Abstract

We describe a general framework for systematically treating halos that are due to refraction in preferentially oriented ice wedges, and we construct an atlas of such halos. Initially we are constrained neither by the interfacial angles nor the orientations of real ice crystals. Instead we consider “all possible” refraction halos. We therefore make no assumption regarding the wedge angle, and only a weak assumption regarding the allowable wedge orientations. The atlas is thus a very general collection of refraction halos that includes known halos as a small fraction. Each halo in the atlas is characterized by three parameters: the wedge angle, the zenith angle of the spin vector, and the spin vector expressed in the wedge frame. Together with the sun elevation, the three parameter values for a halo not only permit calculation of the halo shape, they also give much information about the halo without extensive calculation, so that often a crude estimate of the halo’s appearance is possible merely from inspection of its parameters. As a result, the theory reveals order in what seems initially to be a staggering variety of halo shapes, and in particular it explains why halos look the way they do. Having constructed and studied the atlas, we then see where real or conceivable refraction halos, arising in specific crystal shapes and crystal orientations, fit into the atlas. Although our main goal is to understand halos arising in pyramidal crystals, the results also clarify and unify the classical halos arising in hexagonal prismatic crystals.

© 1999 Optical Society of America

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References

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  1. F. Pattloch, E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A 1, 520–526 (1984).
    [CrossRef]
  2. R. A. R. Tricker, “Arcs associated with halos of unusual radii,” J. Opt. Soc. Am. 69,1093–1100 and 1195 (1979).
    [CrossRef]
  3. Half of SO(3) and the partition can be visualized as follows. The orientations in SO(3) whose (one-point) zenith loci are on the upper hemisphere are pictured as the upper hemisphere with a line segment 0 ≤ ϕ < 360 attached at each point and pointing radially outward—a boy’s head with a brush haircut. The segment—strand of hair—attached to the sphere at Y is the coset Zu = {zrot(ϕ) · u}, where u, corresponding to ϕ = 0 and thought of as a point on the sphere, is the rotation that takes Y to k and that has horizontal rotation axis. The halo-making sets are the sets consisting of entire segments—no hairs are split. The zenith locus of a halo-making set is the portion of scalp from which its hair is growing. (The brush cut picture can be extended to most of SO(3) by extending the hemisphere downward toward the South Pole -k. But the South Pole cannot be added to complete the picture, since there the above description of u is not enough.)
  4. S. W. Visser, “Die Halo-Erscheinungen,” in Handbuch der Geophysik, F. Linke, F. Möller, eds. (Gebrüder Borntraeger, Berlin-Nikolassee, 1942–1961), Vol. 8, pp. 1027–1081.
  5. The group W of orthogonal transformations w satisfying wk = ±k is a subgroup of the group O(3) of all orthogonal transformations, and Z is a normal subgroup of W. The quotient group W/Z consists of the four cosets Z · e, Z · xrot, Z · yref, and Z · zref, which correspond to the four lines of Table 1. The homomorphism of W onto {e, -e, yref, -yref} that is implicit in the theorem can be expressed as w = zrot(ϕ) · yrefi · zrefj → Z · yrefi · zrefj → (-1)jyrefi+j, the first mapping being the quotient map onto W/Z. The quotient group formally captures our intuition that we often do not worry about zrot(ϕ).
  6. A subtle and logically important point that can nevertheless be ignored on a first reading is that whether a given rotation u is a plate (Parry, alternate Parry, etc.,) orientation depends on the wedge under consideration. Technically, this is because the vector N1(u) depends on the wedge as well as on u, since u starts with the crystal oriented so that the specified wedge is in standard orientation. For example, the plate orientations for wedge 1 6 have the form u = zrot(ϕ)yrot(-45), whereas the plate orientations for wedge 3 1 have the form u = zrot(ϕ)yrot(45). They are not the same.
  7. J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).
  8. Thus Par ij = Pu, where P = N3 and where the matrix u gives the orientation, or frame, of wedge i j. In Section 1 we said that since the spin vector P is a wedge vector, then Pu is independent of u. That, however, assumed there was only one wedge under consideration (and one spin vector). Now Pu depends on u. The technical explanation is that P(u) [e.g., Eq. (24)] depends not only on u but on the wedge under consideration. But it is probably best to think less technically: At any moment there is the vector P, and there are the wedge frame vectors A, B, C for the wedge under consideration. Then Pu is given by Eq. (21) as always. Equation (23) is also correct, with A, B, C being the columns of the matrix u. Of course Pu depends on the choice of spin vector as well as on the frame.
  9. G. P. Können, “Identification of odd-radius halo arcs and of 44°/46° parhelia by their inner edge polarization,” Appl. Opt. 37, 1450–1456 (1998).
    [CrossRef]
  10. G. P. Können, “Polarization and intensity distributions of refraction halos,” J. Opt. Soc. Am. 73, 1629–1640 (1983).
    [CrossRef]
  11. W. Tape, Atmospheric Halos (American Geophysical Union, Washington, D.C., 1994).
    [CrossRef]
  12. R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980).
  13. F. Schaaf, “A field guide to atmospheric optics,” Sky Telesc. 77(3) , 254–259 (1989).
  14. M. Pekkola, “Viimeisten halojen ensimmäiset valokuvat,” Tähdet ja Avaruus 20(1) , 31–36 (1990).
  15. M. Pekkola, “Harrastajan palstat—Kustavin halonäytelmä,” Tähdet ja Avaruus 22(2) , 36–37 (1992).
  16. M. Pekkola, “Harrastajan palstat—Kolme komeaa halonäytelmää,” Tähdet ja Avaruus 23(6) , 40–41 (1993).
  17. O. R. Norton, Science Graphics, Bend, Oregon 97708 (personal communication, 1996).
  18. T. Kobayashi, “Vapour growth of ice crystals between -40 and -90 C,” J. Meteorol. Soc. Jpn. 43, 359–367 (1965).
  19. T. Kobayashi, K. Higuchi, “On the pyramidal faces of ice crystals,” Contrib. Inst. Low Temp. Sci. Hokkaido Univ. Ser. A, No. 12, 43–54 and 13 plates (1957).
  20. If there are at least three linearly independent vectors among Pv1, … , Pvk, then there is at most one induced pole symmetry w* for the given w, P, and V. Otherwise there may be more than one w*.

