X and the City: Modeling Aspects of Urban Life

by Adam, John A.

  And now, to geometry!

  X = rmax: “CIGARS AND APPLES” IN THE CITY

  An important theorem in plane geometry states that the angle subtended by a chord (CE) at the center of a circle (2α) is twice that subtended at any point (F) on the circumference. This is easily established from Figure 22.2 by adding the radius OF to form three isosceles triangles, and noting that the sum of the angles around O must be 360°. Now let C represent a light source, and E represent the observer’s eye. If the volume of space surrounding the observer and source contains either raindrops or hexagonal ice crystals (and is sufficiently large), then for raindrops, α ≈ 42° and the arc CFE is the locus of all raindrops in that plane scattering light into the observer’s eye. But by symmetry this is also true for all rotations of that arc around the line CE, forming, as it turns out, an apple-shaped surface. For reasons that will become obvious below, this surface is known as “Minnaert’s cigar.”

  Figure 22.2. Angle subtended by a chord (CE) at the center of a circle (2α) is twice that subtended at any point (F) on the circumference.

  Reference has already been made to Marcel Minnaert, the author of a famous and widely quoted book Light and Colour in the Open Air [38]. In that book (pp. 206–207) he made the following observation:

  One very cold evening (17° F) beautiful halo phenomena could be seen in the steam from a train in the railway station. Near one of the lamps, where the cloud of steam was blown in every direction, a cigar-shaped surface of light could be seen, having one end near the eye, the other near the lamp; all little crystals traversing this surface were lit up, but the space inside was quite dark; the cone tangential to the surface had an angle at the vertex of about 44°. It is at once clear that the cigar-shaped surface is simply the locus of all those points P such that the sum of the angles subtended by EP and PL at L and E respectively is 22°.

  Minnaert then goes on to note that the remarkable three-dimensional nature of this observation is only possible because (i) the light source is so near the eyes, and (ii) of the stereoscopic effects associated with both eyes viewing the crystals (see Figure 22.3).

  Figure 22.3. Sketch of “Minnaert’s cigar” (Redrawn from Minnaert 1954).

  Figure 22.4. Minnaert”s cigar for the 22° halo, illustrating the three-dimensional form (Redrawn from Mattsson et al. 2000).

  Figure 22.5. Minnaert’s “apple-shaped cigar” for the 22° halo, generated by rotating the larger arc around the chord CE in Figure 22.2 (or chord LO in Figure 22.6). The light source is at the top of the “core” and the observer”s eye is at the bottom. It is drawn for both the red and blue ends of the visible spectrum. Redrawn from a diagram by Christian Fenn (see [39]).

  Figure 22.4 is a perspective rendering of the Minnaert 22° cigar, a spindle–shaped surface, truncated here to emphasize its three-dimensional nature. And the apple-shaped surface in Figure 22.5 is also referred to as Minnaert’s cigar (even though it looks like no cigar I’ve ever seen!). It arises from rotating the smaller arc CE about the chord in Figure 22.2, and the apple surface arises from rotating the larger arc CFE about the same chord.

  In fact there are several other intermediate surface shapes that can arise from similar considerations. In addition to the “classical” 22° cigar halo surface (with supplementary angle 158°), there is a less common 46° one (with supplementary angle 134°). Furthermore, as also noted in Appendix 11, there is a secondary rainbow which can appear at an angle of 51° from the anti-solar direction (or as a circle of angular radius 129° around the sun). When rotated about the corresponding chord LO (see Figure 22.6, where the segments LO correspond to the chord CE in Figure 22.2), the appropriate “cigar surfaces” are generated (Figures 22.4 and 22.7). In each case, the vertex angles marked in Figure 22.6 are the angles measured from the anti-solar direction. The scattering angle—the angle through which light from the source has been deviated—is the supplement of these, thus 22° and 46° for the halo surfaces, 138° and 129° for the rainbow surfaces.

  An enlarged version of the small 22° “cigar” surface of Figure 22.6 is shown in Figure 22.7; as in Figure 22.4 it can be interpreted as a surface of revolution.

