"Why this absurd concern with clocks, my friend? Watching time waste will bring no more to spend. Nor can retard the inevitable end."

Walter de la Mare, "Winged Chariot", from Frank Cousins's book *Sundials*

"The tarnished dial, its gnomon shorn away."

Walter de la Mare, "Winged Chariot"

# Theory of sundials

Sundials are self-projecting, automatically updated maps of the sun's position in the celestial sphere. The sun's coordinates (and therefore the time and date) can be read off a network of hour and date curves, the projected graticule of the celestial sphere. In this page, I investigate spherical sundials and gnomonic, stereographic, and pseudo-orthographic map projections of spherical sundials.

I assume familiarity with elementary spherical trigonometry and perspective azimuthal map projections. Chapter 1 of Green's *Spherical Astronomy* and chapter 4 of *Map Projections For Geodesists, Cartographers and Geographers* by Richardus and Adler are useful references. I also refer to Rene Rohr's book *Sundials: History, Theory, and Practice*.

## Sundials as globes: spherical sundials

The sun's hour angle and declination give the time and date almost uniquely. I say almost uniquely since the Sun can have the same declination on two different dates. I gloss over a number of similar technicalities in this page. Most importantly, I ignore the complexities of zone time and assume that "time" means local apparent time. Accordingly, I'll parametrize the celestial sphere by solar hour angle, not hour angle or right ascension.Visual estimates of the sun's hour angle and declination (the time and date) are not very precise; I estimate I can visually determine the time within two hours and the date within three months. *A sundial is an astronomical instrument that allows us to precisely determine the coordinates of the Sun (and therefore the time and date) at a glance.* How can this be done?

The sun's coordinates and the time and date could be determined precisely from a globe of the celestial sphere showing the sun's position and a graticule with curves of constant hour angle and declination. (We have to read the coordinates from a model since it's impossible to mark a graticule directly on the celestial sphere.)

How can a correspondence be set up between the sun in the celestial sphere, which constantly appears to move through the sky, and the image of the sun on a model of the celestial sphere? Projection: Projection makes sundials self-updating maps.the Sun is extremely bright and, from Earth, relatively small in angular extent; the shadow cast by an object illuminated by sunlight records the sun's position.

The previous two paragraphs suggest the spherical sundial (Figure 1), a model of the celestial sphere marked with a graticule of hour angle and declination curves; the shadow of the sun is cast on the spherical surface by a projecting point at the center of the sphere. Notice that projection inverts celestial coordinates. (Noon on a sundial is north of the nodus.)

Solid material blocks the sun's rays, so the spherical sundial is designed as two separate hemispheres. (The sun is so distant, and most sundials are so small, that any parallax between the two halves—or, for that matter, between the center of the Earth and its surface—is ignored.) Each sundial half is called a hemispherium. Hemispheria are mentioned by Vitruvius. How did Eratosthenes measure an angle with a gnomon casting a shadow on a flat surface before the tangent function was tabulated? I think he measured an arc on a hemispherium.

*The spherical sundial does not have a gnomon.* It has a shadow-casting point, which I call the nodus. The concept of a gnomon (which I understand to mean a shadow-casting line parallel to the north/south axis of the celestial sphere) is not fundamental to sundials, since, as I will show, it is only applicable to dials based on the gnomonic projection. The shadow-casting object is better considered to be a point than a line.

Try this thought experiment if you're not convinced: imagine a horizontal sundial with a gnomon. Remove every point on the gnomon except for one. This point must cast a shadow that falls on the sundial's hour lines, since it was once part of the gnomon; however, since it's a point, it's not parallel to any axis and, in particular, not the N/S axis of the celestial sphere. The horizontal sundial must work with just a single shadow-casting point.

## Sundials as plane maps

There are obvious disadvantages to spherical sundials. They're bulky, hard to make, and difficult to read from a distance. Terrestrial globes share these disadvantages, so maps of the Earth are usually flat: coordinates on a globe are converted to coordinates on a plane, cylinder, or cone by the transformation equations of a particular map projection. How can plane maps that show the Sun's position in the celestial sphere be made?

First, I'll provide some map projection background. Map projections on a plane are called azimuthal projections. This website is a good introduction to azimuthal projections. Some azimuthal projections are perspective projections. Perspective projections have a simple geometric interpretation: points on the mapped sphere are joined to points on the projection plane by rays emerging from a center of perspective. Each point on the plane is an image of some point on the sphere. Assuming that the plane is tangent to the sphere, the position of the point of projection totally determines the mapping.