1998

1993

M. Pekkola, “Harrastajan palstat—Kolme komeaa halonäytelmää,” Tähdet ja Avaruus 23(6) , 40–41 (1993).

1992

M. Pekkola, “Harrastajan palstat—Kustavin halonäytelmä,” Tähdet ja Avaruus 22(2) , 36–37 (1992).

1990

M. Pekkola, “Viimeisten halojen ensimmäiset valokuvat,” Tähdet ja Avaruus 20(1) , 31–36 (1990).

1989

F. Schaaf, “A field guide to atmospheric optics,” Sky Telesc. 77(3) , 254–259 (1989).

1984

1983

1979

R. A. R. Tricker, “Arcs associated with halos of unusual radii,” J. Opt. Soc. Am. 69,1093–1100 and 1195 (1979).
[CrossRef]

1965

T. Kobayashi, “Vapour growth of ice crystals between -40 and -90 C,” J. Meteorol. Soc. Jpn. 43, 359–367 (1965).

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980).

Kobayashi, T.

T. Kobayashi, “Vapour growth of ice crystals between -40 and -90 C,” J. Meteorol. Soc. Jpn. 43, 359–367 (1965).

Können, G. P.

Moilanen, J.

J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).

Norton, O. R.

O. R. Norton, Science Graphics, Bend, Oregon 97708 (personal communication, 1996).

Pattloch, F.

Pekkola, M.

M. Pekkola, “Harrastajan palstat—Kolme komeaa halonäytelmää,” Tähdet ja Avaruus 23(6) , 40–41 (1993).

M. Pekkola, “Harrastajan palstat—Kustavin halonäytelmä,” Tähdet ja Avaruus 22(2) , 36–37 (1992).

M. Pekkola, “Viimeisten halojen ensimmäiset valokuvat,” Tähdet ja Avaruus 20(1) , 31–36 (1990).

J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).

Riikonen, M.

J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).

Schaaf, F.

F. Schaaf, “A field guide to atmospheric optics,” Sky Telesc. 77(3) , 254–259 (1989).

Tape, W.

W. Tape, Atmospheric Halos (American Geophysical Union, Washington, D.C., 1994).
[CrossRef]

Tränkle, E.

Tricker, R. A. R.

R. A. R. Tricker, “Arcs associated with halos of unusual radii,” J. Opt. Soc. Am. 69,1093–1100 and 1195 (1979).
[CrossRef]

Visser, S. W.

S. W. Visser, “Die Halo-Erscheinungen,” in Handbuch der Geophysik, F. Linke, F. Möller, eds. (Gebrüder Borntraeger, Berlin-Nikolassee, 1942–1961), Vol. 8, pp. 1027–1081.

Appl. Opt.

J. Meteorol. Soc. Jpn.

T. Kobayashi, “Vapour growth of ice crystals between -40 and -90 C,” J. Meteorol. Soc. Jpn. 43, 359–367 (1965).

J. Opt. Soc. Am.

R. A. R. Tricker, “Arcs associated with halos of unusual radii,” J. Opt. Soc. Am. 69,1093–1100 and 1195 (1979).
[CrossRef]

G. P. Können, “Polarization and intensity distributions of refraction halos,” J. Opt. Soc. Am. 73, 1629–1640 (1983).
[CrossRef]

J. Opt. Soc. Am. A

Sky Telesc.

F. Schaaf, “A field guide to atmospheric optics,” Sky Telesc. 77(3) , 254–259 (1989).

Tähdet ja Avaruus

M. Pekkola, “Viimeisten halojen ensimmäiset valokuvat,” Tähdet ja Avaruus 20(1) , 31–36 (1990).

M. Pekkola, “Harrastajan palstat—Kustavin halonäytelmä,” Tähdet ja Avaruus 22(2) , 36–37 (1992).

M. Pekkola, “Harrastajan palstat—Kolme komeaa halonäytelmää,” Tähdet ja Avaruus 23(6) , 40–41 (1993).

Other

O. R. Norton, Science Graphics, Bend, Oregon 97708 (personal communication, 1996).

T. Kobayashi, K. Higuchi, “On the pyramidal faces of ice crystals,” Contrib. Inst. Low Temp. Sci. Hokkaido Univ. Ser. A, No. 12, 43–54 and 13 plates (1957).

If there are at least three linearly independent vectors among Pv1, … , Pvk, then there is at most one induced pole symmetry w* for the given w, P, and V. Otherwise there may be more than one w*.