  The detailed geometry of the situation is governed by Figure 22.8, where the observer’s eye is at B and the light source is at D. We see from triangle BCD that rc + rd = rmax. From this it follows that

  Figure 22.6. Generalization of Minnaert”s cigar for (a) the 22° halo, (b) the 46° halo, (c) the secondary rainbow, and (d) the primary bow. The light source is at L and the observer”s eye is at O. The angles stated are measured from the “anti-source” point (as in the daylight anti-solar point). Thus the supplement of the 22° halo angle is 158°, etc. (Based on Figure 1 in Mattsson and Barring, 2001).

  Figure 22.7. Profile of Minnaert’s cigar as defined by equation (22.5).

  First, we do a little geometry. The unit of length is taken as the distance BD, that is, BD = 1. This is the length of the “cigar.” The radius of the circle is R = BD/(2 sin rmax) = (2 sin rmax)−1; the distance dc = BC between the crystals scattering light into the observer’s eye and his eye at B is

  The quotient q (or “aspect ratio”) of the cigar’s short and long axes is

  Let’s now find the equation of this cigar-shaped halo. The center of the circle, O′, in Figure 22.8 is located at the point (0, −R cos rmax) or (0, −(cot rmax)/2). The equation of the circle is therefore

  For the 22° halo, this angle is just rmax, so equation (22.4) can be rewritten as approximately

  Figure 22.8. Relationship between the angular radius of a halo (rd) with its center at a distance d from the light source. The length of the “cigar” is distance BD = 1, ED = d, angle BCD = 180° − α, O′ is the center of the circle and O is the coordinate origin. Note from triangle O′OD that α = rmax.

  This is the equation of the profile illustrated in Figure 22.7, noted earlier in “solid” form in Figure 22.4.

  As noted above, Figure 22.6 illustrates a cross section of the complementary cigar shape for other situations with much smaller “scattering angles” relative to the anti-source point (as opposed to the anti-solar point during the daytime). This includes both primary and secondary divergent-light rainbows. Each apple-shaped rainbow surface (c) and (d) can be regarded more accurately as a degenerate form of a torus, missing the central hole because the rotating circle intersects itself. This is well illustrated in the next chapter (Figure 23.10).

  Complicated as these surfaces may seem, there is a very interesting two--dimensional subset of them that may be seen when the ground is covered with raindrops or dew (for “dewbows”) and frost or other ice crystals, such as diamond dust (ground-level cloud composed of tiny ice crystals; see Appendix 11). Recall that a corresponding daytime phenomenon—“roadbows” (glass bead bows)—has been mentioned in chapter 20. As there, we consider first the simpler case of (nearly) parallel light from the sun (or occasionally the moon for “moonbows”). As noted above and in Appendix 11, sunlight scattered from raindrops can produce rainbows, and the scattered light appears to lie on the surface of a cone of semi-angle 42° with vertex at the observer’s eye. Now if the scattering droplets lie on the ground (as with dew), then that cone is effectively intercepted by a horizontal plane—the flat ground! If the solar altitude exceeds 42° the observer will see an elliptical “rainbow”; if it is exactly 42° the shape will be parabolic, but for an altitude less than 42° a hyperbolic rainbow or dewbow arc will be seen. This follows directly from the geometric definition of the conic sections. These same effects can be witnessed when flying above a cloud deck, by the way, for which the rainbow becomes a “cloudbow.”

  X = (d/2)csc α: DEWBOWS IN THE CITY

  Now we will think about rainbows produced by sources of divergent light and scattered by droplets on the ground, that is, by dew. It is evident from Figures 22.2 and 22.6 that the locus of scattering drops will be the major arc of a circle (such as CFE in Figure 22.2) in the plane containing the observer’s eye and
the source of light. The chord CE will generally not be horizontal unless the lamp and eyes are at the same level. Therefore the intersection of this circular arc with the ground will be another circular arc of larger radius. If the distance CE between the light source and the observer’s eye is d, then the locus of all points subtending an angle α < π/2 is the major arc of a circle of radius r = (d/2)csc α. For generic primary and secondary rainbow angles of 42° and 51° these are approximately 0.75d and 0.64d respectively. Interestingly, therefore, in divergent light the radius of the secondary bow arc is smaller than that for the primary bow.