The azimuthal perspective projections I will consider are vertical; i.e., the point of perspective lies on the line that connects the center of the sphere to the sphere's intersection with the plane. There are an infinite number of possible vertical projections, but only three are important here: the gnomonic projection, where the point of projection coincides with the center of the sphere; the stereographic projection, where the point of projection sits on the far side of the sphere; and the orthographic projection, where the point of projection is at infinity.

Finally, if the projection plane is tangent to the sphere's poles or equator, the projection is said to have a polar or equatorial aspect respectively. In the general case where the projection plane is tangent to the sphere elsewhere, the projection is said to have an oblique aspect. Mapping equations are simpler in the polar and equatorial case than in the oblique case.

Back to sundials: the nodus of a sundial is a point of projection that maps the Sun's position on the celestial sphere (or, more correctly, a shadow on an auxiliary spherical sundial) to a shadow on a dial plate. The projection is perspective azimuthal; however, projecting this single point does not constrain the type of perspective azimuthal projection. The Sun's coordinates are read off hour and date curves marked on the dial plate; these curves are the gnomonic, stereographic, or orthographic projection of the graticule of the celestial sphere (again, more correctly, the graticule of an auxiliary spherical sundial).

Different types of plane sundials use different projections for their graticules: almost all familiar sundials have gnomonic graticules, a few rare sundials have stereographic graticules, and analemmatic sundials have orthographic graticules. I now investigate these sundial types in order.

## Gnomonic sundials

In Figure 2 the nodus of a sundial is interpreted to coincide with the center of an auxiliary spherical sundial tangent to the dial plate. (This spherical sundial is not physical—it's an aid to understanding.) The coordinates of the Sun's shadow on the dial plate projected from this nodus can be read from the hour lines and date curves of the projected graticule. What is the image of the graticule, or, more generally, of any point?

The mapping equations for the oblique gnomonic projection and other perspective azimuthal projections are derived in chapter 4 of *Map Projections For Geodesists, Cartographers and Geographers* by Richardus and Adler. I now adapt these to gnomonic sundials. In the following equations, \(\delta\) is the declination of a point on the auxiliary spherical sundial, \(\phi\) is the latitude of the nodus, and \(H\) is the hour angle of a point on the auxiliary spherical sundial. Take the radius of the auxiliary spherical sundial (the height of the nodus) to be unity, and the positive \(x\)-direction and \(y\)-direction on the projected plane to be east and north respectively. Then the image point of a point on the auxiliary spherical sundial with coordinates \((\delta, H)\) and nodus at latitude \(\phi\) has coordinates \((x,y)\), where:
\[x=\frac{cos~\delta~sin~H}{sin~\phi~sin~\delta+cos~\phi~cos~\delta~cos~H}\]
\[y=\frac{cos\phi_0~sin\phi-sin\phi_0~cos\phi~cos~H}{sin\phi_0~sin\phi+cos\phi_0~cos\phi~cos~H}\]

Projected constant-\(\delta\) loci are called date curves. Date curves are conic sections. (The equations for date curves are given parametrically above; I have never seen them anywhere else!) Projected constant \(H\) loci are called hour curves. An important property of the gnomonic projection is that all great circles on the sphere map to straight lines. Curves of constant hour angle on the sphere are half-great circles; therefore, they map to straight lines. The hour curves of gnomonic sundials are therefore lines, so gnomonic sundials are particularly easy to design. (Hour curves on other types of sundials are in general not lines.) Figure 3 shows the graticule of a gnomonic sundial at middle northern latitudes.

Notice that in Figure 3 all the hour linesRohr derives the azimuth of these hour lines. Notice that \(CM\) and \(AM\) in this figure are \(Y\) and \(X\) in the projected plane respectively. It's simpler to find the ratio of these lengths than either length individually. converge at a common point, set back from the nodus, where the N/S axis of the auxiliary spherical sundial intersects the dial plate. A shadow-casting line along the N/S axis of the auxiliary spherical sundial (parallel to the N/S axis of the celestial sphere) connecting the nodus to this common point is called a gnomon. The gnomon has the property that any point along its shadow falls on the same hour line. Figure 4 shows intuitively why this is true: points along the gnomon are the nodi of other auxiliary spherical sundials.

Finally, the mapping equations above give a horizontal sundial; there's no loss of generality, however, since all inclining and declined sundials are horizontal sundials at some longitude and latitude. Rohr has a good treatment of this topic at the end of chapter 4 of his book. Sometimes, gnomonic sundials are inclined so that they have a polar aspect, since this aspect of the gnomonic projection is very simple. Oddly, these sundials are called equatorial sundials.The nomenclature is perfectly backwards: sundials with an equatorial aspect, somewhat rarer, are called polar sundials.