Half of SO(3) and the partition can be visualized as follows. The orientations in SO(3) whose (one-point) zenith loci are on the upper hemisphere are pictured as the upper hemisphere with a line segment 0 ≤ ϕ < 360 attached at each point and pointing radially outward—a boy’s head with a brush haircut. The segment—strand of hair—attached to the sphere at Y is the coset Zu = {zrot(ϕ) · u}, where u, corresponding to ϕ = 0 and thought of as a point on the sphere, is the rotation that takes Y to k and that has horizontal rotation axis. The halo-making sets are the sets consisting of entire segments—no hairs are split. The zenith locus of a halo-making set is the portion of scalp from which its hair is growing. (The brush cut picture can be extended to most of SO(3) by extending the hemisphere downward toward the South Pole -k. But the South Pole cannot be added to complete the picture, since there the above description of u is not enough.)

S. W. Visser, “Die Halo-Erscheinungen,” in Handbuch der Geophysik, F. Linke, F. Möller, eds. (Gebrüder Borntraeger, Berlin-Nikolassee, 1942–1961), Vol. 8, pp. 1027–1081.

The group W of orthogonal transformations w satisfying wk = ±k is a subgroup of the group O(3) of all orthogonal transformations, and Z is a normal subgroup of W. The quotient group W/Z consists of the four cosets Z · e, Z · xrot, Z · yref, and Z · zref, which correspond to the four lines of Table 1. The homomorphism of W onto {e, -e, yref, -yref} that is implicit in the theorem can be expressed as w = zrot(ϕ) · yrefi · zrefj → Z · yrefi · zrefj → (-1)jyrefi+j, the first mapping being the quotient map onto W/Z. The quotient group formally captures our intuition that we often do not worry about zrot(ϕ).

A subtle and logically important point that can nevertheless be ignored on a first reading is that whether a given rotation u is a plate (Parry, alternate Parry, etc.,) orientation depends on the wedge under consideration. Technically, this is because the vector N1(u) depends on the wedge as well as on u, since u starts with the crystal oriented so that the specified wedge is in standard orientation. For example, the plate orientations for wedge 1 6 have the form u = zrot(ϕ)yrot(-45), whereas the plate orientations for wedge 3 1 have the form u = zrot(ϕ)yrot(45). They are not the same.

J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).

Thus Par ij = Pu, where P = N3 and where the matrix u gives the orientation, or frame, of wedge i j. In Section 1 we said that since the spin vector P is a wedge vector, then Pu is independent of u. That, however, assumed there was only one wedge under consideration (and one spin vector). Now Pu depends on u. The technical explanation is that P(u) [e.g., Eq. (24)] depends not only on u but on the wedge under consideration. But it is probably best to think less technically: At any moment there is the vector P, and there are the wedge frame vectors A, B, C for the wedge under consideration. Then Pu is given by Eq. (21) as always. Equation (23) is also correct, with A, B, C being the columns of the matrix u. Of course Pu depends on the choice of spin vector as well as on the frame.

W. Tape, Atmospheric Halos (American Geophysical Union, Washington, D.C., 1994).
[CrossRef]

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980).

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Figures (59)

Fig. 1
Fig. 1

Wedge and spin vector P. The vector P is fixed in the wedge and has constant zenith angle ψ. The wedge is otherwise unconstrained; it is free to rotate about P, and P is free to rotate about the vertical.

Fig. 2
Fig. 2

Left, light ray proceeding from left to right, from medium with refractive index n 1 to medium with refractive index n 2. The vector I, with length n 1, is the incident ray, and J, with length n 2, is the refracted ray. The vector N is the unit normal vector. Right, same but with the addition of the light points S = -I and T = -J, which give the directions from which the rays I and J appear to be coming. The point T is the N-projection of S to the sphere of radius n 2. The relation between S and T, like that between I and J, is parallel projection between concentric spheres of radii n 1 and n 2.

Fig. 3
Fig. 3

Sun S and corresponding halo point H, both on the inner sphere. The halo point is the apparent position of the sun when one looks at the sun through the wedge. To find it geometrically, one N-projects point S to the outer sphere, getting point T, and then X-projects back to the inner sphere, getting H. The vectors N and X are the entry and exit face normals of the wedge. The two spheres are concentric and have radii 1 and n, the refractive indices outside and inside the wedge. The outer sphere is cut away completely, leaving only a skeleton, and the inner sphere is cut away at the left to reveal N and X. The wedge should be thought of as at the center of the spheres, as in Fig. 10.

Fig. 4
Fig. 4

Wedge frame vectors A, B, C and entry and exit normal vectors N and X. Note that the light ray (not shown) proceeds approximately opposite to the directions of N and X.

Fig. 5
Fig. 5

Wedge in standard orientation. In standard orientation the wedge frame vectors A, B, C coincide with the standard Cartesian coordinate vectors i, j, k. The left-hand diagram, with the vector B = j pointing into the paper, also shows the entry and exit normals N and X as well as the wedge angle α. The right-hand diagram is the more conventional view used in this paper, with the x-axis pointing more or less toward the reader.