  This is illustrated in Figure 22.9 for an observer at O and a light source at L. When a nonvertical chord LO lies completely above the ground, and the circular arc is rotated around this chord, there is a point at which the arc is tangential to the ground, and thereafter the intersection with the ground splits into two points as illustrated in Figure 22.9. These two points of intersection move apart and then together, eventually tracing out the kidney-shaped curves illustrated in figures (a)–(d). The traces (e) and (f) correspond to the observer being directly below the light source, and above ground. In (e) the observer is low enough that the Minnaert “apple” surface traces out two concentric circular intersections; in (f) the surface is tangential to the ground because the observer is slightly higher above ground than in case (e). According to a report by Mattsson (1998), bows of the form (a)–(d) have been reported, but those like (e) and (f) have not. This is a challenge for the interested and observant reader!

  Figure 22.9. Intersections of various “Minnaert cigar rainbow surfaces” with the ground for different configurations of observer (O) and light source (L). Redrawn from Mattsson (1998).

  Regarding the corresponding “ground halos” produced by frost or diamond dust, I have been informed by Alexander Hauβmann that as young scientists, he and his friend Richard Löwenherz observed that in frost, only the 22° was frequently seen, whereas in diamond dust both the 22° and 46° halos were often present. They concluded that frost crystal faces were generally not well enough developed to ensure the consistent presence of the larger halo.

  Chapter 23

  LIGHTHOUSES IN THE CITY?

  Lighthouses in a town or city, you ask? Certainly; there are over a thousand in the United States (though many of them are no longer in use), and Michigan has the most lights of any state, with more than 150 past and present lights. A state-by-state listing of all U.S. lighthouses may readily be found online, as well as listings for lighthouses in Europe and elsewhere. Many of these are close to or within city boundaries. London’s only lighthouse (no longer functioning as such) is located at Trinity Quay Wharf in London’s docklands. By contrast, New York City has several. According to one online account [40]

  When we think of lighthouses, we often conjure up images of majestic white towers perched on rocky outcroppings dotting the New England coastline, or a stolid monolith announcing your arrival to the Outer Banks. But lighthouses in New York City? A city of soaring skyscrapers and hard concrete? What business do these anachronistic buildings of olden days have in New York, a city forever living on the cutting edge? Well, the truth is lighthouses do exist in New York City, and they have rightfully earned their place here, having helped build up this great city into what it has become.

  Oh, by the way. If you still find the idea of lighthouses in a city too much of a stretch, consider searchlights instead. Once used primarily for military purposes, they now are widely used for advertising, fairs, festivals, and many other public events. Most of the features discussed below also apply to searchlight beams.

  X = θ: RAINBOWS IN LIGHTHOUSE BEAMS

  Again, we can have some fun examining the “geometry of light” in this rather unusual context, and as in previous sections, we find some quite surprising results (see Figure 23.1). As a lighthouse (or searchlight) beam sweeps a rain-filled sky, a primary and often a secondary rainbow slice may often be noticed. As with a rainbow during the day, they are separated by a dark band (see Figure 23.2). Naturally, only those drops in the beam will scatter light into the observer’s eye. Subtle differences occur between the two types of beam, however, because a searchlight is close to the ground with a beam at an angle to the vertical direction, whereas the lighthouse is elevated relative to the observer, and has a horizontal beam. In each case, as the beam rotates about a vertical axis, the bows appear to slide up and down the beam.

  Figure 23.1. Primary (and faint secondary) bow in the beam of the Westerhever Lighthouse in Nordfriesland (Germany). Photo by Achim Christopher.

  Figure 23.2. Rainbow “slices” seen in a lighthouse beam. The primary bow (or bright section) is at d, and if color is seen at all, the point c will be tinged with red. Alexander”s dark band is the segment at b, and the fainter secondary bow, if visible, will be at point a. (Redrawn from Floor (1982).

