not perpendicular to the ecliptic, since otherwise it would not be possible to relate Sun’s declination and time of year, and there would be no mean solar time sundials (and hardly any seasons). Even an angle not exactly 90° but close to it would mean problems. But then, to think of it, what is really fortunate is that Earth’s axis is not in the ecliptic.
References [A]
The Astronomical Almanac for the Year 1998. Government Printing Office. Washington (1997) 13
[F]
F REEMAN , J.G.: A Latitude-Independent Sundial. Journal of the Royal Astronomical Society of Canada 72 No. 2 (1978), 69–80 [L] L OSKE , L.M.: Die Sonnenuhren. Kunstwerke der Zeitmessung und ihre Geheimnisse. Springer–Verlag. Berlin (1959) [P] P EITZ , A.: Sonnenuhren 2. Tabellen und Diagramme zur Berechnung. Callwey–Verlag. Munich (1978) [Ro] ROHR, R.R.J.: Sundials: History, Theory, and Practice. Dover. New York (1996) [Ru] RUOHONEN, K.: Designing Sundials by MATLAB and MAPLE. Tampere University of Technology. Department of Information Technology. Mathematics Software Report No. 9. Tampere (1995) (Available in PDF form in http://matwww.ee.tut.fi/~ruohonen/koti.html .) [Sa] S ADLER , P.M.: An Ancient Time Machine: The Dial of Ahaz. American Journal of Physics 63 No. 3 (1995), 211–216 [Sch] S CHUMACHER , H.: Sonnenuhren 1. Gestaltung–Konstruktion–Ausführung. Callwey–Verlag. Munich (1978) [SP] SCHUMACHER, H. & PEITZ, A.: Sonnenuhren 3. 303 Beispiele aus 12 Ländern. Callwey–Verlag. Munich (1982) [Se] S EIDELMAN , P.K.: Explanatory Supplement to the Astronomical Almanac. University Science Books. Mill Valley CA (1992)
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