Fig. 6
Fig. 6

Left, wedge together with the wedge frame vectors A, B, C, outward normal N to the entry face, and sun vector S, all shown for a particular wedge orientation u. This is the view from space. Middle, corresponding wedge coordinate vectors A u , B u , C u , N u , and S u . This is therefore the view from the wedge. The wedge is shown here in standard orientation in order to emphasize the geometric meaning of the subscripted vectors. The rotation u takes the wedge here to the wedge in the left-hand diagram, and it takes the vectors A u = i, B u = j, C u = k, N u , S u to A, B, C, N, S [Eq. (23)]. Right, vectors A, B, C, N, and S as seen from the wedge.

Fig. 7
Fig. 7

Left, Bravais coordinate grid. Bravais coordinates (θ, δ) are spherical coordinates centered at (0, 1, 0). The solid dots, all of which are on the front hemisphere (x ≥ 0), mark the location of poles of halos shown in Appendix A. [Figure 37 of Appendix A also shows halos with poles on the rear hemisphere (x ≤ 0).] Right, same, but shown in stereographic projection from (-1, 0, 0). This is the layout used in Appendix A.

Fig. 8
Fig. 8

Left, circle K = K(ψ, P u ) with radius ψ and center P u . The circle is the zenith locus of the halo-making set determined by ψ and P u according to the Spin Vector Assumption. Here ψ = 30. Right, same, but with ψ = 90, so that K is a great circle.

Fig. 9
Fig. 9

Above left, entry and exit regions and typical sun point S and halo point H. The entry region of the sphere is the lower of the two regions enclosed by the heavy curves—it consists of the directions from which light can originate and then pass through the wedge; when S lies outside the entry region there is no H. The exit region is the upper of the two regions enclosed by the heavy curves—it consists of the directions from which the outgoing light can appear to come; H will always be in the exit region. Since from Fig. 3 the points S and H are on the same Bravais circle, then they differ by a rotation through an angle d = d(S) about B. Right, same, but seen in stereographic projection, as in Fig. 7, and with the addition of level curves of the deviation Δ, in the entry region, and their images in the exit region. The vectors D and E are the minimum deviation entry and exit vectors; Δ is minimum when S = D or, equivalently, H = E. The apparent difference in sizes of the entry regions in the two diagrams is due to the stereographic projection. The wedge at lower center belongs at the center of the sphere. Here α = 80.2.

Fig. 10
Fig. 10

Finding the entry region. Above left, reference diagram showing concentric spheres of radii 1 and n, with both spheres cut away completely to show the wedge with entry and exit normals N and X. Compare Fig. 3. Above right, N-projection of the inner sphere to the outer. The region of the outer sphere within the cylinder consists of light points of rays that have entered at the entry face. Below left, same, but with X-projection of the outer sphere to the inner. The region of the outer sphere within the new cylinder consists of light points of rays that originate within the wedge and that can exit the exit face. The region T on the outer sphere and common to the two cylinders therefore consists of light points of rays within the wedge that have entered the entry face and that can exit the exit face. Below right, the entry region—the result of N-projecting T back to the inner sphere. This region consists of light points of rays outside the wedge that can enter the entry face and exit the exit face. Similarly, X-projecting T would give the exit region. See also Fig. 9. Here α = 80.2.

Fig. 11
Fig. 11

Creation of a point halo. Above left, view from the wedge. The halo has constant k u —zenith point as seen from the wedge—as shown. The points S u and H u are the sun and corresponding halo point as seen from the wedge. The curve S is the sun locus, the path traced out by S u ; it is a circle with center k u and radius σ = 90 - Σ, the zenith angle of the sun. The curve H is the halo point locus, the path traced out by H u . The wedge at lower left belongs at the center of the left-hand sphere. The wedge is shown in standard orientation and is included to emphasize the geometric meaning of the subscripted vectors. Above right, view from space, with the halo at the far upper left of the sphere. Both upper diagrams show the triangle whose vertices are the zenith, the sun, and the halo point; the left diagram is the view from the wedge, the right is the view from space. The meaning of Δ, τ, η, ϕ, σ at left is therefore the same as at the right, namely, Δ, τ, η, ϕ are the deviation, bearing, zenith angle, and azimuth of the halo point, and σ is the zenith angle of the sun. Note that Δ and τ are sufficient to locate H with respect to S. Below right, same halo but seen from inside the celestial sphere and looking directly toward the sun, as in the halo diagrams of Appendix A. [k u = P u = B(30, -45), α = 90, σ = 65 (hence Σ = 25)]

Fig. 12
Fig. 12

Creation of another point halo. As in Fig. 11, the point S u traverses the sun locus S, and H u simultaneously traverses the halo point locus H. In this figure, however, the arc of S within the entry region subtends a much larger angle from k u , thus producing a wider variation in τ as S u varies, so that the halo is no longer confined to a narrow sector but instead extends more than 90° along and just outside the circular halo. [ k u = B(71, 41), α = 80.2, σ = 30 (Σ = 60)]

Fig. 13
Fig. 13

Creation of another point halo. The upper diagram shows the entry region together with the sun locus S and level curves of the deviation Δ. Tangencies of S with the level curves give relative extrema of Δ, which can then be located on the halo itself, in the lower diagram. Here there are two relative minima and two relative maxima. [ k u = B(95, 16), α = 60, σ = 25 (Σ = 65)]