  We will use Figure 23.3 to explain this phenomenon for a lighthouse. Its beam, parallel to the ground, comes from a lamp L at height h above the ground. The observer at O is a distance d from the base of the lighthouse, standing directly under the beam at a particular instant of time. It is raining! The point O′ is a distance h vertically above O. From what we already know about rainbow formation, we know that if the angle θ ≈ 42°, a primary bow (a better word might be slice) will be seen at BP on the beam; similarly, if θ ≈ 51°, a secondary bow will be located at BS, though it will be fainter (and may not be visible at all). Certainly the beam will by contrast be brighter there than for the angular range 42° < θ < 51°, corresponding to the dark region between bows (Alexander’s dark band). This may be more readily seen in a lighthouse beam than in sunlight. The reddish edges of the bow define the ends of the dark band, though the colors may well be less evident because the eye is less sensitive to color at the (relatively) low light intensities present in a lighthouse beam.

  Figure 23.3. Geometry for rainbow formation in a stationary lighthouse beam.

  When rainbows are produced by sunlight (or moonlight) the secondary bow appears higher in the sky than the primary. As such, it is often perceived to be farther away than the primary bow, but this is erroneous. While raindrops contributing to the secondary bow may or may not be farther away than those for the primary, the case for the lighthouse bows is definite and clear-cut: while the secondary bow appears higher in the sky, the drops giving rise to it are nearer the observer. This can be seen from Figure 23.3. The drops at BP are a distance OBP = h csc 42° ≈ 1.5h from the observer; those at BS are at a distance OBS = h csc 51° ≈ 1.3h. But this is for the case of a beam that is stationary with respect to the observer, so unless he or she runs in circles to stay directly under the beam, we need to consider a more general case!

  X = R(): THE THREE-DIMENSIONAL CASE

  As the beam rotates, the bows will appear to slide back and forth along the beam to maintain the rainbow angles of 42° and 51°, respectively. Let’s examine the geometry for an observer at O positioned outside the plane containing the lighthouse and its beam, as shown in Figure 23.4. For simplicity we consider that the beam in direction LR does not diverge significantly, and thus retains its intensity over its visible length. R is the location of a patch of rain scattering the light back toward the eye of the observer. O′ is a point directly above the observer at the same height as the light source. The beam is at an angle from the line joining O to the base of the lighthouse. δ is the angle through which the light is deviated at R to be seen as a rainbow “slice” at O. For a primary bow, δ ≈ 138°. Finally, angle OR = α.

  Figure 23.4. Geometry for “sliding rainbow” slices.

  From Figure 23.4, applying the law of sines to triangle LOR, we find that

  Application of the rule of cosines to triangles LOR and LO’R respectively yield

  Noting that triangle OO′R is a right triangle, it follows from these equations that

  Now we are in a position to express the distance R of the scattering drops, relative to the height of the light source (so R = r
/h) in terms of the relative distance of the observer, D = d/h, and the angles and δ after some algebra as

  Note both from the equation and the polar plots in Figure 23.5 that R is symmetric about the polar axis, = 0, and that if = 180°, then R = −(D + cot δ). Also, if D = 0, the plot reduces to a circle of radius R = −cot δ ≈ 1.11).

  What exactly do these polar graphs tell us? That’s a good question. They show the distance R of the rainbow (or more accurately, that of the corresponding raindrops) in units of h along the beam as a function of beam orientation for given values of D. This is the distance (also in units of h) of the observer from the base of the lighthouse. Simply put, the graphs show the points (in the horizontal plane through L) where the observer sees the primary bow. (A similar set of graphs can be drawn for the secondary bow.) To interpret this, note that the observer is on the line = 0° (the positive x-axis) at a distance D from the origin at L. For the case of D = 0.5 the rainbow lies on a slightly squashed circle as the beam rotates; for D = 1.11 the rainbow moves along the beam toward L and eventually coalesces with the light source when = 180°. The case for D >