There are a variety of sundial designs that incorporate the solar analemma in one or another form, so that they can provide the precise time according to current civil time standards — which hold that each day and each second should have the same length despite the eccentricity and consequent inconstant speed of the Earth’s orbit and, thus, the solar day.
I was thinking in particular of using the sunbeam reflected from a small round mirror to illuminate a spot on a wall; the point illuminated on the wall will vary according to both the time of day and the angle of elevation of the sun.
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| light |w
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mirror |l
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Each day the sunbeam will sweep a (noncircular) arc across this wall, but from day to day the arc will vary as the elevation angle of the sun does. The total variation in elevation from solstice to solstice is about 47°; it is fastest near the equinoxes and slowest near the solstices. I roughly guess that that means that, at its fastest, it moves about half a degree per day, which is rather pleasantly the size of the sun as seen from Earth.
This means that, if the mirror is small enough, the spot on the wall will also take up about half a degree (32 arcminutes) as seen from the mirror. So, in theory, you could mark the arcs on the wall for each day of the year; they will gradually be displaced by a quarter of a day each year until being reset by the leap year.
This means that, in theory, you could mark two times of day on the wall at each point, one for when the sun crosses that point moving north and one for when it is moving south. Or you could switch out the wall twice a year, on the solstices, and mark one time of day on the wall at each point.
Since an hour is 15° (360÷24), half a degree is about two minutes, so each point along the centerline of the spot’s track will be illuminated for about two minutes. This would seem to pose some difficulties for telling time with a sundial with less error than two minutes, but if the mirror is small, the spot is a well-defined circle with the size of the sun’s disc, and you can see the location of its center to a precision of something like a tenth of its width; this should permit a timekeeping precision of something like ten seconds.
For those ten seconds to constitute about a millimeter of motion, the distance from the small mirror to the wall needs to be about 1.4 meters, although perhaps this can be productively folded up using additional mirrors. At this distance, the 32-arcminute sun disk will project as a 13-mm-diameter circle convolved with the shape of the mirror. (If you want a sharp boundary on it without sacrificing as much brightness, you might consider using a mirror in the shape of an annulus; an annulus convolved with a solid circle has the same diameter, but a much sharper boundary, than two solid circles convolved.)
Suppose that the wall is itself a mirror, but not a flat one. Then it will reflect the spot elsewhere, for example onto a screen, but distorted and possibly changed in direction. It can form caustics in the reflection, and these caustics can have stronger contrasts than mere solar caustics, because the light falling on the wall comes from a smaller point source than the sun’s disc. (It might be worthwhile to make the mirror subtend, for example, 8 arcminutes; at the 1.4-meter distance suggested above, this is about 3.3 millimeters. (Assuming we’re using a circular mirror, not an annulus.)
To focus the spot back to a 3.3-mm point at the same distance would merely require a radius of curvature of that same 1.4 meters. The versine of half of 32 arcminutes is about 1.1 × 10⁻⁵, so if you made the 13-mm spot a spherical reflector, its center would need to be cut deeper than its edges by about 15 microns, a number which varies only a little as the focal length and direction vary. See Caustics for some notes on how to shape nearly-flat surfaces to arbitrary shapes with this kind of precision.
In particular, it wouldn’t be that hard for a series of facets to reflect the beam to the same place on the screen as the sun’s image passed over them, so that instead of scanning across the screen, the projected image stayed in the same place; but it could vary from one facet to the next. And if the facet is convex rather than concave, it could be larger than the 13-mm illuminated area on the wall, rather than smaller, though at the cost of brightness.
(And there don’t need to be actual facets; you can use a smooth curve only occasionally interrupted with the kind of discontinuity you see in Fresnel lenses. Facets are just a crude discrete approximation of the problem.)
Unavoidably, though, since each point along the center of the track is illuminated for two minutes, there will be a certain amount of fading from one image to the next over the course of those two minutes.
One particularly attention-getting image to project might be the current time, written in Arabic numerals, with a colon, like the various “digital sundial” projects that exist.
This poses the problem of how to avoid a vague superposition of numerals during the two-minute transition from one facet to the next. A possible solution is to use a larger number of smaller facets, so that the facets close to the transition zone are projecting not just the current time but the negative of the adjacent time; on one side of the boundary of the 12:46 to 12:48 transition, for example, you would project a mostly gray image with “12:4” in white, “6” in white, and “8” in black, while on the other side, you would project the “8” in white and the “6” in black. Thus, as the preponderance of light shifted from one side of the boundary to the other, the “6” would fade to gray and be replaced by the “8”.
(To keep the black image from being obtrusive a bit further over, you’d want to counterbalance it with a dimmer and perhaps blurrier white image, etc.; I think the brightness curve ends up looking something like the derivative of sinc. Essentially you’re trying to Wiener-filter out the low-pass temporal filter imposed by the sun's nonzero width in order to get a sharper transition.)
This approach to getting faster transitions by counterbalancing with inverse images probably precludes the use of caustics in the sense of places where the Jacobian determinant (of the position of the beam on the screen as a function of its position on the mirrored wall) vanishes, since that could easily create more brightness than you could counterbalance, but you can still vary the magnitude of that determinant substantially to vary the brightness. But your contrast ratio might be limited to 2:1, which sucks.
This poses the additional question of whether the facets would need to be so small that diffraction would pose a problem. If the individual facets were 1 mm across and were effectively planar at the level of 100-μm-diameter “microfacets”, which seems feasible, the Airy limit (1.220λ/D for a circular aperture, as explained in Caustic business card) would be, say, 1.22·555 nm / 100 μm, about 23 arcminutes of diffraction-limited divergence. So, yes, diffraction would start to pose a problem; the wall might need to be larger and further away, and you might need to use larger microfacets. But it’s not so overwhelming that I think it makes the problem infeasible, just challenging.
Suppose that instead of using caustics, you use Bill Beaty’s scratch holograms. You stick a bronze plaque on a wall, paint it with clear polyurethane, and put a peephole nearby. The reflection off the scratches on the plaque from the sun when you’re looking through the peephole displays the current time.
A simple approximation, which is easy to improve on, is to divide the plaque into pixels, and add scratches to each pixel to reflect the sun at every angle where it should be lit up. As long as the scratches aren’t too dense, the scratches at different angles will only interfere a little bit with each other, but it still might be a good idea to display different times on different parts of the display to reduce the “burn-in” effect of too many scratches in the same place. If the plaque is facing north (or south, if you’re in the northern hemisphere like a sucker) and the peephole is in front of and below it, the sun will move through nearly a whole 180° arc each day, but faster close to noon.
Correcting for the Equation of Time can’t be done by displaying different images at different times of year depending on the elevation, but it could be done to some extent by moving the peephole; the angle at which a point P on a scratch reflects is when it is perpendicular to the plane including your eye, the sun, and P. So moving the sun a little to the left is equivalent to moving your eye a little to the right, and vice versa. So it might be adequate to mark dates along the bottom of a viewing slit to show you where to position your eye. (Maybe a part of the plate you view from the side instead of from below could tell you what the sun’s elevation is and thus what the date is.)
As with the analemma, I’m not going to do the math for the angles right now.
The scratches, though, I will. The scratch depth needs to be at least on the order of a wavelength of light (say, half a wavelength) in order to scatter incoming light properly — the ray entering at a point should leave as a plane. It is unreasonably challenging to make the scratch walls much steeper than 45°, and indeed with the usual kind of abrasive scratches, you’ll get