Fig. 14
Fig. 14

Creation of a great circle halo. Above, left and right, view from the wedge. The zenith locus is the great circle K with pole P u as shown, and the sun locus is the large annular region S consisting of circles. Each point k u on K is the center of a circle of radius σ which is traced out by S u , and S is the union of all such circles. In the exit region are the corresponding curves traced out by H u , which make up the halo point locus H. See also Fig. 12, in which k u , S u , and H u are the same as the particular k u , S u , and H u shown here; the point halo in that figure is a subset of the great circle halo here. Below right, the great circle halo, shown as a union of point halos corresponding to the curves at upper right. A more conventional depiction of the halo is given in Fig. 18. [ P u = B(60, -60), α = 80.2, σ = 30 (Σ = 60)]

Fig. 15
Fig. 15

Minimum and maximum angular distances s 1 and s 2 from P u = B(30, -45) to the α = 90 entry region. Here s 1 = 58 and s 2 = 117. The point halo with pole P u and wedge angle α is nonempty for s 1 ≤ σ ≤ s 2. The great circle halo with this pole and wedge angle is never empty, since s 1 ≤ 90 ≤ s 2, so that the zenith locus passes through the entry region.

Fig. 16
Fig. 16

Left, D-centered coordinates (s, t). Middle, S-centered coordinates (Δ, τ). Right, S-centered coordinates seen from inside the celestial sphere and looking directly at the sun.

Fig. 17
Fig. 17

Above left, the points k u , Su , and H u —the zenith point, sun point, and halo point as seen from the wedge—when u is a minimum deviation orientation, that is, when S u = Du . Compare with Fig. 12, in which S u D u . [α = 80.2, σ = 30 (Σ = 60), τ = 140] Above right, the same three points but seen from the space frame. Right, same, but viewed from inside the celestial sphere. Note the identical bearings τ of H here and of k u in the upper left diagram.

Fig. 18
Fig. 18

Above left, contact circle C = C(α, σ), with radius σ and center D u . For the given wedge angle α and solar zenith angle σ, a halo with zenith locus K will contact the circular halo if and only if K intersects C. Below left, point halo with K consisting of the single point k u shown at upper left. Since k u is on C, the halo contacts the circular halo. The contact point H has the same bearing τ as k u . See also Fig. 17, which has the same α, σ, τ, and k u , and see Fig. 12, which has the same α, σ, and k u . Above right, same, but now K is the great circle with pole P u as shown. Below right, the great circle halo with zenith locus K at upper right. Since K intersects C, the halo contacts the circular halo. The contact points H and H′ with the circular halo correspond to the intersection points k u and k u of K with C, with corresponding points having the same bearing. This is the same halo as in Fig. 14. The circular halo is the 35° halo.

Fig. 19
Fig. 19

Left, cube consisting of triples (σ, s, ψ), 0 ≤ σ ≤ 180, 0 ≤ s ≤ 180, 0 ≤ ψ ≤ 180. The cube represents halos that satisfy the Spin Vector Assumption, with each triple (σ, s, ψ) representing all halos having solar zenith angle σ and zenith locus K(ψ, Pu ), where P u is at angular distance s from D u . All of the halos on the inner circle in Fig. 20, for example, would be represented by the single point (σ, s, ψ) = (80, 30, 90). The regular tetrahedron having vertices (0, 0, 0), (180, 180, 0), (180, 0, 180), (0, 180, 180), consists of halos that contact the circular halo. Here the cube is truncated to show the tetrahedron, and the near upper face of the tetrahedron has been removed to expose the tetrahedron’s ψ = 90 section, which consists of great circle halos that contact the circular halo. The ψ = 0 section, which consists of point halos that contact the circular halo, is the tetrahedron’s bottom edge σ = s. Right, the ψ = 90 section in detail. Contours are lines of constant Δτ—the half-spread between contact points. The bearing of the contact points from the sun is τ = t ± Δτ, where t is the bearing of P u from D u . The horizontal line shows the evolution of the contact points for a great circle halo as σ increases from 0 to 180.

Fig. 20
Fig. 20

Dependence of contact points on (s, t), the D-centered coordinates of poles P u (Fig. 16, left). Great circle halos are shown for P u having s = 0, 30, 60, 90, and t = 0, 45, … , 315. For each halo the contact points are located symmetrically at angle Δτ on either side of the direction with bearing t, which is the direction from D u (s = 0, center of figure) to P u . For a given ψ and Σ, as here, the half-spread Δτ depends only on s, the distance from D u to P u . See the Contact Point Theorem, part (ii). [α = 80.2 (Δ m = 35) and Σ = 10]

Fig. 21
Fig. 21

Dependence of contact points on sun elevation Σ. The halo has pole P u with D-centered coordinates (s, t) = (79, 31). The halo has two contact points for each Σ with -s < Σ < s, one contact point for Σ = ±s, and none otherwise. The contact points are located symmetrically at angle Δτ on either side of the direction with bearing t, which is the direction from D u to P u . As Σ decreases from s to -s, the half-spread Δτ of the contact points decreases from 180 to 0 as shown. Here α = 80.2 (Δ m = 35), but the results are essentially independent of α. Figure 18 shows the same halo for Σ = 60.