## Stereographic sundials

In Figure 5 the nodus of a sundial is now interpreted to be at the *top* of an auxiliary spherical sundial that sits tangent to the dial plate.

Taking the diameter of the sphere (or the height of the nodus) to be unity, the image point of a point on the auxiliary spherical sundial with coordinates \((\delta, H)\) and nodus at latitude \(\phi\) has coordinates \((x,y)\), where: \[x=\frac{cos~\delta~sin~H}{1+sin~\phi~sin~\delta+cos~\phi~cos~\delta~cos~H}\] \[y=\frac{cos~\phi~sin~\delta-sin~\phi~sin~\delta~cos~H}{1+sin~\phi~sin~\delta+cos~\phi~cos~\delta~cos~H}\]

The stereographic projection has the important property that circles on a sphere are stereographically projected to circles on a plane. Therefore, the hour and date curves of the stereographic sundial are circular arcs (Figure 6).

I don't know of many examples of stereographic sundials. This may be because their hour curves are circular arcs, which are harder to mark on a dial plate than the hour lines of gnomonic sundials. However, their date curves are also circular arcs, which are actually easier to mark on a dial plate than the hyperbolas and ellipses of the gnomonic projection. Again, inclining and declined stereographic sundials can be designed by changing the effective longitude and latitude of the nodus. Finally, I haven't had the fortitude to think very hard about whether there's a gnomon analog for stereographic sundials: let me know if you have any thoughts (my email's at the end of the page).

## Analemmatic (pseudo-orthographic) sundials

Analemmatic sundials are subtler than gnomonic or stereographic sundials. Analemmatic sundials are unrelated to the figure-eight analemma. "Analemma" in these terms originally had the meaning of "auxiliary graphical construction" (think lemma, an auxiliary result in a mathematical proof).The graticule of an analemmatic sundial is an orthographic projection of an auxiliary spherical sundial. However, the shadow of the Sun cast on the dial plate is *not* an orthographic projection: the nodus would have to be at infinity for this to be the case.

How can analemmatic sundials possibly work? Azimuthal projections are so named because they preserve azimuths around the point on the the projection plane that is tangent to the projected sphere. Analemmatic sundials work because the azimuth of the line connecting the orthographic projection of the nodus on the dial plate and the orthographic projection of the sun on the dial plate is the same as the azimuth of the line connecting the orthographic projection of the nodus on the dial plate and the shadow cast by the nodus on the dial plate. I therefore call analemmatic sundials *pseudo-orthographic* projections of the auxiliary spherical sundial.

Taking the radius of the auxiliary spherical sundial to be unity, the image point of a point on the auxiliary spherical sundial with coordinates \((\delta, H)\) and nodus at latitude \(\phi\) has coordinates \((x,y)\), where: \[x=cos~\delta~sin~H\] \[y=cos~\phi~sin~\delta-sin~\phi~cos~\delta~cos~H\]

Curves of constant hour angle and declination on the celestial sphere are projected to ellipses on the dial plate (Figure 7). All constant declination ellipses are similar (although not concentric), as can be confirmed from the projection equations.

Next, I'll make a working sundial from part of the orthographic graticule using the fact that the azimuth of the line connecting the orthographic projection of the nodus on the dial plate and the orthographic projection of the sun on the dial plate is the same as the azimuth of the line connecting the orthographic projection of the nodus on the dial plate and the shadow cast by the nodus on the dial plate. I hope this derivation of the analemmatic sundial equations is clearer and more motivated than those presented elsewhere.

The following refers to Figure 8 above. The ellipse centered on \(O\) is the orthographic projection of the zero declination curve of the auxiliary spherical sundial. Simple trigonometry shows that the minor axis of this ellipse is \(sin~\phi\) times the length of the major axis. The smaller ellipse centered on \(O'\) is the orthographic projection of the small circle of declination corresponding to the solar declination on some other date (here nearly a solstice). \(H\) and \(H'\) are corresponding hour points on these ellipses (orthographic projections of points on the same curve of constant hour angle). As mentioned above, these ellipses are similar, so lines \(OH\) and \(O'H'\) have the same azimuth.

The nodus is positioned vertically above \(O\); its shadow falls along \(OH'\), where \(H'\) is taken generally as the orthographic projection of the sun's shadow on the auxiliary spherical sundial at that date and time. (This should be clear if you draw the auxiliary spherical sundial.) At the equinox, a shadow-casting object above \(O\) casts a shadow passing through \(OH\) and the time can be read directly off the central ellipse; in Figure 8, the hour and date are such that the shadow passes through \(O'H'\) and the time cannot be read accurately off the central ellipse. In short, the sun's azimuth at a certain time of day changes through the year, which makes it complicated to tell the time using a single azimuth dial.

