Our program is developed in two branches: one which uses only standard
linux tools (bash, awk, etc.), and another one which instead uses only
Python3. Both versions will need
gmt. To be able to draw the geographic
map within the sundial, be sure to also have the gmt-gshhg package
installed.
Apart from gmt, you will need Python ≥ 3.6 to ble able to run the
program.
Apart from gmt, you will need the following (standard) Linux tools:
- bash
- date
- sed
- awk
- bc
- paste
The literature on sundials is huge (and with some surprises even in recent times, see bifilar sundials for example). The basic ideas behind the gnomonic sundial, that we will present shortly here, are thousands of years old, and can be understood with some basic knowledge of the trigonometric functions. Nonetheless, the ability to create them easily, automatically and complete with a full geographic map is recent, thanks to current computers.
We will assume in the rest of this section that our location is in the northern hemisphere and above the tropic of cancer.
A gnomonic sundial is, in its essence, a miniature reproduction of the earth, projected onto a surface. Let's take a globe, orient its axis along the earth's axis (i.e., pointing to Polaris star, i.e., north with angle equal to the local latitude), and roll the globe so to have your location facing upwards, to your zenith. You now have small copy of the earth, with the same exact orientation w.r.t. to the sun. If we assume the sun to be at infinite distance (since the distance sun-earth is much larger than the earth's radius), we can use our globe as an exact reproduction of the earth, to understand our local time.
We say that we are at (local-time) midday when the sun is exactly at south, which means that it is at the zenith on some place on earth which is south of us, between the tropics (on the summer solstice the sun travels right above the tropic of Cancer, on the winter solstice above the tropic of Capricorn, and on the two equinoxes above the equator). Which point has the sun exactly at its zenith? If we assume the earth to be spherical (which we will) then it's easy: it's just the intersection between the line going from the sun to the center of the earth and its spherical surface.
Now imagine to take our globe, make it semi-transparent and invert each point on it with its antipode: in our new globe our current location will be facing downward, looking at nadir. Let's make the center point of our globe opaque, so that it casts a visible shadow on the semi-transparent inverted globe: the point shown now on the semi-transparent globe will be the one which originally had the sun at its zenith, thus indicating the midday meridian.
Instead of having only the globe center opaque, we could have the whole axis: in this way the shadow would cover the whole midday meridian (but we would lose the exact position of the zenith point).
One can construct a sundial using the semi-transparent globe we outlined above. This type of sundial is called equatorial, and it is typically realized with the globe axis casting its shadow on a circular stripe (or a disk) representing the equator.
What if we want to be able to see the same information on a flat surface instead? Remember that we want to see which point on the sphere is along the line from the sun to the globe's center. What we can do is project, starting from the globe's center, every point on the globe's surface until we reach the flat surface. This projection is called gnomonic, and dates back to Thales, in the 6th century BC. If we draw our map on a flat surface using the gnomonic projection, then we can get rid of the globe, leaving only its center or its axis (which will be the gnomon of our sundial) and read the zenith location directly on the flat surface.
We'll now focus on horizontal surfaces, but if you want to build a vertical or an inclining sundial, we can simply change the point of the globe projection accordingly. Let's take as an example Cagliari, in Italy, which has coordinates LON = 9.133 and LAT = 39.248. In our hypothetical, semi-transparent sphere our location will be facing down, thus being the intersection point of a horizontal plane tangent to the globe. Instead, if we want to project our globe to a vertical plane facing south, the tangent point on the globe will have coordinates LON = 9.133 and LAT = 39.248 - 90 = -50.752. If your wall is not vertical and/or not facing south, you just need to compute the tangent point of wall plane with the semi-transparent globe to generate the correct sundial map. Our program can take care of these computations.
If the inverted globe is projected onto a horizontal surface, then north and south are inverted in the map, hence the line from the north pole the chosen location should point to (the local) north.
Once we have the map set up, we just need the gnomon, i.e., a point (nodus) or a rod (style), to represent the center of the globe and/or its axis. Since the projection is tangent to our location on the map, the easiest way to add the nodus is just by putting it on top of a vertical style, standing at our location in the map, of length r. Let d be the distance in the map between the north pole and our location, then by simple trigonometry we have r = d * tg(LAT), where LAT is our location's latitude. Alternatively, one might measure directly r as the distance between the center of the projection and some point which is 45 degree from it (e.g., from (LON, LAT) and (LON, LAT±45), as it's drawn by default by our program).
By using a vertical gnomon pointed on our location we must read the time using only the tip of the shadow. We can have the whole shadow being more informative by making the gnomon polar, i.e., parallel to the globe's axis (i.e., starting at the north pole and pointing to Polaris, i.e., north, and up of an angle equal to LAT). Some simple computations show that the length g of the polar gnomon, reaching the center of the globe, is g = d / cos(LAT) = r / sin(LAT).
Until the 19th century each city used its own local time, i.e., based on the solar position: when the sun is exactly at south then it is midday. Until transport and communication were slow, this was not a problem, but when railways became more and more widespread the time difference between cities started posing serious challenges to the design and supervision of train schedules. E.g., traveling from Venice to Turin required passengers and train personnel to adjust their clocks by 18 minutes.
At the end of 19th century the world was divided in 24 time zones, each large 15 degrees, to mitigate the effects of the different local times and accumulate the differences in multiple of one hour. E.g., Italy has adopted the time of the 15 degrees meridian, which goes roughly through the mount Etna, so that there are no time differences when using national trains.
Adopting the reference time in the gnomonic sundial is quite simple, you just consider it midday when the sun crosses your reference meridian, instead of your local one.
Until now we have considered the days to be 24 hours long, but actually solar days (i.e., time difference between local middays) are quite variable throughout the year, with a difference between shortest and longest one of about half an hour. This is due both to the eccentricity of the earth's orbit and to the difference between the axes of rotation and revolution (obliquity of the ecliptic).
However, a mean time of exactly 24 hours has been adopted, to avoid adjusting clocks everyday. The difference in time between this mean time and the real one is described by the equation of time.
Using the equation of time we can plot on our map, for each day at noon, the coordinates of the points which will have the sun at the zenith. Because of the equation of time this curve will not be a segment, but a 8 shaped curve, known as lemniscate. This curve is the earth equivalent of the analemma. Note, e.g., that in November the difference between mean and real solar time reaches 16.5 minutes. Considering again Cagliari as our location, that means that when the official time is noon, our real, local time is actually 11:20, off by almost 40 minutes (23.5 minutes from the reference meridian + 16.5 minutes from the equation of time).
To incorporate also this correction to the local time in our sundial, we draw the analemma corresponding to the midday meridian (or we could also draw one for each meridian) and we read the midday when the shadow touches the lemniscate. Since the lemniscate has two intersection point with almost all parallels, we need to know which one to consider when reading. The analemma starts clockwise at the tropic of Capricorn on winter solstice, crosses the equator (west of midday meridian) on spring equinox and so on, changing season each time it touches a tropic or the equator. Thus, when, e.g., reading a position south of the equator, if it's winter we should consider the western branch of the analemma, if its autumn the eastern one.
TBD
TBD
TBD
See the directory of examples, with descriptions and pictures.
Geographic sundials, showing on the dial the positions of several cities, were used throughout the 19th century. See, e.g., this particularly rich one, which dates back to 1851.
A more modern example, built in 1997, with a complete map is shown here.
gnomonic-sundial is developed by
- Francesco Versaci [email protected]
I'd like to thank Gian Casalegno for his high quality educational materials, which have inspired this work, and for tirelessly providing valueble answers to my questions.
This is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
See COPYING for further details.