Fig. 22
Fig. 22

Dependence of contact points on wedge angle α. Great circle halos are shown for wedge angle α = 28, 60, 80.2, and 90. Each halo has pole P u with the same D-centered coordinates (s, t) = (79, 31), the same as in Fig. 21. The bearing of the contact points is independent of α. Note, however, that the four poles are not the same, since D-centered coordinates depend on α. (Σ = 50)

Fig. 23
Fig. 23

Left, halo with the sun at S. Also shown are the halo’s x-rotation when the sun is at xrot S, its y-reflection when the sun is at yref S, and its z-reflection when the sun is at zref S. The four halo shapes do not appear in the sky simultaneously. This view is of the rear of the celestial sphere, seen from inside. Above right, the same four halos but with the sun at S. The x-rotation is no longer a rotation of the original halo, and the z-reflection is no longer a reflection of the original halo. Below right, zenith loci of the halos. Here K consists of the single point k u = B(30, -45); the sets K and yref K are on the front hemisphere, and -K and -yref K are on the rear. (α = 40, Σ = 25)

Fig. 24
Fig. 24

Geometry of Rule 2, showing views from the wedge at left and from space at right. For each orientation u, the orientation v = xrot · u satisfies - k v = ku and Sv = S u , where S′ = xrot S. At left, therefore, H(e, Sv) = H(e, S u ), and at right, H(v, S′) = xrot H(u, S). The right-hand diagram is drawn for the special case where S is in the plane y = 0.

Fig. 25
Fig. 25

Geometry of Rule 3, showing views from the wedge at left and from space at right. For each orientation u the orientation v = yref · u ·yref satisfies k v = yref k u and S v = yref S u , as shown. At left, therefore, H(e, S v ) = yref H(e, S u ), and at right, H(v) = yref H(u). The orientations u and v differ by a rotation whose axis, as seen from the wedge, is the solid dot shown on y = 0. Here S is in the plane y = 0, so that yref S = S.

Fig. 26
Fig. 26

Hexagonal prismatic crystal and crystal frame vectors N3, N1 × N3, N1.

Fig. 27
Fig. 27

Left, geometric derivation of the pole Par 3 1, the coordinate vector of the Parry spin vector P = N3 with respect to wedge 3 1. The crystal is oriented so that wedge 3 1 is in standard orientation (Fig. 5), and in this orientation the vector P coincides with the desired pole. The pole Par 3 1 is therefore (1/√2, 0, 1/√2) = B(90, -45). Right, same crystal orientation but with the x-axis pointing nearly out of the paper, so that the diagram can be more easily related to Fig. 59. The spin vectors N1 and N1 × N3 for plate orientations and alternate Parry orientations, respectively, have been added. From the diagram here, the pole Plate 3 1 is (-1/√2, 0, 1/√2) = B(90, -135) and the pole AP 3 1 is (0, 1, 0) = B(0, δ).

Fig. 28
Fig. 28

Pyramidal crystal and crystal frame vectors N3, N1 × N3, N1.

Fig. 29
Fig. 29

Geometric derivation of the pole Par 3 16, the coordinate vector of the Parry spin vector P = N3 with respect to wedge 3 16. The crystal is oriented so that wedge 3 16 is in standard orientation, and in this orientation the vector P coincides with the desired pole. The wedge angle is 28°, so face 3 is inclined 14° from the vertical. The pole Par 3 16 is therefore (cos 14, 0, sin 14) = B(90, -14), in agreement with the table in Fig. 52.

Fig. 30
Fig. 30

The 24° arcs Col 13 5, Col 13 7, Col 3 25, and Col 3 27, shown individually and then as a composite. The composite is mmm symmetric. At this sun elevation it has a total of eight contact points with the 24° circular halo, one pair of contact points coming from each of the four component halos. Also see Fig. 57 and recall that the pole for Col i j is the same as that for Plate i j. (Σ = 40)

Fig. 31
Fig. 31

Monte Carlo simulation made using pyramidal crystals having column orientations. The crystals have pyramidal faces and prism faces, but not basal faces; all the faces in Fig. 28 are present except faces 1 and 2. With study, the halos of Fig. 30 can be discerned among the many halos here. The tick marks are at 1° intervals. (Σ = 40, the same as for Fig. 30)

Fig. 32
Fig. 32

Wooden model of the pyramidal crystal in Fig. 28. If the model is at the origin of coordinates, then the x-axis is pointing toward and below the camera. The wedge 13 5 is in standard orientation. The dowel with flag N1 therefore points in the direction of Plate 13 5, and the dowel with flag N3 points in the direction of Par 13 5. Compare with Fig. 57. The model was made by Jack Corbin.

Fig. 33
Fig. 33

Point halos (ψ = 0) with wedge angle α = 60 and sun elevation Σ = 0. The corresponding circular halo is the common circular halo, with radius Δ m = 22. Halos are shown for the 27 poles P u shown in Fig. 7, all of which are on the front hemisphere. In the halo diagrams the inner circle is the circular halo, and the line within it is part of the parhelic circle, which is included as a reminder of sun elevation. Some halos can be empty, depending on sun elevation.

Fig. 34
Fig. 34

Point halos with wedge angle α = 60 (Δ m = 22) and sun elevation Σ = 20. Same as Fig. 33 except for Σ.

Fig. 35
Fig. 35

Point halos with α = 60 (Δ m = 22) and Σ = 50. Same as Figs. 33 and 34 except for Σ.

Fig. 36
Fig. 36

Point halos with α = 60 (Δ m = 22) and Σ = 80. Same as Figs. 3335 except for Σ. The small circle in each halo diagram is the parhelic circle, which for the sun elevation here has a radius of only 10°.

Fig. 37
Fig. 37

Point halos with α = 60 (Δ m = 22) and Σ = 20, and with poles P u on the rear hemisphere (x ≤ 0). Compare Fig. 34, which shows halos with poles on the front hemisphere (x ≥ 0).