How can a sundial compensate for this change? A number of different sundials could be made for different times of year, or a dial with many different projected ellipses could be used. I have never seen sundials like these (although they would be neat). Analemmatic sundials save on additional sundials and markings by changing the position of the nodus relative to the ellipse centered on \(O\).

The distance between \(O\) and \(O'\) is \(sin~\delta~cos~\phi\). This can be calculated from the projection equations. Suppose we move the gnomon to position \(-O'\), which is the distance \(OO'\) taken opposite \(O'\) from the origin, \(O\). (The negative sign is a play on notation taking \(O\) as \(0\).) Contemplate the figure. The shadow cast by the nodus along the segment \(OH'\), a distance \(OO'\) away from the center of the ellipse along the minor axis, falls on the correct hour point \(H\). *Therefore, if the ellipse centered on \(O\) were the same size as the ellipse centered on \(O'\), the shadow cast with the nodus positioned above \(-O'\) would fall on the line \(OH\), giving the correct time.*This suggests an analemmatic sundial with concentric ellipses corresponding to different months and offsets given by \(sin~\delta~cos~\phi\).

The ellipse centered on \(O'\) must be "stretched" so that it is the same size as the ellipse centered on \(O'\). Construct a line through \(H\) parallel to \(OO'\). Extend \(OH'\) to intersect this line at \(G\). Extend \(OO'\) and construct a line through \(G\) parallel to \(OH\) and \(O'H'\) to intersect the extended \(OO'\) at \(O''\). I now say that the shadow cast with the nodus positioned above \(-O''\) falls on the line \(OH\), giving the correct time! This is clear: the ellipse centered on \(O'\) passing through \(G\) has the same dimensions as the ellipse centered on \(O\); the shadow cast by the nodus along the segment \(OG\), a distance \(OO''\) away from the center of the ellipse along the minor axis, falls on the correct hour point \(H\).

How long is \(OO''\)? Simple trigonometry shows that the ellipse centered on \(O'\) has linear dimensions \(cos~\delta\) those of the ellipse centered on \(O\); all the linear dimensions of this ellipse stretched to size are equal to the previous dimensions times a scaling factor of \(1/cos~\delta\); therefore, \(OO''\) is \(sin~\delta~cos~\phi\) times \(1/cos~\delta\) or \(tan~\delta~cos~\phi\). The length \(OO''\) is measured from the center of the ellipse along the minor axis *away from* the orthographic projection of the curve with the current solar declination.

The nodus of the analemmatic sundial can be considered to be a vertical stick, since the stick collapses to a point in the orthographic projection and every point on the stick casts a shadow that falls along the line connecting \(O\) to the projection of the nodus.A person makes a good movable vertical stick. See Figure 9. This stick is not a gnomon, since it's not parallel to the N/S axis of the celestial sphere.

Figure 9 (reproduced from Denis Savoie's book *Sundials: Design, Construction, and Use*) shows the consequence of unclear markings on analemmatic sundials that use a person as a nodus: people stand on the month markings, which are to the side of the minor axis, and consequently read an inaccurate local apparent time.

## Sundials combined with terrestrial maps

Sundials can show the ground point of the Sun on the Earth. Return to Figure 2. The auxiliary spherical sundial is gnomonically projected on the dial plate. What if the Earth were *also* gnomonically projected on the dial plate? The shadow of the nodus would show the ground point of the Sun on the Earth, the position where the Sun is directly overhead. (The Earth must be rescaled and its coordinates inverted to agree with those of the auxiliary spherical sundial.) Figure 10 shows a horizontal gnomonic sundial I created for my location, Providence, RI, using the mapping software ArcGIS.

## Future work and conclusions

There's much more work to be done in understanding sundials in terms of map projections. One major group of sundials remains uninterpreted: I don't see how altitude dials, such as ring and cylinder dials, are map projections. I haven't explored dials on a cylinder or cone: these would lend interest to the perspective cylindrical and conic projections, which are not useful for terrestrial mapping and are glossed over in map projection books. Finally, sundials combined with terrestrial maps should be promoted: these sundials are visually and intellectually interesting.

To sum up: *plane sundials are best understood as gnomonic, stereographic, and pseudo-orthographic projections of spherical sundials.* I hope I've convinced you, and I'm interested to hear what you think. Email me comments and corrections at william.m.vaughan@gmail.com.