Fig. 38
Fig. 38

Point halos with wedge angle α = 28 (Δ m = 9) and sun elevation Σ = 20. Same as Fig. 34 except for α.

Fig. 39
Fig. 39

Point halos with α = 52.4 (Δ m = 18) and Σ = 20. Same as Figs. 34 and 38 except for α.

Fig. 40
Fig. 40

Point halos with α = 80.2 (Δ m = 35) and Σ = 20. Same as Figs. 34, 38, and 39 except for α.

Fig. 41
Fig. 41

Point halos with α = 90 (Δ m = 46) and Σ = 20. Same as Figs. 34 and 3840 except for α.

Fig. 42
Fig. 42

Great circle halos (ψ = 90) with wedge angle α = 60 (circular halo radius Δ m = 22) and sun elevation Σ = 0.

Fig. 43
Fig. 43

Great circle halos with α = 60 (Δ m = 22) and Σ = 20. Same as Fig. 42 except for Σ.

Fig. 44
Fig. 44

Great circle halos with α = 60 (Δ m = 22) and Σ = 50. Same as Figs. 42 and 43 except for Σ.

Fig. 45
Fig. 45

Great circle halos with α = 60 (Δ m = 22) and Σ = 80. Same as Figs. 4244 except for Σ. The small circle in each halo diagram is the parhelic circle, which for the sun elevation here has a radius of only 10°.

Fig. 46
Fig. 46

Great circle halos with wedge angle α = 28 (Δ m = 9) and sun elevation Σ = 20. Same as Fig. 43 except for α.

Fig. 47
Fig. 47

Great circle halos with α = 52.4 (Δ m = 18) and Σ = 20. Same as Figs. 43 and 46 except for α.

Fig. 48
Fig. 48

Great circle halos with α = 80.2 (Δ m = 35) and Σ = 20. Same as Figs. 43, 46, and 47 except for α.

Fig. 49
Fig. 49

Great circle halos with α = 90 (Δ m = 46) and Σ = 20. Same as Figs. 43 and 4648 except for α.

Fig. 50
Fig. 50

Point halos with α = 60 (Δ m = 22) and Σ = 20. Same as Fig. 34 except that here each simulation, located at P u , is the mmm-symmetric composite consisting of the halos with poles ± P u , ±yref P u, ±zref P u , and ±xrot P u . For each halo in the simulation, the x-rotation, the y-reflection, the z-reflection, and the face interchange are also present, although not necessarily nonempty at the given sun elevation. It is the composite, rather than any of the components separately, that would be seen in most real halo displays.

Fig. 51
Fig. 51

Great circle halos with α = 60 (Δ m = 22) and Σ = 20. Same as Fig. 43 except that here each simulation, located at P u , is the mmm-symmetric composite consisting of the halos with poles P u , yref P u , zref P u , and xrot P u . For each halo in the simulation, the x-rotation, the y-reflection, the z-reflection, and the face interchange are also present, although not necessarily nonempty at the given sun elevation. The x-rotation of a great circle halo, however, is the halo itself, and the z-reflection is the same as the y-reflection. It is the composite, rather than any of the components separately, that would be seen in most real halo displays.

Fig. 52
Fig. 52

Poles of 9° arcs (α = 28). There are four arcs from plate orientations and hence two from column orientations. There are twelve arcs from Parry orientations and hence six from alternate Lowitz orientations. For Σ = 20 the appearance of each 9° arc having a pole on the front hemisphere can be estimated from Figs. 38 and 46.

Fig. 53
Fig. 53

Poles of 18° arcs (α = 52.4). For Σ = 20 the appearance of each 18° arc having a pole on the front hemisphere can be estimated from Figs. 39 and 47.

Fig. 54
Fig. 54

Poles of 20° arcs (α = 56). For Σ = 0, 20, 50, and 80, the appearance of each 20° arc having a pole on the front hemisphere can be estimated from Figs. 3336 and 4245, which all have α = 60 ≈ 56.

Fig. 55
Fig. 55

Poles of 22° arcs (α = 60). Halo names are given when they exist, with names in parentheses referring to great circle halos and the remaining names referring to point halos. For Σ = 0, 20, 50, and 80, the appearance of each 22° arc having a pole on the front hemisphere can be estimated from Figs. 3336 and 4245.

Fig. 56
Fig. 56

Poles of 23° arcs (α = 62). For Σ = 0, 20, 50, and 80, the appearance of each 23° arc having a pole on the front hemisphere can be estimated from Figs. 3336 and 4245, which all have α = 60 ≈ 62.

Fig. 57
Fig. 57

Poles of 24° arcs (α = 63.8). For Σ = 0, 20, 50, and 80, the appearance of each 24° arc having a pole on the front hemisphere can be estimated from Figs. 3336 and 4245, which have α = 60 ≈ 63.8. The tables are continued on the following page.

Fig. 58
Fig. 58

Poles of 35° arcs (α = 80.2). For Σ = 20 the appearance of each 35° arc having a pole on the front hemisphere can be estimated from Figs. 40 and 48.

Fig. 59
Fig. 59

Poles of 46° arcs (α = 90). Here halo names are given when they exist, with names in parentheses referring to great circle halos and the remaining names to point halos. For ∑ = 20 the appearance of each 46° arc having a pole on the front hemisphere can be found from Figs. 41 and 49. The tables are continued on the following page.

Tables (3)

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Table 1 Orthogonal Transformationsa w such that wk = ±k

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Table 2 Spin Vector P and Its Zenith Angle ψ for the Six Classes of Crystal Orientations

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Table 3 Wedge Angles and Corresponding Circular Halosa

Equations (81)

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P·k=cos ψ=constant.
prS, N, n2=S+λN,
λS, N, n2=S·N2+n22-n121/2-S·N.
J=prI, -N, n2.
T=prS, N, n2,
T=prS, N, n.
H=FN, X, S=prT, X, 1=prprS, N, n, X, 1.
A=N+X/N+X,  B=N×X/N×X,  C=N-X/N-X,
H=FN, X, S=FuN0, uX0, S,
N0=N0α=cosα/2, 0, sinα/2,  X0=X0α=cos-α/2, 0, sin-α/2,
Hu, α, S=FuN0, uX0, S.
FwN, wX, wS=wFN, X, S,
Hwu, α, wS=wHu, α, S.
wu=w uif det w=1,w u yrefif det w=-1,
Hwu, α, wS=Hwu, α, wS,
Hwu, α, wS=wHu, α, S.
SHU, α, S,
HU, α, S=Hu, α, S : uU, Hu, α, Sno ray.
V=V·AA+V·BB+V·CC.
Vu=V·A, V·B, V·C
=V·Ai+V·Bj+V·Ck.
V=uVu.
Ve=Vu  if V is a wedge vector.
uPu·k=cos ψ.
Bθ, δ=sin θ cos δ, cos θ, -sin θ sin δ.
V=uVu=u Bθ, δ=usin θ cos δ, cos θ, -sin θ sin δ
=sin θ cos δA+cos θB-sin θ sin δC.
uU implies zrotϕ·uU for all ϕ.
ZUU.
Zu=ζu : ζZ.
UKα=UKα,
UKα=UKα,
UK1UK2 if and only if K1K2.
pp-1K=K,
p-1pU=U  if U is a halo-making set.
U=p-1K=u : kuK=u : uY=k for some Y in K,
K=pU=ku : uU=Y : uY=k for some u in U.
HU1, α1, SHU2, α2, S for all S,
cos Δ=S·H.
sinα+Δm/2=n sinα/2.
D=cos-Δm/2A+sin-Δm/2C
Du=cos-Δm/2, 0, sin-Δm/2.
Eu=cosΔm/2, 0, sinΔm/2.
ΔS is minimum  iff ΔS=Δm  iff S=D  iff T=nA  iff H=E.
u Bθ, α/2-90,
Txθ=u Bθ, α/2-90+n2-11/2 X,
sin θ0=n2-11/2 tanα/2.
Sxθ=prTxθ, N, 1, θ0θ180-θ0.
Snθ=u Bθ, -α/2+90, θ0θ180-θ0.
Hu=FN, X, Su=u-1FN, X, S=Fu-1N, u-1X, u-1S=FNu, Xu, Su.
Ds, t=rot-t, Du·Ds, 0
SΔ, τ=rotτ, S·SΔ, 0,
uτ=rotτ, S u0
=u0 rotτ, Du.
uτDu=S,
uτEu=SΔm, τ.
kuτ=Dσ, τ,
Huτ=SΔm, τ,
uDu=S,
uEu=SΔm, τ.
if Hu=SΔm, τ then u=uτ.
Pu=cos sDu+sin s sin tj+sin s cos tDu×j.
cos σ cos s+sin σ sin s cosτ-t=cos ψ.
τ=t±Δτ,
Δτσ, s, ψ=arccoscos ψ-cos σ cos s/sin σ sin s.
HUK, zrotϕS=zrotϕ HUK, S.
HU-K, xrot S=xrot HUK, S,
HUK, wS=wHUK, S  for all S,
HUyref K, yref S=yref HUK, S,
HUyrot K, zref S=zref HUK, S.
K=Kif wk=kanddet w=1e.g., w=zrotϕ, Rule 1-Kif wk=-kanddet w=1e.g., w=xrot, Rule 2yref Kif wk=kanddet w=-1e.g., w=yref, Rule 3-yref Kif wk=-kanddet w=-1e.g., w=zref, Rule 4
HUK, wS=wHUK, S for all S.
Hwu, wS=wHu, S.
kwu=wu-1k=yref·u-1·w-1k=-yref·u-1k=-yref ku,
wuU-yref K  uUK.
HwU, wS=wHU, S,
N1=N1, N2=-N1, N3=N3,  N4=cos 60 N3+sin 60 N1×N3,  N5=cos 120 N3+sin 120 N1×N3,  N6=cos 180 N3+sin 180 N1×N3,  N7=cos 240 N3+sin 240 N1×N3,  N8=cos 300 N3+sin 300 N1×N3.
N13=cos 28N3+sin 28N1,N23=cos 28N3+sin 28N2,N14=cos 28N4+sin 28N1,N24=cos 28N4+sin 28N2,N15=cos 28N5+sin 28N1,N25=cos 28N5+sin 28N2,with N1, N2, , N8 as before.
Pwv=   wv-1 Pvif det w=1,yref wv-1 Pvif det w=-1,
w*Pv=Pwv  for all vV,
w*Pv =   wv-1Pvif det w=1,yref wv-1Pvif def w=1,

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